Linear and Nonlinear Aspects of Vortices :

the Ginzburg-Landau Model

F. Pacard T. Riviere

January 28, 2004

Contents

1 Qualitative Aspects of Ginzburg-Landau Equations 11.1 The integrable case . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The strongly repulsive case . . . . . . . . . . . . . . . . . . . . . 31.3 The existence result . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Uniqueness results . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Elliptic Operators in Weighted Holder Spaces 212.1 The function spaces . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Mapping properties of the Laplacian . . . . . . . . . . . . . . . . 24

2.2.1 Rescaled Schauder estimates . . . . . . . . . . . . . . . . 262.2.2 Mapping properties of the Laplacian in the injectivity range 282.2.3 Mapping properties of the Laplacian in the surjectivity

range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Applications to nonlinear problems . . . . . . . . . . . . . . . . . 42

2.3.1 Minimal surfaces with one catenoidal type end . . . . . . 432.3.2 Semilinear elliptic equations with isolated singularities . . 452.3.3 Singular perturbations for Liouville equation . . . . . . . 47

3 The Ginzburg-Landau Equation in C 493.1 Radially symmetric solution on C . . . . . . . . . . . . . . . . . . 493.2 The linearized operator about the radially symmetric solution . . 50

3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.2 Explicit solutions of the homogeneous problem . . . . . . 51

3.3 Asymptotic behavior of solutions of the homogeneous problem . 523.3.1 Classification of all possible asymptotic behaviors at 0 . . 523.3.2 Classification of all possible asymptotic behaviors at ∞ . 54

3.4 Bounded solution of the homogeneous problem . . . . . . . . . . 583.5 More solutions of the homogeneous equation . . . . . . . . . . . . 663.6 Introduction of the scaling factor . . . . . . . . . . . . . . . . . . 67

4 Mapping Properties of Lε 694.1 Consequences of the maximum principle in weighted spaces . . . 69

4.1.1 Higher eigenfrequencies . . . . . . . . . . . . . . . . . . . 694.1.2 Lower eigenfrequencies . . . . . . . . . . . . . . . . . . . . 76

v

4.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3 A right inverse for Lε in B1 \ 0 . . . . . . . . . . . . . . . . . . 81

4.3.1 Higher eigenfrequencies . . . . . . . . . . . . . . . . . . . 824.3.2 Lower eigenfrequencies . . . . . . . . . . . . . . . . . . . . 86

5 Families of Approximate Solutions with Prescribed Zero set 965.1 The approximate solution u . . . . . . . . . . . . . . . . . . . . . 96

5.1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.1.2 The approximate solution near the zeros . . . . . . . . . . 985.1.3 The approximate solution away from the zeros . . . . . . 104

5.2 A 3N dimensional family of approximate solutions . . . . . . . . 1085.2.1 Definition of the family of approximate solutions . . . . . 108

5.3 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Existence of Ginzburg-Landau Vortices 1166.1 Statement of the result . . . . . . . . . . . . . . . . . . . . . . . . 1166.2 The linear mapping DM(0,0,0) . . . . . . . . . . . . . . . . . . . 1206.3 Estimates of the nonlinear terms . . . . . . . . . . . . . . . . . . 120

6.3.1 Estimates of Q1 . . . . . . . . . . . . . . . . . . . . . . . 1216.3.2 Estimates of Q2 . . . . . . . . . . . . . . . . . . . . . . . 1226.3.3 Estimates of Q3 . . . . . . . . . . . . . . . . . . . . . . . 123

6.4 The fixed point argument . . . . . . . . . . . . . . . . . . . . . . 1266.5 Further information about the branch of solutions . . . . . . . . 127

7 Elliptic Operators in Weighted Sobolev Spaces. 1317.1 General overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.2 Estimates for the Laplacian . . . . . . . . . . . . . . . . . . . . . 1337.3 Estimates for some elliptic operator in divergence form . . . . . . 142

8 Generalized Pohozaev Formula for ρ−Conformal Fields 1548.1 The Pohozaev formula in the classical framework . . . . . . . . . 1548.2 Comparing Ginzburg-Landau solutions through Pohozaev’s argu-

ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1578.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1578.2.2 The comparison argument in the case of radially symmet-

ric data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1588.3 ρ−conformal vector fields . . . . . . . . . . . . . . . . . . . . . . 1618.4 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.4.1 Comparing solutions through a Pohozaev type formula :the general case . . . . . . . . . . . . . . . . . . . . . . . . 165

8.4.2 Conservation laws for Ginzburg-Landau equation . . . . . 1668.4.3 The Pohozaev formula . . . . . . . . . . . . . . . . . . . . 1678.4.4 Integration of the Pohozaev formula . . . . . . . . . . . . 170

8.5 Uniqueness Results . . . . . . . . . . . . . . . . . . . . . . . . . . 1738.5.1 A few uniqueness results . . . . . . . . . . . . . . . . . . . 173

vi

8.5.2 Uniqueness results for semilinear elliptic problems . . . . 1758.6 Dealing with general nonlinearities . . . . . . . . . . . . . . . . . 179

8.6.1 A pohozaev formula for general nonlinearities . . . . . . . 1798.6.2 Uniqueness results for general nonlinearities . . . . . . . . 1818.6.3 More about the quantities involved in the Pohozaev identity182

9 The role of the zeros in the uniqueness question 1859.1 The zero set of solutions of Ginzburg-Landau equations . . . . . 1859.2 A uniqueness result . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.2.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . 1949.2.2 The proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . 207

10 Solving uniqueness questions 21210.1 Statement of the uniqueness result . . . . . . . . . . . . . . . . . 21210.2 Proof of the uniqueness result . . . . . . . . . . . . . . . . . . . . 213

10.2.1 Geometric modification of the family vε . . . . . . . . . . 21310.2.2 Estimating the L2 norm of |uε| − |vε| . . . . . . . . . . . . 21610.2.3 Pointwise estimates for uε − vε and |uε| − |vε|. . . . . . . 22610.2.4 The final arguments to prove that uε = vε. . . . . . . . . 231

10.3 A conjecture of F. Bethuel, H. Brezis and F. Helein . . . . . . . . 235

11 Towards Jaffe and Taubes conjectures 23711.1 Statement of the result . . . . . . . . . . . . . . . . . . . . . . . . 237

11.1.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . 23811.1.2 The uniqueness result . . . . . . . . . . . . . . . . . . . . 240

11.2 Gauge-invariant Ginzburg-Landau critical points with one zero. . 24111.3 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 254

11.3.1 The Coulomb gauge . . . . . . . . . . . . . . . . . . . . . 25411.3.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . 25711.3.3 The Pohozaev formula . . . . . . . . . . . . . . . . . . . . 26411.3.4 The end of the proof . . . . . . . . . . . . . . . . . . . . . 266

vii

This book has expanded from our attempt to understand the geometry ofGinzburg-Landau vortices. First in trying to construct these vortices and thenin trying to show that they enjoy some uniqueness properties. In addition tothe fact that the result are interesting for their own sake, we hope that thisbook will help to promote the use of weighted spaces and conservation laws inthe study of nonlinear problems arising in geometry or mathematical physics.

The Ginzburg-Landau vortices are naturally associated to variational tech-nics, however, in the construction of such vortices we have chosen to use somegluing technics based on a fixed point argument. Gluing theorems are by nowcommon place in many problems arising from geometry, such as minimal sur-faces and constant mean curvature surfaces. One of the main ingredient is theextensieve use of weighted Holder spaces which happen to be a very powerfultool in the analysis of linear differential operators over manifolds with corners.

Among the advantages of this gluing technic is that one can obtain somedeep understanding of the nonlinear partial differential equation associated withGinzburg-Landau vortices. The interaction between the linear analysis and thenonlinearities of the problem plays a central role and allows us to shed new lighton the geometry of vortices.

We further investigate the geometry of vortices by mean of purely non-linear tools such as conservation laws, Pohozaev formula for conformal fields,nonlinear comparison arguments, . . . Our understanding of the uniqueness typeproperties of Ginzburg-Landau vortices relies on a delicate analysis of the in-teraction between linear estimates in weighted spaces for linear operators withregular singularities and the nonlinar tools described above.

These existence and uniqueness results yield to a precise description of themoduli space of Ginzburg-Landau vortices in the strongly repulsive case. There-fore, reducing the study of this infinite dimensional problem to the study of afinite dimensional one.

viii

Chapter 1

Qualitative Aspects ofGinzburg-Landau Equations

The Ginzburg-Landau functional has been introduced by V.L. Ginzburg andL.D. Landau in [27] as a model for superconductivity. If Ω is a domain of Rn

which is diffeomorphic to the unit ball B1 ⊂ Rn the functional has the followingform

Gκ(u,A) :=∫

Ω

|∇u− iAu|2 +κ2

2

∫

Ω

(1− |u|2)2 +∫

Ω

|dA− hext|2. (1.1)

Here the condensate wave function u is defined from Ω into C and A is a 1-formdefined in Ω which represents the potential associated to the induced magneticfield h = dA in the material. The quantity |u|2 is nothing but the density ofcooper pairs of electrons which produce the superconductivity. Finally, hext

denotes the external magnetic field which is applied and then which is givenin the problem. The parameter κ > 0 is usually called the Ginzburg-Landauparameter.

Let us mention that, in Abelian gauge theory, this functional is defined notonly on domains of Rn but also on manifolds, in which case the function u andthe 1-form A have to be replaced respectively by sections and connections of agiven hermitian complex line bundle over the manifold. And ∇u − iAu has tobe replaced by the covariant derivative of u relative to A.

Critical points of the functional G are known as Ginzburg-Landau vortices,they solve the following system of equations

− ? dA ? dAu = κ2 u (1− |u|2)− ? d ? (dA− hext) = (i u, dA u),

(1.2)

where ( , ) is by definition the real part of the hermitian product (on C or onthe given hermitian complex line bundle). We have set

dA u := d u− i Au,

1

and where ? is the Hodge star operator (see Chapter 12 for a precise definition).Granted these definitions, we obtain in local coordinates

? dA ? dA u =2∑

k=1

(∂k − iAk)(∂k − iAk)u.

Naturally, these objects can be defined in an intrinsic way, we refer to [36] forfurther details.

Depending on κ and the applied field hext, a precise description of the min-imizers of Gκ (or more generally a description of the critical points of Gκ) isa very rich mathematical problem which remains still open in most of the sit-uations. Surprisingly, the set of critical points of Gκ is highly sensitive to theGinzburg-Landau parameter κ.

1.1 The integrable case

When Ω = R2 (which we will identify with C), when there is no exterior mag-netic field hext = 0 and when the Ginzburg-Landau parameter is given byκ = 1√

2, solutions of (1.2) have been intensively studied by A. Jaffe and C.

Taubes [36]. The value κ = 1/√

2 is a critical value of the Ginzburg-Landauparameter since, in this case, solutions of (1.2) whose energy Gκ(u,A) is finite,are also solutions of a first order system of equations. This is the reason why thisspecial case is known as the integrable or self dual case. A. Jaffe and C. Taubeshave given a complete description of the moduli space of Ginzburg-Landau vor-tices in that case. More precisely, they proved that :

(i) Given any set of points a1, . . . , aN ∈ C2 and given any set of integersn1, . . . , nN which either all belong to n ∈ Z : n < 0 or all belong ton ∈ Z : n > 0, there exists a solution such that

u−1(0) = a1, . . . , aN and deg(

u

|u| , aj

)= nj .

(ii) Any Ginzburg-Landau vortex with finite energy is one which is describedin (i).

(iii) If u and v are two Ginzburg-Landau vortices, then the following propertyholds

zeros of u = zeros of v =⇒ u = v. (1.3)

Of course, In the right hand side equality, one has to take into account themultiplicity of the zeros of u or v, moreover, the zeros have to be countednegatively for negative degree. The zeros are called the vortices of u.

2

This result, which holds in the integrable case κ = 1√2, serves to illustrate

to what extend is the information about a solution u of the Ginzburg-Landauequation included in of the zero set of u.

In the literature, this integrable case is also known as the non interactingsituation. Indeed, in this case, vortices can exist anywhere in the plane, namelythere is no constraint on the location of the vortices and, moreover, the energyof a solution only depends on

∑j nj the sum of the degrees at each vortex and

does not depend on the location of the vortices. Concerning what happens inthe other situations κ < 1/

√2 and κ > 1/

√2 A. Jaffe and C. Taubes have

stated some conjectures (see [36] pages 58-59), but as far as we know nothinghas been rigorously proved until now.

Qualitatively speaking, the case where κ < 1/√

2 is known as the attractivecase in which vortices tend to attract each other and to concentrated at oneplace, while the case κ > 1/

√2 is known as the repulsive case in which vortices

of the same sign tend to repulse each other. In the first case, supraconduc-tor materials are known as Type I supraconductors while, in the second casesupraconductor materials are known as Type II supraconductors.

1.2 The strongly repulsive case

In this book, we will focus our attention on the strong repulsive case whichcorresponds to the κ −→ +∞ (equivalently, the parameter ε = 1/κ −→ 0).When Ω = C2 and hext = 0, it has been conjectured by A. Jaffe and C. Taubesthat there should exist only 3 stable critical points of (1.2). Moreover, thesecritical points should be axially symmetric of vorticity (topological degree) ±1or 0. We refer to [64] and [28] for a proof of the stability of the degree ±1 axiallysymmetric solutions. Roughly speaking, all other vortices are sent at infinity,thus one may work on a bounded domain where vortices are be confined.

We will pay a special attention to properties like (1.3) in the strong repulsivesituation and we will investigate the consequences of this property when we willdescribe the set of solutions in this situation where vortices are confined.

In [11], it was proposed by F. Bethuel, H. Brezis and F. Helein to begin withthe analysis of a simpler problem we describe now. Assume that Ω is a simplyconnected bounded domain in C. For all ε > 0, we consider the non-gaugeinvariant Ginzburg-Landau functional

Eε(u) :=∫

Ω

|∇u|2 +1

2ε2

∫

Ω

(1− |u|2)2. (1.4)

In order to have a relevant problem, the absence of external magnetic field hasto be compensated by a prescription of a Dirichlet boundary condition u = gon ∂Ω, where

g : ∂Ω −→ S1,

has some topological degree (or vorticity on the boundary) deg g = d > 0.Critical points of this problem are solutions of the following semilinear elliptic

3

equation

∆u +1ε2

u (1− |u|2) = 0 in Ω

u = g on ∂Ω.

(1.5)

Although the boundary condition is not physically relevant, this model has beenvery successful in isolating and in rigorously describing how

vorticity =⇒ formation of vortices.

Otherwise stated F. Bethuel, H. Brezis and F. Helein give a proof of how aglobal topological obstruction generates vortices.

In [12], the corresponding gauge invariant prescribed Dirichlet boundary condi-tion problem was considered for the functional (1.1) instead of (1.4), and it wasproved that the absence of gauge invariance in the simple model above does notchange qualitatively the result about the formation of vortices.

Going back to the original model (1.1) without prescribing the Dirichletboundary data anymore, one would like to understand where this global topo-logical obstruction should arise naturally :

Why should there be vorticity ?

It was first observed by physicist, and then verified by formal computations (seefor instance [85]), that if one applies a uniform external magnetic field hext ona supraconductor material, then as long as hext is less than some critical valueHc1 , the fundamental state (that is the one which minimizes the energy) has novorticity while if

hext ≥ Hc1 =⇒ vorticity.

Moreover, in this later case, the number of vortices depends on the value of hext.There are recent progresses in the mathematical understanding of this question.For example, some answers are sketched in [13] and [83]. It is proven in [86] thatthere exists a critical value of hext bellow which the absolute minimizer of (1.1)has no vortex. Finally, in [88] and in [89], a family of stable solutions havingall possible vortices has been investigated and most of the physical observationsconcerning the existence or the non existence of vortices are rigorously estab-lished (see also the recent contribution of [23]). Let us just observe that all theabove mentioned works rely on the analysis which is carried on for the modelstudied by F. Bethuel, H. Brezis and F. Helein.

We now go back to the simpler model for which we wish to study of theasymptotic behavior of solutions of (1.5) as ε tends to zero. This analysis is farfrom being straightforward. Indeed, in the case where the boundary data g hasa non zero degree, then on the one hand it is well known that g does not admitan interior extension u : Ω −→ S1 with finite norm in W 1,2(Ω) (otherwise, bystandard density argument, g would also admit a regular extension), on theother hand the presence in the definition of Eε of the potential

1ε2

∫

Ω

(1− |u|2)2,

4

somehow forces the function u to take values into S1. Thus, even if we restrictour attention to the case of minimizers of Eε, as ε tends to 0, the infimumof the Ginzburg-Landau energy tends to +∞ and this makes it hard (evenimpossible) to deduce directly some a priori bounds for the solutions of (1.5) insome reasonable space from which one would be able to deduce a convergenceresult.

Nevertheless, the following result of [11] is the first step towards the completedescription of the asymptotic behavior of solutions of (1.5) as ε tends to 0.

Theorem 1.1 [11] Assume that Ω is a starshaped bounded regular domain ofC, let εn be a sequence of positive numbers tending to zero and let uεn

be asequence of solutions of (1.5). Then there exists a subsequence (still denoteduεn

), there exist N points a1, . . . , aN in Ω and N integers d1, . . . , dN such that

uεn−→ u? in Ck,α

loc (Ω \ a1, . . . , aN,C),

where u? is the harmonic map defined from Ω \ a1, . . . , aN into S1 given by

u? = eiφN∏

j=1

(z − aj

|z − aj |)dj

(1.6)

and where φ is a harmonic function determined by the condition u? = g on ∂Ω.

In some sense, the loss of compactness of the sequences of critical points uεn

in W 1,2(Ω) induces a point concentration phenomenon which is similar to theconcentration compactness phenomenon for critical powers (see [49]), exceptthat here the energy also blows up. The locus where concentration of energyoccurs corresponds to the limit of zero set of the sequence uε. Let us mentionthat the asymptotic behavior stated above was first established for the gaugeinvariant Ginzburg-Landau functional restricted to axially symmetric couples(u, A) (modulo Gauge action) on all of C by M. S. Berger and Y. Y. Chen in[9].

Among the ingredients of the proof of Theorem 1.1, the following boundsare essential :

|u| ≤ 1 in Ω, (1.7)

‖∇u‖L∞(Ω) ≤c

ε, (1.8)

and1ε2

∫

Ω

(1− |u|2)2 ≤ c. (1.9)

The derivation of these a priori bounds relies on the fact that the domain Ω isassumed to be starshaped. Using these bounds one can already obtain a roughpicture of u the solution of (1.5). Indeed, let aj be one of the points where u

5

cancels. Assume for simplicity that aj = 0 Then (1.8) implies that |u| ≤ 1/2 inthe ball Bε/(2c), and an easy computation shows that

1ε2

∫

Bε/(2c)

(1− |u|2)2 ≥ (3/8)2 π c2.

Now, because of the uniform bound (1.9), we can conclude that the zero setof u is included in a uniformly bounded number of balls of size ε and thataway from these balls |u| ≥ 1/2 (of course the value 1/2 is arbitrary and couldhave been replaced by any fixed positive number strictly less than 1). Theparameter ε is then the characteristic length for the concentration phenomenon.So the problem reduces to the determination of where this zero set is going toconcentrate. The answer to this last question is given in the following :

Theorem 1.2 [11] Under the assumptions of the previous Theorem, the N -tuple (aj)j=1,...,N is a critical point of a function which is defined on ΩN \ ∆and which is commonly called the renormalized energy

Wg((aj)j) := −2π∑

k 6=l

dkdl log |ak − al|+∫

∂Ω

Φ(

g × ∂g

∂τ

)

−2π

N∑

j=1

R(aj),(1.10)

where Φ is the harmonic conjugate of u?, i.e. this is the unique solution of

∆Φ = 2π

N∑

j=1

djδaj in Ω

∂Φ∂ν

= g × ∂g

∂τon ∂Ω

(1.11)

And where, by definition

∆ := (a1, . . . , aN ) ∈ ΩN : ∃j 6= k : aj = ak.

Here ν denotes the outward unit normal to Ω and τ denotes the unit tangentvector to ∂Ω such that the orthonormal basis (ν, τ) is direct and

R := Φ−N∑

j=1

log |z − aj |. (1.12)

Though this does not appear explicitely in the notation, observe that therenormalized energy depends on the choice of (dj)j=1,...,N . The denominationrenormalized energy can be justified in the following way. Take any N -tuple ofdistinct points (b1, . . . , bN ) in Ω and construct Φ the harmonic conjugate of the

6

harmonic map v? : Ω \ b1, . . . , bN −→ S1 which has degree dj at bj . Namely,Φ is a solution of (1.11) where the points aj are now replaced by bj . Finally,denote Φj(z) := log |z − bj | and define R := Φ −∑

j Φj to be the regular partof Φ. Clearly ∫

Ω

|∇v?|2 =∫

Ω

|∇Φ|2 = +∞.

Thus, in order to define some energy one subtracts to the above integrals someinfinite energy which does not depend on Ω, neither on g nor on the bj . Moreprecisely, the function Φ is the sum of R and of all the Φj . The expression ofWg is then obtained through the following procedure :

First expand |∇R+∑

j Φj |2, integrate each term on Ω and perform a formalintegration by parts of |∇Φj |2 on Ω. Now subtract to that expression the infinite

quantity −∑

j

∫

Ω

Φj · ∆Φj , which is a quantity which is independent of Ω, g

and bj .

It follows from a short computation that Wg, defined by (1.10), is exactlygiven by

Wg((bj)j) =∑

k 6=l

∫

Ω

∇Φk · ∇Φl + 2∫

Ω

∇R ·∑

j

∇Φj

+∫

Ω

|∇R|2 +∫

∂Ω

∑

j

Φj · ∂Φj

∂ν

=∫

Ω

|∇v?|2 +∑

j

∫

Ω

Φj ·∆Φj .

The renormalized energy can also be understood as the sum of terms mod-eling the interaction between the different vortices −2π dk dl log |bk − bl| (theinteraction is attractive if sign dk 6= sign dl and repulsive otherwise) and termsmodeling the interaction between the boundary and each vortex (repulsive).

A very useful characterization of the renormalized energy is provided by thefollowing :

Proposition 1.1 [11] The N -tuple (aj)j=1,...,N is a critical point of Wg if andonly if for all j = 1, . . . , N , the map u? defined by (1.6) can be written as

u? =(

z − aj

|z − aj |)dj

eiHj ,

in some neighborhood of aj. With

∇Hj(aj) = 0.

7

In the particular case where the solutions uε are a not only critical points butalso minimizers, the above results have been extended and made more preciseby M. Struwe in [96], under the assumption that Ω is simply connected (see alsothe approach in [12]).

Theorem 1.3 [96] Assume that Ω is simply connected and that uεn is a se-quence of minimizers then the conclusions of Theorem 1.1 and Theorem 1.2hold with

dj = +1,

for all j = 1, . . . , N . Furthermore, the N -tuple (aj)j=1,...,N is a minimizer ofWg and the following asymptotic expansion of the energy holds

Eεn(uεn

) = −2 π deg g log εn + Wg((aj)) + c deg g + δ(εn)

where c is some universal constant and where δ(s) −→ 0 as s → 0.

Since then, this result has been generalized to situation where the potential inthe functional also depends on the point in the domain (see [7], [8] and [1]).Higher dimensional versions of this Theorem have been established in [82] and[46]. Let us mention that a similar phenomenon has been observed by R. Hardtand F.H. Lin for the p-energy of maps defined from domains into S1, as theparameter p tends to 2 (see [31], [32]). Finally, in the self-dual case, this kindof asymptotic has been investigated in [35].

So Theorem 1.1, Theorem 1.2 and Theorem 1.3 provide a complete descrip-tion of the limit of Ginzburg-Landau vortices as the parameter ε tends to 0. Inparticular, if one knows the critical points of the function Wg, using (1.6), onededuces almost explicitly the possible limits of the critical points of the func-tional Eε. Stated differently, one can obtain information about the solutions ofan infinite dimensional problem by first solving a finite dimensional one. Themain purpose of this book is to further investigate the relation between thecritical points of Eε, for ε small, and the critical points of Wg. More preciselywe would like to answer the following natural question :

Assume that the N -tuple (aj)j=1,...,N is a critical point of Wg and considerthe corresponding u? defined in (1.6). How many branches of critical points ofEε converge to u? as ε tends to 0 ?

Among others, topological methods have been very efficient in providing apartial answer to this question and F.H. Lin and T.C. Lin in [45], have success-fully investigates the links between the level sets of Eε and the level sets of Wg.In particular they have proved the :

Theorem 1.4 [45] Assume that (aj)j=1,...,N is a non degenerate critical pointof Wg, where all dj = +1. Then, for all ε small enough, there exists uε a criticalpoint of Eε, which converges to the corresponding u?.

Still using topological tools, L. Almeida and F. Bethuel have also obtained someexistence results [3].

8

Although the Ginzburg-Landau equations are Euler-Lagrange equations ofsome variational problem, our approach to the above stated question will bebased more on PDE and functional analysis tools versus topological and varia-tional tools ! For instance, one of our first task will be to recover and generalizeF.H. Lin and T.C. Lin’s result by the mean of some local implicit functionTheorem in weighted Holder spaces instead of topological methods.

In doing so, our motivation if twofold : First this approach will provide avery precise description of the solution and in particular of its zero set. Second,this implicit function Theorem approach leads to local uniqueness results whichwill be the first step towards a precise description of the branches of solutions.

Before we go further into the description of the branches of solutions, weneed to have a precise picture of the shape of the solutions when ε is small,ε 6= 0. Many works have been devoted to this question. Let us concentrate onthe case where uε is a minimizer since in this case the results are more precise.We already know that, as ε tends to 0, uε converges strongly to u? in Ck norm,away from the points aj . Thus, it remains to understand what happens near theset where uε cancels. The following result has been established by P. Bauman,N. Carlson and D. Philips :

Theorem 1.5 [6] Assume that ε is sufficiently small. Then, for any minimizeruε of Eε, 0 is a regular value of uε moreover

u−1ε (0) = a1(ε) . . . , ad(ε),

where d = |deg g| and|aj(ε)− aj | = o(1),

where (aj)j=1,...,N is a minimizer of Wg.

Making use of very precise estimates of the energy of minimizers, M. Comteand P. Mironescu [21] have been able to prove that there exists α ∈ (0, 1) suchthat if (aj)j=1,...,N is a non degenerate minimizer of Wg, then

|aj(ε)− aj | = O(εα).

They further conjecture that the same estimate should be true for α = 1 butour construction seems to show that this conjecture a little too optimistic.

Granted the result of P. Bauman, N. Carlson and D. Philips, it is naturalto look at the tangent map (or the limit profile) of uε at aj(ε). More precisely,assume that aj(ε) is a vortex of uε (i.e. a zero of uε) and assume that c > 0 ischosen small enough so that Bc(aj(ε)) ⊂ Ω for all ε small enough and also thataj(ε) is the only zero of uε in Bc(aj(ε)) . We then define

uε := uε(ε ·).Clearly, this function satisfies

∆uε + uε (1− |uε|2) = 0 in Bc/ε(0)

uε(0) = 0.

9

Moreover, it can also be proved that∫

Bc/ε(0)

(1− |uε|2)2 ≤ C

deg(

uε

|uε| ; ∂Bc/ε(0))

= +1.

As ε → 0, we may pass to the limit in the above equation and obtain that thelimit map u solves

∆u + u (1− |u|2) = 0 in C

u(0) = 0∫

C(1− |u|2)2 ≤ C.

(1.13)

It is proven in [15] that :

Theorem 1.6 [15] [Quantization result] Assume that u is a solution of

∆u + u (1− |u|2) = 0,

in C, such that ∫

C(1− |u|2)2 ≤ C.

Then, the degree d of u/|u| at infinity is well defined and we have∫

C(1− |u|2)2 = 4 π d2.

In particular, if d = 0, then u is constant.

It is therefore important to classify all the solutions of (1.13) since they providethe local profile of uε near each 0.

Among the possible solutions of (1.13), the axially symmetric ones are mostnatural to look at since, if we write u = S(r) ei(θ+θ0) for θ0 ∈ R, then we areleft to solve a second order ordinary differential equation

d2S

dr2+

1r

dS

dr− 1

r2S + S (1− S2) = 0

S(0) = 0

limr→∞

S = 1.

(1.14)

It is proven by R. M. Herve and M. Herve that :

Theorem 1.7 [34] There exists a unique solution of (1.14) which is defined onall R+, is not constant and whose limit at ∞ is 1. This solution S is strictlyincreasing and we have 0 ≤ S < 1.

10

Moreover, it is proved in [34] that∫

R2(1− S2)2 < ∞.

The study of such special solutions of (1.13) in general was undertaken in [15],where it was proven that the asymptotic behavior at infinity of any solution isthe one of S (see also [90]). The question whether all solutions of (1.13) areaxially symmetric remained an open problem for some years, until this problemwas finally solved by P. Mironescu using a short and surprising argument whichwill be largely commented and generalized in Chapter 9 of this book.

Theorem 1.8 [65] Any solution of (1.13) coincides with S ei(θ+θ0), where S isthe solution defined in Theorem 1.6 and θ0 ∈ R.

To summarize, we have obtained a fairly precise picture of the minimizersof Eε since we know that, away from the points aj , they look like u? and neareach zero aj(ε) (which is known to converge to aj with a speed at least εα for

some α ∈ (0, 1)) they looks approximately like S(|z−aj(ε)|

ε

)ei(θ+θ0). There is

a quantitative version of this last assertion [91], [65] : There exists a harmonicfunction ψ : Ω −→ R such that

∥∥∥∥∥∥uε − eiψ

d∏

j=1

S

( |z − aj(ε)|ε

)ei arg(z−aj(ε))

∥∥∥∥∥∥L∞(Ω)

→ 0. (1.15)

This estimate can be complemented by other pointwise estimates [21]

|∇uε| ≤ c

L|∇|uε|| ≤ c

ε2

L3(1.16)

where L = max(ε, dist (z, a1(ε), . . . , ad(ε))).Even though the asymptotics of the minimizers of the Ginzburg-Landau

functional are well understood, some completely new arguments are neededin order to establish some uniqueness result which would provide a bijectionbetween the minimizers of Eε (or the critical points of Eε ) and the minimizersof Wg (or critical points Wg).

For example, let us consider the simple situation where Ω = B1 and g = eiθ.In this case, it was conjectured by F. Bethuel, H. Brezis and F. Helein that,for all ε > 0, the minimizer of Eε is axially symmetric, i.e. has the form Sε eiθ

(such a Sε is known to be unique [34]). When ε is large enough, the fact thatthe conjecture is true is a direct consequence of the convexity of the functional(see [11] Theorem VIII.7). While, in the case where ε is small, it is proven in[10] that any sequence of minimizers converges to u? = x

|x| . In other words,0 is the unique minimum of Weiθ ). Several works have been devoted to thisseemingly simple uniqueness question [44] and [20]. We shall see that the tools

11

P. Mironescu has used for proving Theorem 1.8 can be adapted to this situationand if one assumes that uε vanishes at 0. Hence, this already proves that

zero uε = 0 =⇒ uε = Sε(r) eiθ. (1.17)

Unfortunately, even for a minimizer, one does not know that uε vanishes at 0.For the time being the most refined energy estimates which are available [66]yields |zerouε| ≤ c ε2, but no direct arguments are known to show zero uε = 0.However, we will see that a contraction mapping argument, which is justifiedby the previous observations, allows to solve Bethuel-Brezis-Helein’s conjecturefor small values of ε.

This property also serves to illustrate, in the axially symmetric setting, thestrength of the information which is carried by the zero set of the solution. Thisis in fact a special case of a much general property we are currently investigating,which should extend to the strongly interacting case on bounded domain, theproperty (1.3) which has been established by A. Jaffe and C. Taubes.

The case where the limit vortices can have degree dj 6= ±1 seems much morecomplicated, let us mention the very interesting comments on this question in[69]. When dj 6= ±1 it is not known if there exist limit profiles which are differentfrom the axially symmetric one, for example, a limit profile with splitted zeros(however, if there exists a zero of degree dj , Mironescu’s result still holds andsays that the solution is also axially symmetric see [65]). It is also not known ifone can expect intermediate configurations, say with different zeros distant onefrom the other by a distance >> ε, but converging to the same limit vortex.For instance in the case where Ω = B1 and g = e2iθ, 0 is still a critical pointof We2iθ (although it is not a minimum), and there exists a unique axiallysymmetric solution which converges to the u? = e2iθ. Is the axially symmetricsolution the only one converging to e2iθ ? Formal computations seem to militatefor the existence of a critical point having 3 vortices of degree +1 at the verticesof a (small) equilateral triangle centered at 0 and one of degree −1 close to thiscenter. Therefore, it is far from being impossible that limit vortices of higherdegree (dj 6= ±1) induces a degeneracy of the number of solutions associated tosome critical point of Wg.

1.3 The existence result

In this book, we will restrict ourselves to the case where the degree of the limitvortices is either +1 or −1. In Chapters 3 through 7, we construct solutionsusing some fixed point Theorem for contraction mappings. Precisely we provethe :

Theorem 1.9 Assume that dj = ±1, for all j = 1, . . . , N . Let (aj)j=1,...,N bea nondegenerate critical point of Wg. Then, there exists ε0 > 0 and, for allε ∈ (0, ε0), there exists a solution uε of (1.5) with exactly N isolated zeros aj(ε)such that

12

(i) limε→0

aj(ε) = aj,

(ii) limε→0

uε = u? in C2,αloc (Ω \ a1, . . . , aN),

where u? is given by (1.6).

In particular, we recover and slightly generalize Theorem 1.4 of F.H. Lin andT.C. Lin, which has been obtained first by variational methods. As a byprod-uct of our construction, we obtain more precise informations about the criticalpoints we construct. For example, we know the exact number of zeros and alsothe rate of convergence of the zeros towards the aj .

Proposition 1.2 For all α ∈ (0, 1) there exists a constant cα > 0 independentof ε ∈ (0, ε0) such that

|aj(ε)− aj | ≤ cα εα. (1.18)

We also have the following information about what is usually called the densityof area

Proposition 1.3 There exists ε0 > 0, σ > 0 and c > 0 depending only on Ωand g such that for all ε ∈ (0, ε0)

det (∇uε) ≥ cε2

sup(ε4, r4),

in each Bσ(aj(ε)), where r := |z − aj(ε)|.But probably the most important byproduct of this approach is a local unique-ness property for the solutions we have constructed, in a space which will bemade precise later on. This will be one of the main tool for proving our mainuniqueness result. Let us now sketch the strategy of the proof of Theorem 1.9.

The proof of Theorem 1.9 relies on the use of a fixed point theorem, forcontraction mappings, in weighted Holder spaces. Such an approach is stronglyinfluenced by the work of the first author and R. Mazzeo [54] concerning theexistence of complete conformally flat metrics with constant scalar curvature onSn \ p1, . . . , pN for any set of points p1, . . . , pN ∈ Sn.

The general strategy is fairly natural and standard. We first construct ε →uε a one parameter family of approximate solutions of equation (1.5) by patchingtogether several concentrated profiles S(r/ε) eiθ centered at the limit vorticesaj , together with the limit solution u?. Naturally, as ε tends to 0, the quality ofthe approximate solution improves. Then, we study the mapping properties ofLε = DNuε , the linearized Ginzburg-Landau operator about uε, where we haveset

N (u) := ∆u +u

ε2(1− |u|2).

Finally, the solution is obtained as a perturbation of uε, applying some con-traction mapping argument.

13

Even if one can prove that, for ε small enough, the linearized operator Lε

is bijective when defined between standard spaces (e.g. Lε : W 2,2 −→ L2 orLε : C2,α −→ C0,α), the norm of its inverse blows up as ε tends to 0. Reducingto nothing the hopes to find the solution using such a crude approach.

Therefore, we have to be more careful and our first task will be to identifywhat is responsible for this loss of invertibility as ε tends to 0. The mainobservation is that, if one performs a space dilation about one of the blow-uppoints at the scale which makes appear the limit profile (in our case 1 −→ 1/ε),the rescaled linearized operator Lε converges to L the linearized operator aboutthe limit profile solution of the limit equation (in our case ∆u+u(1−|u|2) = 0 onall C). This limit equation has a group of invariance (in our case, the isometriesof the plane) and the infinitesimal action of this group gives rise to Jacobi fieldsof L which, in some sense, are responsible for the loss of invertibility of Lε

as ε tends to 0. This loss of invertibility is by now well understood since asimilar phenomenon appears in many other problems, for instance in the theoryof critical points at infinity for the Yamabe problem and its generalizations in[78], [79], [80], [4] or in the construction of constant mean curvature surfaces in[37], [55]or also in the singular Yamabe problem in [87], [54].

In C, there exists of course an infinite dimensional space of solutions ofLw = 0, but only a 3 dimensional space of solutions coming from geometricperturbation of the radially symmetric solution S eiθ :

• Φ0 which corresponds to the action of rotations,

• Φ+1 which corresponds to the action of translation along x-axis,

• Φ−1 which corresponds to the action of translation along y-axis.

Somehow surprisingly, these Jacobi fields generate exactly the space of solutionsof the homogeneous problem Lw = 0 which are bounded in L∞ norm.

Now, performing the dilation backward, we consider the 3N following mapsΦ0( z−aj

ε ), Φ+1( z−aj

ε ) and Φ−1( z−aj

ε ) which (after multiplication by a cutofffunction η away from the aj in order to ensure these functions are equal to 0 on∂Ω) are responsible for the norm of the inverse of Lε to blow up.

We also have to deal with an additional problem which comes from thefact that, when we correct uε by adding to it a function v to obtain an exactsolution to (1.5), we would like to preserve the zero set of uε. Since at eachvortex |uε| ∼ |z − aj |, we can only allow perturbations v satisfying, for someµ > 1

|v| ≤ c |z − aj |µ, (1.19)

at each vortex.

We set Σ := a1, . . . , aN. In order to overcome the second problem, itis fairly natural to work on the weighted Holder space C2,α

µ (Ω \ Σ), which canroughly be defined to be the space of C2,α functions which are bounded by|z − aj |µ near each aj . Unfortunately, on this space, the operator Lε is not

14

surjective anymore, but has finite dimensional cokernel. This is the reason whywe will work with a finite dimensional extension of this weighted Holder spaceby adding at least the space Kε which is generated by the functions Φ0( z−aj

ε )and Φ±1( z−aj

ε ) near each aj . Notice, and this is very important, that none ofthe functions in the finite dimensional extension belong to the weighted spaceC2,α

µ (Ω \Σ) provided µ > 1 since they behave like |z− aj |0 or like |z− aj |1 neareach aj .

When µ ∈ (1, 2), we will prove that the cokernel of Lε is exactly 6N dimen-sional. And, beside the above mentioned functions, we also have to consider theextension Hε generated by 3N other functions Ψ0( z−aj

ε ) and Ψ±2( z−aj

ε ) (seeChapter 3). We can now consider the space

C2,αµ (Ω \ Σ)⊕Hε ⊕Kε.

We will prove that Lε is an isomorphism from this space into C0,αν−2(Ω \ Σ), but

again, as ε tends to zero, the norm of its inverse blows up. This time this phe-nomenon can be identified explicitly and geometrically since it is intimately con-nected with the definition of the finite dimensional extension mentioned above.

There is still one phenomenon we have not taken into account yet : thescaling effect. This phenomenon is related to the fact that u is not scalar butvector (complex) valued and that the modulus |uε| and the phase of uε behavein a different way as ε tends to zero.

Given the precise description of the convergence of uε, critical point of theGinzburg-Landau functional, towards u? (which is established for minimizersbut should also hold for critical points in general) and given estimates like (1.16),it follows that the characteristic length for |uε| is ε although the characteristiclength for the phase is 1 (more details will be given in Chapter 4). This is oneof the reasons why, we have to distinguish the perturbations of the modulusof uε from the perturbations of the phase of uε and instead of considering thelinearized operator acting on Ck,α

µ (Ω\Σ)⊕Hε⊕Kε, we will work with Ek,αµ (Ω\

Σ) ⊕ Hε ⊕ Kε which an altered version of the above defined spaces in whichthe real part and imaginary parts of the functions have distinct behaviors. Theexact definition of this space will be given in Chapter 4. With these definitions,we will prove that

Lε : E2,αµ (Ω \ Σ)⊕Hε ⊕Kε −→ E0,α

µ−2(Ω \ Σ), (1.20)

is uniformly invertible independently of ε, provided µ ∈ (1, 2).

In the final step of the construction of the solution uε, we will perturb uε byadding to it a function w ∈ E2,α

µ ⊕Hε and then by composing the result withthe action of a diffeomorphisms on the domain whose infinitesimal action on uε

generates exactly Kε (namely, small rotations around each aj , small translationsin a neighborhood of each aj). An application of a standard fixed point theoremfor contraction mappings will ensure both the existence and the local uniquenessof a solution of ∆u +

u

ε2(1− |u|2) = 0 in a neighborhood of uε.

15

To conclude, let us mention two open problems :

It should be possible to apply our construction not only when the vorticeshave degrees dj ∈ ±1 but also for arbitrary degrees dj ∈ Z. For the time being(at least) one preliminary result seems to be missing. Indeed, let ud = eidθ Sd(r)denote the radially symmetric solution of

N (u) := ∆u + u(1− |u|2) = 0,

in C, which tends to eidθ at ∞. Let Ld = dNuddenote the linearized operator

about ud, we would like to say that the only bounded solutions of Ldw = 0which are defined in C are linear combinations of the following three Jacobifields i ud, ∂xud and ∂yud. Unfortunately we have not been able to prove sucha result.

It is known that Ld necessary has negative eigenvalues (see [2]) but a betterunderstanding of the behavior of the eigenvalues close to 0 as ε tends to 0 isstill missing.

Another direction were the above described technique could be successful isthe construction of solutions of the Gross-Pitaevskii equation which models thedynamic of quantum fluids

i∂u

∂t−∆u = u (1− |u|2). (1.21)

As a first approximation, the dynamic of the vortices can be taken to be

daj

dt(t) = −∇⊥j W ((aj)), (1.22)

where

∇⊥j W :=

(∂W

∂a2j

,−∂W

∂a1j

),

when working in the adiabatic limit case (i.e. |ak − al| ' 1/ε). The problemwould be to apply a result like (1.20) in this time dependent situation in orderto construct the solutions announced in [70].

1.4 Uniqueness results

In the second part of the book, we explain how the local uniqueness resultinduces a complete description of the branches of solutions converging to a limitconfiguration all of whose vortices have degree ±1.

One of the key points is the following property which is proved in Chapter10. Let v be one of the solutions constructed in the first half of the book andassume that dj = ±1 for all j, let u be any solution of (1.5), then the followingholds

zeros of u = zeros of v =⇒ u = v, (1.23)

16

provided ε and ‖u− u?‖W 1,1 are sufficiently small (the smallness which is nec-essary only depends on Ω and g). This property is exactly the generalization of(1.17) to the non necessarily axially symmetric setting. In order to prove (1.23),we first establish some generalized Pohozaev formula, where the usual conformalvector field is replaced by more general vector fields which we call ρ-conformalvector field (see Chapter 9). This yields some important estimates concerningthe difference between u and v in weighted Sobolev space (see Chapter 11).

In general though, we do not know a priori that the two solutions u and v,which do converge to the same u?, have the same zeros, we only know thattheir respective zeros converge to the same limit vortices. Nevertheless, usingthe approach we have used to prove property (1.23), we obtain an informationabout the closeness between u and v. Mixing this upper bound and the localuniqueness result we have already established for v, we get that u = v. Thisestablishes one of the main results proved in this book :

Theorem 1.10 Let εn be a sequence tending to 0 and un, u′n be critical pointsof Eεn which converge to the same u?. Assume that all dj are equal to ±1 andthat (aj)j=1,...,N , the set of singularities of u?, is a nondegenerate critical pointof Wg. Further assume that there exists c0 > 0 such that

12

∫

Ω

|∇u|2 +1

4 ε2

∫

Ω

(1− |u|2)2 ≤ c0 log 1/εn,

holds for un and u′n. Then, for n large enough, un = u′n.

Our technics give the same result in the simpler situation where deg g = 0 andwhere there are no vortices in the limit : assume that un and u′n converge tothe same u? which has no vortex, then for n large enough un = u′n.

In fact our result is slightly stronger since we will prove that, for n largeenough, un = u′n is equal to the solution given by Theorem 1.9. Thus, wehave as a byproduct a nice description of all the critical points of the Ginzburg-Landau functional which are sufficiently close to their limit u?.

Corollary 1.1 Let εn be a sequence tending to 0 and let un be a sequence ofcritical points of Eεn . Assume that the zeros of un converge, as n tends to +∞,to (aj)j=1,...,N which is a nondegenerate critical point of Wg and also that alldj = ±1. Further assume that there exists c0 > 0 such that

12

∫

Ω

|∇un|2 +1

4 ε2

∫

Ω

(1− |un|2)2 ≤ c0 log 1/εn.

Then, for n large enough, un has exactly N zeros (aj(εn))j=1,...,N moreover thefollowing bound holds

|aj(εn)− aj | ≤ εα ∀j = 1, . . . , N,

where α ∈ (0, 1).

17

In the particular case of minimizers the result can be stated in the followingway

Corollary 1.2 Assume that all minimizers of Wg are nondegenerate. Let usdenote by A the set of all minimizers of Wg. Then, for all 1 > α > 0, thereexists ε0 > 0 such that, for all ε ∈ (0, ε0), the distance between the zero set of aminimizer uε of Eε and A is bounded by εα.

Another interesting consequence of the above Theorem is concerned with thesign of the area density

Corollary 1.3 Under the assumptions of Corollary 1.1. There exist some con-stants ε0 > 0 and σ > 0 such that, for all ε ∈ (0, ε0) and for all z ∈ Bσ(aj)

det (∇uε) ≥ 0.

From our main result, we may also deduce the following result in the casewhere the minimizers of Wg is unique.

Corollary 1.4 Assume that all dj = +1, that (aj)j=1,...,N is the unique mini-mizer of Wg and further assume that it is a nondegenerate critical point of Wg.Then there exists ε0 > 0 such that, for all ε ∈ (0, ε0) there exists one and onlyone minimizer uε of Eε.

In the case where deg g = 0, that is when there are no vortices in the limit, theuniqueness of the minimizer for ε small was proved by D. Ye and F. Zhou in[100].

Finally in the special case where Ω = B1 and g = eiθ we are able to solveBrezis’s conjecture for ε sufficiently small

Corollary 1.5 Any critical point of the Ginzburg-Landau functional on the unitdisk which is equal to eiθ on the boundary is, for ε sufficiently small, the axiallysymmetric solution.

In the last Chapter of the book we extend the use of the techniques developedabove to the case of the gauge invariant functional. The combination of aPohozaev type argument and estimates in weighted Sobolev spaces which arisein a crucial matter yields to the following result :

Theorem 1.11 Let (u,A) be a solution of the free Yang-Mills-Higgs equations(1.2) in C, with hext = 0, verifying the energy upper-bound

Gκ(A, u) ≤ 2 π log κ + c0, (1.24)

for some constant c0 > 0 independent of κ. Further assume that

u(0) = 0.

Then, for κ large enough, (u, A) is gauge equivalent to one of the two axiallysymmetric solutions of degree +1 and −1.

18

Notice that similar technics were used to make some progresses toward theA. Jaffe and C. Taubes conjectures in the strong repulsive case : κ >> 1 :

Theorem 1.12 [84] For κ sufficiently large, the only homotopy classes contain-ing a minimizer of the Abelian Yang-Mills-Higgs functional (1.1) in C are the 0and ±1 classes, moreover any global minimizer is one of the axially symmetricsolutions (modulo gauge invariance) described in [9].

As a final remark let us mention that all the above techniques can be modifiedto prove a result corresponding to Theorem 1.10 for (un, An) and (u′n, A′n),minimizers of the gauge invariant functional (1.1), on a compact surface, for agiven C−bundle. More precisely, adapting the different steps of our proof ofTheorem 1.10 to the gauge invariant situation by the mean of the argumentsdeveloped in Chapter 12, the following statement can be proved :

Theorem 1.13 Let εn be a sequence tending to 0 and (un, An), (u′n, A′n) be twosolutions of the free Yang-Mills-Higgs equations (1.2), hext = 0, on a compactsurface with no boundary for a given C-bundle. Assume these two solutionsverify the energy upper-bound

G1/εn≤ c0 log 1/εn,

and that they both converge to a singular section and connection (u?, A?) whosesingularities is a non-degenerate critical point of the renormalized energy Wgiven in [75], with multiplicities ±1. Then for n large enough (un, An) and(u′n, A′n) are gauge equivalent.

In other words, we have a complete description of the moduli space of so-lution of the Abelian Yang-Mills-Higgs equations on a surface in the genericstrongly repulsive case, starting from the critical points of the function W .

19

Chapter 1

20

Chapter 2

Elliptic Operators inWeighted Holder Spaces

Given a regular open subset Ω of Rn and given a1, . . . , aN ∈ Ω, we define someweighted spaces as the set of functions defined in Ω \ a1, . . . , aN which decayor blow up near each puncture ai at most at a certain prescribed rate. Then, weproceed to the investigation of the mapping properties of some class of ellipticoperators which are defined between these spaces.The material and the results of this chapter are well known and one can finda thorough study of this problem in [63], [50], [51] and also in [52]. Therefore,in this first Chapter, our intention is not to give a complete description of thetheory of elliptic operators between weighted spaces but rather to expose, ina simple setting, the technics which will be extensively used in the subsequentChapters. In particular, the proofs we give all rely on very simple tools whichseem more flexible and which can be easily adapted to other contexts (sucha simple approach was probably originated in [18]). Moreover, we hope thatthis introduction will help the interested reader to adapt the theory to his ownproblem.We will assume that the reader is familiar with the basic theory of the Laplacianin the framework of Holder and Sobolev spaces (as it can be found, for example,in [26]). In order to illustrate our purpose, we will concentrate on the case ofthe Laplacian in some bounded open subset of Rn, since we believe that thereader will be more familiar with the arithmetic in this simple case.We end this Chapter by three, somehow academic, applications of this theoryin nonlinear settings.

2.1 The function spaces

In all this section, and even through all this chapter, Ω will be a regular opensubset of Rn, n ≥ 2. And Σ := a1, . . . , aN will be a finite set of points in Ω.

21

Let us chose σ > 0 in such a way that, if ai 6= aj ∈ Σ, then B2σ(ai) andB2σ(aj) are disjoint and both included in Ω. For all s ∈ (0, σ), we define

As := x ∈ Ω : dist(x, Σ) ∈ [s, 2s] . (2.1)

We also setΩσ := x ∈ Ω : dist(x, Σ) > σ . (2.2)

Finally, we chose some ”regularized” distance function to the set Σ. Thisregularized distance function is a function d ∈ C∞loc(Ω \ Σ), which satisfies thefollowing properties :

(i) d(x) := dist(x,Σ) for all x ∈ Ω \ Ωσ.

(ii) d(x) ≥ σ for all x ∈ Ωσ.

Given k ∈ N, α ∈ [0, 1) and given a regular subset ω ⊂ Rn, the space Ck,α(ω)denotes the usual Holder space, which is endowed with the usual Holder norm‖ ‖k,α,ω. We refer to [26] for a precise definition of these notions.For any function w ∈ Ck,α

loc (Ω\Σ), we consider the family of semi-norms indexedby s ∈ (0, σ)

[w]k,α,s :=k∑

j=0

sj supAs

|∇jw|+ sk+α supx,y∈As

|∇kw(x)−∇kw(y)||x− y|α . (2.3)

We are now in a position to define the weighted Holder spaces we will workwith in this chapter :

Definition 2.1 Given k ∈ N, α ∈ [0, 1) and ν ∈ R, the space Ck,αν (Ω \ Σ) is

defined to be the set of functions w ∈ Ck,αloc (Ω \ Σ) for which the following norm

is finite‖w‖Ck,α

ν (Ω\Σ) := ‖w‖Ck,α(Ωσ) + sups∈(0,σ)

s−µ [w]k,α,s. (2.4)

For example, for all k ∈ N, α ∈ [0, 1) and all ν ∈ R, the function

x ∈ Ω \ Σ −→ dν(x)

belongs to Ck,αν (Ω \ Σ).

With a slight abuse of notation, we will write for some vector valued functionw (having values in Rm, m ≥ 1) that w ∈ Ck,α

ν (Ω \ Σ) if all the coordinatefunctions of w belong to Ck,α

ν (Ω \ Σ).

Remark 2.1 More generally, one can define analogously some weighted spacesfor which the weight parameter ν is different at each puncture ai.

As a first property of these spaces, we have the following :

22

Lemma 2.1 Endowed with the above defined norm, the space Ck,αν (Ω \ Σ) is a

Banach space.

We now give a short list of useful properties which allow one to do calculus inthese spaces and also to determine easily whether or not a function belongs toCk,α

ν (Ω \ Σ).

Proposition 2.1 The following properties hold.

(i) Assume that w ∈ Ck+1,αν (Ω \ Σ), then ∇w ∈ Ck,α

ν−1(Ω \ Σ).

(ii) Assume that w ∈ Ck+1,0ν (Ω \ Σ) and that ∇w ∈ Ck,0

ν−1(Ω \ Σ), then, for allα ∈ [0, 1), the function w belongs to Ck,α

ν (Ω \ Σ).

(iii) Assume that, for i = 1, 2, wi ∈ Ck,ανi

(Ω \ Σ), then w1 w2 ∈ Ck,αν1 ν2

(Ω \ Σ)and

‖w1 w2‖Ck,αν1 ν2

≤ c ‖w1‖Ck,αν1‖w2‖Ck,α

ν2,

for some constant c > 0 independent of w1 and w2.

(iv) Assume that w ∈ Ck,αν (Ω \ Σ) and further assume that w > 0 in Ω \ Σ,

then, for all p > 0, wp ∈ Ck,αp ν (Ω \ Σ) and in addition

‖wp‖Ck,αp ν

≤ c ‖w‖p

Ck,αν

,

for some constant c > 0 which does not depend on w.

When working in weighted Holder spaces, the classical compactness embeddingTheorem in Holder spaces takes the following form :

Lemma 2.2 Assume that k + α < k′ + α′ and that ν < ν′, then the embedding

I : Ck′,α′

ν′ (Ω \ Σ) −→ Ck,αν (Ω \ Σ)

is compact.

Very frequently, we will have to consider the subspace of functions whichvanish on the boundary of Ω. This is the reason why we define

Ck,αν,D(Ω \ Σ) :=

w ∈ Ck,α

ν (Ω \ Σ) : w = 0 on ∂Ω

. (2.5)

We end up this section with some important remark. This remark is intendedto shed light on our definition of weighted Holder spaces and show that, in somesense, our definition is the natural one.

Remark 2.2 Instead of considering the spaces defined in Definition 2.2, wecould also have considered the spaces

Ck,αν (Ω \ Σ) :=

dν w : w ∈ Ck,α(Ω)

.

23

A closer look at this definition shows that Ck,αν (Ω \ Σ) is a proper subspace of

Ck,αν (Ω \ Σ). For example, the function

x ∈ Ω \ Σ −→ dν ∇d,

belongs to the later space but does not belong to the former. This example showsthat the spaces Ck,α

ν (Ω \Σ) are not well behaved since, for example, property (i)of the previous Proposition does not hold for them.

2.2 Mapping properties of the Laplacian

In this section, we consider the Laplacian as the simplest example of some ellipticoperator acting on the weighted Holder spaces we have just defined.

Let us denote by φj , j ∈ N, the set of eigenfunctions of the Laplacian onSn−1 with corresponding eigenvalues λj , that is

∆Sn−1φj = −λj φj .

We assume that the sequence of eigenvalues is increasing λj ≤ λj+1, and thatthe eigenfunctions are normalized by

∫

Sn−1φ2

j dθ = 1.

Notice that we will always assume that the eigenvalues are counted with multi-plicity, namely

λ0 = 0, λ1 = . . . = λn = n− 1, λn+1 = 2n, . . .

In polar coordinates the Laplacian in Rn reads

∆ = ∂2rr +

n− 1r

∂r +1r2

∆Sn−1 . (2.6)

If we project the operator ∆ over the eigenspace spanned by φj , we obtain theoperator

Lj w := ∂2rr +

n− 1r

∂rw − λj

r2w. (2.7)

If w ∈ C2,α(B1 \ 0) is a solution of ∆w = 0 in B1 \ 0, we may write theeigenfunction decomposition of w as

w(r, θ) =∑

j≥0

wj(r) φj(θ).

Then, the function wj , which is a solution of Lj wj = 0 in (0, 1], is a linear com-bination of two linearly independent solutions of Ljw = 0. Such two solutionsare given by

w+j := rγ+

j and w−j := rγ−j , (2.8)

24

when n ≥ 3 and j ≥ 0 (or when n = 2 and j ≥ 1), and by

w+0 := 1 and w−j := − log r,

when n = 2 and j = 0. Here, we have set

γ±j =2− n

2±

√(2− n

2

)2

+ λj . (2.9)

The key observation is that the coefficients γ±j give, at 0, all the possibleasymptotic behaviors of the solutions of the homogeneous problem ∆w = 0 inthe punctured ball B1 \ 0.Definition 2.2 The numbers γ±j are the indicial roots of the Laplacian at theorigin.

We can be more explicite, since

. . . < γ−n = . . . γ−1 = 1− n < γ−0 = 2− n ≤≤ γ+

0 = 0 < γ+1 = . . . = γ+

n = 1 < . . .

In particular, the set of indicial roots of the Laplacian at the origin is the set ofall integers except −1,−2, . . . , 3− n.

More generally, we can also consider elliptic partial differential operatorswith ”regular singularities”. Namely, operators of the form

L = ∆ +c

r2,

where the function c is assumed to be smooth in B1. In this case, the formula(2.9) for the indicial roots of L has to be replaced by

γ±j =2− n

2±

√(n− 2

2

)2

+ c(0)− λj ∈ C. (2.10)

Again, all these coefficients do determine, at 0, all the possible asymptoticbehaviors of the solutions of the homogeneous problem Lw = 0 in B1 \ 0.Notice that the indicial roots only depend on the value of the function c at thepuncture 0.

Granted the definition of the weighted Holder spaces, we easily see that, forall k ≥ 2, α ∈ [0, 1) and ν ∈ R,

∆ : Ck,αν,D(Ω \ Σ) −→ Ck−2,α

ν−2 (Ω \ Σ), (2.11)

is a well defined bounded linear operator. Similarly, if the function c is smoothin Ω, then

L : Ck,αν,D(Ω \ Σ) −→ Ck−2,α

ν−2 (Ω \ Σ)

w 7−→ ∆w +c

d2w,

25

is also a well defined and bounded operator.

The main problem we are interested in can be stated as follows : For whichvalues of the parameter ν the above defined operators are injective ? surjective ?isomorphisms ?In the case where the operator is not surjective, what is the dimension of thecokernel of the operator ? And, in the case where the operator is not injective,what is the dimension of the kernel of the operator ? How does this depend onν ?

In the remaining of this section, we will give an answer to these questionsin the case of the Laplacian. We will take this opportunity to introduce someuseful technics which, later on, will be applied to analyze the Ginzburg-Landauvortices. Since we confine ourselves to study the Laplacian, some of the proofsbelow can be simplified. Nevertheless, we have chosen not to give the mostefficient proofs but the proofs which can be easily adapted to the analysis ofGinzburg-Landau vortices.

2.2.1 Rescaled Schauder estimates

To go further in the investigation of the mapping properties of our operators, wewill extensively use what we call ”rescaled Schauder estimates”. The startingpoint is the well known result [26] :

Lemma 2.3 Assume that α ∈ (0, 1) is fixed. There exists a constant c > 0 suchthat, if w ∈ C2,α(B1) satisfies ∆w = f in B1 and if f ∈ C0,α(B1), then

‖w‖C2,α(B1/2) ≤ c(‖w‖L∞(B1) + ‖f‖C0,α(B1)

).

From now on, we will always assume that α ∈ (0, 1) and shall do it withoutfurther comment. Using the above Lemma, we immediately obtain the followinglocal result :

Corollary 2.1 Assume that f ∈ Ck−2,αν−2 (B1 \ 0) and that w solves ∆w = f

in B1 \ 0. Further assume that

‖r−ν w‖L∞(B1) ≤ c ‖r2−ν f‖L∞(B1), (2.12)

for some constant c > 0 which does not depend on f , nor on w. Then, thereexists c′ > 0 which does not depend on f , nor on w, such that

‖w‖Ck,αν

≤ c′ ‖f‖Ck−2,αν−2

, (2.13)

where the norm of w is taken in B1/2 \ 0, while the norm of f is taken inB1 \ 0.

26

Proof : Let us choose x0 ∈ B1/2 \ 0. We set R = |x0|/2 and define thefunction v in B1 by

v(x) := w(x0 + Rx).

We have ∆v = g in B1, where by definition g(x) := R2 f(x0+Rx). Furthermore,the effect of the scaling yields

‖g‖C0,α(B1) ≤ cRν ‖f‖C0,αν−2

,

as well as‖v‖L∞(B1) ≤ cRν‖r−ν w‖L∞(B1).

Using (2.12), the last estimate becomes

‖v‖L∞(B1) ≤ cRν ‖f‖C0,αν−2

.

We may now apply Lemma 2.3 to obtain

‖v‖2,α,B1/2 ≤ cRν ‖f‖C0,αν−2

.

Performing the scaling backward, we conclude that

2∑

j=0

Rj supBR/2(x0)

|∇jw|+ R2+α supx,y∈BR/2(x0)

|∇2w(x)−∇2w(y)||x− y|α≤ c ‖f‖0,α,ν−2.

This estimate, together with a simple covering argument, yields

‖w‖C2,αν

≤ c′ ‖f‖C0,αν−2

,

where the norm of w is taken in B1/2 \ 0. The proof of the result is thereforecomplete when k = 2. The general case, when k ≥ 2, follows easily by induction.2

In other words, in order to obtain an inequality such as (2.13), it is sufficientto prove (2.12) since all the other estimates are consequences of Lemma 2.2.

The previous local result has a global counterpart, which can be stated asfollows :

Corollary 2.2 Assume that f ∈ Ck−2,αν−2 (Ω \ Σ) and that w solves ∆w = f in

Ω \ Σ, with w = 0 on ∂Ω. Further assume that

‖d−ν w‖L∞(Ω) ≤ c ‖d2−ν f‖L∞(Ω),

for some constant c > 0 which does not depend on f , nor on w. Then, thereexists c′ > 0 which does not depend on f , nor on w, such that

‖w‖Ck,αν

≤ c′ ‖f‖Ck−2,αν−2

.

27

We will also need the following classical result [26] :

Lemma 2.4 There exists a constant c > 0 such that, if w ∈ C2,α(B1 \ B1/2)and f ∈ C0,α(B1 \B1/2) satisfy

∆w = f in B1 \B1/2

w = 0 on ∂B1.

Then,‖∇w‖L∞(B1\B3/4) ≤ c

(‖w‖L∞(∂B1/2) + ‖f‖L∞(B1\B1/2)

),

for some constant c > 0 which does not depend on f , nor on w.

2.2.2 Mapping properties of the Laplacian in the injectiv-ity range

We determine for which values of the weight parameter ν, the operator

∆ : C2,αν,D(Ω \ Σ) −→ C0,α

ν−2(Ω \ Σ),

is injective. Once this is done, we look more closely at the mapping propertiesof ∆ when the weight parameter is in the injectivity range.

To begin with, let us prove the following simple result :

Proposition 2.2 Assume that ν > 2 − n, then the operator ∆ defined fromC2,α

ν,D(Ω \ Σ) into C0,αν−2(Ω \ Σ) is injective.

Proof : We argue by contradiction and assume that, for some ν > 2 − n, theoperator ∆ is not injective. There would exist a function w ∈ C2,α

ν (Ω \ Σ) suchthat ∆w = 0 in Ω \ Σ and w = 0 on ∂Ω.

Let ai ∈ Σ be given, for the sake of simplicity in the notations we assumethat ai = 0. We define w to be the unique (smooth) solution of

∆w = 0 in Bσ

w = w on ∂Bσ.

Let us write the eigenfunction decomposition of w − w at 0 as

w − w =∑

j≥0

wj φj .

Since w ∈ Ck,αν (Ω \ Σ) and since, by the maximum principle, w is bounded by

sup∂Bσ|w| in Bσ, we see that, for all r ∈ (0, σ]

|wj(r)| ≤ cn

(‖w‖Ck,α

νrν + sup

∂Bσ

|w|)

, (2.14)

28

where cn := |Sn−1|1/2.Now, observe that in Bσ, wj is a linear combination of the two functions w+

j

and w−j which have been defined in (2.8). Moreover, wj(σ) = 0 by assumption.Thus, there exists αj ∈ R such that

wj = αj

(( r

σ

)γ+j −

( r

σ

)γ−j)

.

However, sinceγ−j ≤ 2− n < ν,

we see, using (2.14), that necessarily wj ≡ 0. In other words, we have provedthat w = w in Bσ. Therefore, w is a smooth solution of ∆w = 0 in Ω withw = 0 on ∂Ω and, as a consequence, w ≡ 0. 2

We now discuss the dimension of the cokernel of the Laplacian, when theweight parameter is chosen in the injective range, namely ν > 2− n. To do so,we need the following local result :

Proposition 2.3 Assume that ν > 2−n and that ν is not an indicial root, i.e.ν /∈ γ±j : j ∈ N. Let j0 ∈ N be the least index for which ν < γ+

j0. Then,

there exists c > 0 such that, for all f ∈ C0,αν−2(B1 \ 0), there exists a unique

w ∈ C2,αν (B1 \ 0) solution of

∆w = f in B1 \ 0,with the boundary data given by

w ∈ Spanφj : j = 0, . . . , j0 − 1 on ∂B1.

In addition, ‖w‖C2,αν

≤ c ‖f‖C0,αν−2

.

Proof : We first concentrate our attention on the existence of w. To this aimwe consider the eigenfunction decomposition of f

f =∑

j≥0

fj φj ,

and we look for w of the form

w =∑

j≥0

wj φj .

We see that, for all j ≥ 0, we have to solve the ordinary differential equationLj wj = fj in (0, 1], with the boundary condition wj(1) = 0 for all j ≥ j0. Wedistinguish two cases.

Case 1. Assume that j < j0 (naturally, if j0 = 0 this case is void). Here wehave an explicit expression for the solution wj which is given by

wj = rγ+j

∫ r

0

s1−n−2γ+j

∫ s

0

tn−1+γ+j fj dt ds.

29

Notice that this expression is well defined since 0 ≤ γ+j < ν, for all j < j0. In

addition, it is an easy exercise to obtain the estimate

sup(0,1]

|r−νwj | ≤ cj sup(0,1]

|r2−ν fj |,

for some constant cj > 0 only depending on j. Thus, we have

supB1\0

∣∣∣∣∣∣r−ν

j0−1∑

j=0

wj φj

∣∣∣∣∣∣≤ cj0 sup

B1\0|r2−ν f |, (2.15)

for some constant cj0 > 0, only depending on j0.

Case 2. Assume that j ≥ j0. This time, for each j, we also have an explicitformula for wj . Namely,

wj = − rγ+j

∫ 1

r

s1−n−2γ+j

∫ s

0

tn−1+γ+j fj dt ds,

and, as above, we observe that this expression is well defined since 2 − n < νand 0 ≤ γ+

j . Furthermore, using the fact that ν ≤ γ+j , we can estimate

sup(0,1]

|r−νwj | ≤ cj sup(0,1]

|r2−ν fj |,

for some constant cj > 0 only depending on j.

At this point, let us observe that, for j ≥ j0, the existence and estimate forwj could have been obtained using the method of sub- and supersolutions oncean appropriate barrier function is constructed. Notice that

Ljrν = (ν (n− 2 + ν)− λj) rν−2.

By assumption ν > 2− n and ν < γ+j , for all j ≥ j0. Hence,

ν(n− 2 + ν)− λj < 0, for all j ≥ j0.

This, together with the fact that Lj satisfies the maximum principle, impliesthat the function

r ∈ (0, 1] −→ (λj − ν(n− 2 + ν))−1

(sup(0,1]

|s2−ν fj |)

rν ,

can be used as a barrier function to obtain both the existence and estimate forwj .

In order to simplify the notations, we set

f ′′ :=∑

j≥j0

fj φj .

30

So far, we know that for all J > j0 there exists some constant cJ > 0, which isindependent of f but may a priori depend on J , such that

supB1\0

∣∣∣∣∣∣r−ν

J∑

j=j0

wj φj

∣∣∣∣∣∣≤ cJ sup

B1\0|r2−ν f ′′|. (2.16)

We claim that the constant cJ does not depend on J . Assuming that we havealready proven this claim, we can pass to the limit J → +∞ and define

w′′ :=∑

j≥0

wj φj ,

which is clearly a solution of our problem. In addition, (2.15) together with(2.16) imply that

supB1\0

|r−ν w| ≤ c supB1\0

|r2−ν f |.

In order to finish the proof of the Proposition, it is enough use the rescaledSchauder estimates of Corollary 2.1. The uniqueness of w follows from Propo-sition 2.2.

Hence, it remains to prove the claim. We argue by contradiction and assumethat the claim is not true. There would exist a sequence Ji ≥ j0 tending to ∞,a sequence of functions

f ′′i =Ji∑

j=j0

f ij φj ,

and a sequence of functions

w′′i =Ji∑

j=j0

wij φj ,

solutions of ∆w′′i = f ′′i in B1 \ 0, with w′′i = 0 on ∂B1 such that

supB1\0

|r−ν w′′i | = 1, (2.17)

whileMi := sup

B1\0|r2−ν f ′′i |,

tends to 0 as i tends to ∞.

Using (2.17), we get the existence of xi ∈ B1 \ 0 such that

|xi|−ν |w′′i (xi)| ≥ 1/2.

Now, it follows from Lemma 2.4 that the sequence (|xi|)i≥0 stays bounded awayfrom 1. Therefore, up to a subsequence, we may assume that the sequence xi

converges to x∞ in B1. We set Ri := 1/|xi|,w′′i (x) := |xi|−ν w′′i (|xi|x),

31

andf ′′i (x) := |xi|2−ν f ′′i (|xi|x).

Notice that we still have

supBRi

\0|r−ν w′′i | = 1, and sup

BRi\0

|r2−ν f ′′i | = Mi.

Let us now distinguish a few cases according to the value if x∞.

1 - Assume that x∞ 6= 0. We set R∞ := 1/|x∞|. Using Schauder’s estimatestogether with Ascoli-Arzela Theorem, and after the extraction of subsequencesif this is necessary, we may pass to the limit in ∆w′′i = f ′′i and obtain a solutionof

∆w′′ = 0,

in BR∞ \ 0 with w′′ = 0 on ∂BR∞ . We also know that

supBR∞\0

|r−ν w′′| ≤ 1, (2.18)

and that w′′ ∈ Spanφj : j ≥ j0. In addition, w′′ is not identically equal to 0.Indeed, for all i ≥ 0, we have |w′′i (xi/|xi|)| ≥ 1/2 and, passing to the limit, weget

supBR∞\0

|w′′| ≥ 1/2. (2.19)

We consider the eigenfunction decomposition of w′′

w′′ =∑

j≥0

wj φj .

Now, for all j ≥ j0, there exists αj ∈ R such that

wj = αj

((r

R∞

)γ+j

−(

r

R∞

)γ−j)

.

However, sinceγ−j < ν,

we see, using (2.18), that necessarily wj ≡ 0. Hence, w′′ ≡ 0 which is a contra-diction.

2 - Assume that x∞ = 0. In this case, and after the extraction of subse-quences if this is necessary, we may pass to the limit in ∆w′′i = f ′′i and obtaina solution of

∆w′′ = 0,

in Rn \ 0. Furthermore, we know that

supRn\0

|r−ν w′′| ≤ 1,

32

and also that w′′ ∈ Spanφj : j ≥ j0. Again, w′′ is not identically equal to0. We consider the eigenfunction decomposition of w′′. This time, wj has tobe a linear combination of rγ+

j and rγ−j and has to be bounded by a constanttimes rν on all R. However, since γ−j < ν < γ+

j , we see that necessarily wj ≡ 0.Hence, w′′ ≡ 0 which is the desired contradiction.

Since we have ruled out every possible case, the proof of the claim is com-plete. 2

Remark 2.3 In dimension n = 2 much softer arguments can be used to reachthe conclusion in Case 2. Indeed, for all j ≥ j0, we have already shown how toprove the estimate

sup(0,1]

|r−ν wj | ≤ (λj − ν(n− 2 + ν))−1 sup(0,1]

|r2−ν fj |. (2.20)

But, in dimension 2, all eigenfunctions φj are uniformly bounded and∑

j≥1

λ−1j ,

converges. Hence, the right hand side of (2.20) is summable and we directlyobtain

supB1\0

∣∣∣∣∣∣r−ν

∑

j≥j0

wj φj

∣∣∣∣∣∣≤ c sup

B1\0|r2−ν f |.

For all i = 1, . . . , N , we will denote by (ri, θi) the spherical coordinates aboutai, namely

ri = |x− ai| and θi :=x− ai

|x− ai| .

Thanks to the result of the previous Proposition, we can now prove the globalresult :

Proposition 2.4 Assume that ν > 2 − n and that ν /∈ γ±j : j ∈ N. Letj0 ∈ N be the least index for which ν < γ+

j0. Then,

∆ : C2,αν,D(Ω \ Σ) −→ C0,α

ν−2(Ω \ Σ),

is injective and the dimension of its cokernel is given by N j0, where we recallthat N is the cardinal of Σ.

Proof : Since we have assumed that ν > 2− n, we see that

C0,αν−2(Ω \ Σ) ⊂ L1(Ω).

Therefore, we already know from standard results [14] that there exists a weaksolution of ∆w = f in Ω with w = 0 on ∂Ω such that w ∈ W 1,p(Ω), for allp < n

n−1 . Furthermore, standard regularity results imply that w ∈ Ck,αloc (Ω \ Σ).

33

Remark 2.4 Observe that, for more general operators, the above argument canno longer be applied since usually there is no available result in the literaturethat ensures the existence of w, to overcome this difficulty one can for exampleuse the ”domain decomposition method” as will be described in the proof ofProposition 2.6.

We now analyze more carefully the behavior of w near any of the ai ∈ Σ.For the sake of simplicity in the notations we assume that ai = 0. Thanks toProposition 2.3, we know that there exists a unique w solution of

∆w = f in Bσ \ 0

w − w ∈ Spanφ0, . . . , φj0−1 on ∂Bσ,

which belongs to C2,αν (Bσ \ 0). In particular, there exist λ0, . . . , λj0−1 ∈ R

such that

w − w =j0−1∑

j=0

λj σγ+j φj on ∂Bσ.

We set

w = w +j0−1∑

j=0

λj rγ+j φj .

By definition, we have ∆(w− w) = 0 in Bσ \ 0 and w− w = 0 on ∂Bσ. Sincew − w ∈ W 1,p(Bσ), for all p < n/(n− 1), the singularity at 0 is removable andwe conclude that w = w. In order to make the distinction between the differentpoints ai, we set (λi

1, . . . , λij0−1) := (λ1, . . . , λj0−1).

Thanks to the above analysis, we can define a continuous linear mapping

Λ : f ∈ C0,αν−2(Ω \ Σ) −→ (λi

j)i,j ∈ Rj0 N ,

where the indices i, j satisfy 1 ≤ i ≤ N and 0 ≤ j < j0 − 1. In other words, themapping Λ identifies the first j0 coefficients of the expansion of w near each ai.

Clearly, a function f belongs to the image of C2,αν,D(Ω \Σ) by ∆ if and only if

all the coefficients λij are equal to 0, i.e. if and only if Λ(f) = 0. This already

proves that the dimension of the cokernel of ∆ is bounded by N j0.

In order to prove that this dimension is indeed N j0, it remains to show thatΛ is surjective. We give the proof of this fact when n ≥ 3. Minor modificationsare needed to handle the case n = 2. Let us assume that ζ is smooth functionwith compact support in [1/2, 1], which is positive in (1/2, 1). For all j ≥ 0 andfor all ε ≤ σ, we set

vj := −εγ+j −2

∫ 2σ

r

s1−n−2γ+j

∫ s

0

tn−1+γ+j ζ(t/ε) dt ds,

and

αj := −∫ +∞

0

s1−n−2γ+j

∫ s

0

tn−1+γ+j ζ(t) dt ds.

34

Notice thatlimε→0

vj = αj 6= 0. (2.21)

Given λ := (λij)i,j ∈ RNj0 , we now define, for all ε ∈ (0, 1], the functions

Wε(λ) =N∑

i=1

η(ri)

j0−1∑

j=0

λij

αjvj(ri) r

γ+j

j φj(θi)

,

and

Fε(λ) =N∑

i=1

η(ri)

j0−1∑

j=0

λij

αjζ(ri/ε)φj(θi)

,

where η is a cutoff function identically equal to 1 in [0, σ] and equal to 0 in[2σ,+∞) and where (ri, θi) are the spherical coordinates about ai. .

Obviously ∆Wε − Fε, defined in Ω \ Σ, is supported in Aσ and there existsa constant c, independent of λ, such that

‖∆Wε − Fε‖0,α,ν ≤ c εn−2 ‖λ‖.

Therefore‖Λ(∆Wε − Fε)‖ ≤ c εn−2 ‖λ‖.

It readily follows from (2.21) and from the definition of Wε that

limε→0

Λ(∆Wε(λ)) = λ.

We define the linear mapping Jε by

Jε : RNj0 −→ RNj0

λ 7−→ Λ(Fε(λ)).

Collecting these results, we find that limε→0 Jε is the identity. Hence, for εsmall enough, Jε is invertible and Λ is surjective. 2

As a consequence we may now state the :

Corollary 2.3 Assume that ν > 2− n and that ν /∈ γ±j : j ∈ N. Let j0 ∈ Nbe the least index for which ν < γ+

j0. Then

∆ : C2,αν,D(Ω \ Σ) −→ C0,α

ν−2(Ω \ Σ),

is Fredholm of indexIndex = −N j0.

35

As in the previous proof, we let η be a cutoff function identically equal to 1in [0, σ] and equal to 0 in [2σ,+∞). We define

Kj0 := Spanη(ri) r

γj

i φj(θi) : j = 0, . . . , j0 − 1, i = 1, . . . , N

,

where (ri, θi) are the spherical coordinates about ai. Granted these definitions,we can restate the last result as :

Corollary 2.4 Assume that ν > 2− n and that ν /∈ γ±j : j ∈ N. Let j0 ∈ Nbe the least index for which ν < γ+

j0. Then

∆ : C2,αν,D(Ω \ Σ)⊕Kj0 −→ C0,α

ν−2(Ω \ Σ),

is an isomorphism.

2.2.3 Mapping properties of the Laplacian in the surjec-tivity range

Paralleling what we have done in the previous section, we determine for whichvalues of the weight parameter ν, the operator

∆ : C2,αν,D(Ω \ Σ) −→ C0,α

ν−2(Ω \ Σ),

is surjective. Once this is done, we look more closely at the mapping propertiesof ∆ when the weight parameter is in the surjectivity range.

Proposition 2.5 Assume that ν < 0 and that ν /∈ γ±j : j ∈ N. Let j0 ∈ Nbe the least index for which ν > γ−j0 . Then, there exists c > 0 such that, for allf ∈ C0,α

ν−2(B1 \ 0) there exists w ∈ C2,αν (B1 \ 0) solution of

∆w = 0 in B1 \ 0

w = 0 on ∂B1.

Furthermore, ‖w‖2,α,ν ≤ c ‖f‖0,α,ν−2.

Proof : The proof of this result is nearly identical to the proof of Proposi-tion 2.4. Here also, we consider the eigenfunction decomposition of the functionf

f =∑

j≥0

fj φj ,

and look for w of the formw =

∑

j≥0

wj φj .

We see that, for all j ≥ 0, we have to solve Ljwj = fj in (0, 1], with theboundary condition wj(1) = 0.

36

Case 1. If j < j0 (if j0 = 0, then this case is void). Here, we choose

wj = rγ+j

∫ 1

r

s1−n−2γ+j

∫ 1

s

tn−1+γ+j fj(t) dt ds.

It is then a simple exercise to see that, first this expression is well defined sinceν < γ−j and γ+

j + γ−j = 2− n. Moreover,

supB1\0

∣∣∣∣∣∣r−ν

j0−1∑

j=0

wj φj

∣∣∣∣∣∣≤ cj0 sup

B1\0|r2−ν f |,

for some constant cj0 > 0, only depending on j0.

Case 2. If j ≥ j0. This time for each j there also exists some explicit formulawhich is given by

wj(r) = − rγ+j

∫ 1

r

s1−n−2γ+j

∫ 1

s

tn−1+γ+j fj(t) dt ds,

and, using the fact that γ−j = 2 − n − γ+j < ν together with ν < 0 ≤ γ+

j , weobtain that

sup(0,1]

|r−νwj | ≤ cj sup(0,1]

|r2−ν fj |,

for some constant only depending on j.

Again, the existence and estimate for wj , for j ≥ j0, could have been ob-tained using the method of sub- and supersolutions once an appropriate barrierfunction is constructed. To this aim, simply observe that

Ljrν = (ν (n− 2 + ν)− λj) rν−2.

By assumption ν < 0 and γ−j < ν, for all j ≥ j0. Hence,

ν(n− 2 + ν)− λj < 0, for all j ≥ j0.

This, together with the fact that Lj satisfies the maximum principle, impliesthat the function

r ∈ (0, 1] −→ (λj − ν(n− 2 + ν))−1

(sup(0,1]

|s2−νfj |)

rν ,

can again be used as a barrier function to obtain both the existence and estimatefor wj .

We keep the notationf ′′ :=

∑

j≥j0

fj φj .

37

It now remains to prove that there exists a constant c > 0 which does notdepend on f ′′ such that

supB1\0

|r−ν∑

j≥j0

wj φj | ≤ c supB1\0

|r2−νf ′′|.

The proof of this fact is similar to the proof of the corresponding point inProposition 2.4, therefore we omit it. 2

Using the result of the above Proposition, we can now prove the followingglobal result :

Proposition 2.6 Assume that ν < 0 and that ν /∈ γ±j : j ∈ N. Let j0 ∈ Nbe the least index for which ν > γ−j0 . Then,

∆ : C2,αν,D(Ω \ Σ) −→ C0,α

ν−2(Ω \ Σ),

is surjective and the dimension of its kernel is given by N j0.

Proof : The proof relies on a domain decomposition method. Recall that, bydefinition Ωσ := x ∈ Ω : dist(x, Σ) > σ. For the sake of simplicity, we restrictour attention to the case where n ≥ 3 since minor modifications are needed tohandle the 2 dimensional case.

Step 1 : We shall introduce what we call the ”exterior Dirichlet to Neu-mann map”. Let Ψ = (ψ1, . . . , ψN ) be a N -tuple of functions in the spaceΠN

i=1C2,α(∂Bσ(ai)). Given these boundary data, we define wΨ ∈ C2,α(Ωσ) to bethe (unique) solution of

∆w = 0 in Ωσ

w = 0 on ∂Ω

w = ψi on ∂Bσ(ai) for all i = 1, . . . , N.

(2.22)

The ”exterior Dirichlet to Neumann map” is defined by

S(Ψ) := σ(∂r1wΨ|∂Bσ(a1), . . . , ∂rN

wΨ|∂Bσ(aN )

). (2.23)

Where ri = |x− ai|. It is easy to check that

S : ΠNi=1C2,α(∂Bσ(ai)) −→ ΠN

i=1C1,α(∂Bσ(ai)),

is a well defined linear bounded operator. Moreover, it is well known thatthis operator is elliptic, with principal symbol −|ξ|. In order to have a betterunderstanding of the operator S, let us assume that ai = 0 and consider theeigenfunction decomposition of ψ ∈ C2,α(∂Bσ)

ψ =∑

j≥0

αj φj .

38

Let vψ be the unique solution of ∆v = 0 in Rn \ Bσ which decays at ∞ andwhich satisfies v = ψ on ∂Bσ. Naturally, we have the explicit formula

vψ =∑

j≥0

αj

( r

σ

)γ−jφj .

And thus, we obtainσ ∂rvψ|∂Bσ

=∑

j≥0

γ−i αj φj .

The fact that each coefficient of ψ is multiplied by γ−j ∼ −λ1/2j reflects the fact

that the operator ψ −→ σ ∂rvψ|∂Bσ is elliptic of principal symbol −|ξ|.Now, for all Ψ ∈ ΠN

i=1C2,α(∂Bσ(ai)), we may then define an ”auxiliaryDirichlet to Neumann” map by

S(Ψ) := σ(∂r1vψ1 |∂Bσ(a1), . . . , ∂rN

vψN|∂Bσ(aN )

). (2.24)

This time, S is decoupled (i.e. is in diagonal form). Let us now compare S andS. Let η be some cutoff function identically equal to 1 in B3σ/2 and equal to 0outside B2σ. We set

vΨ := wΨ −N∑

i=1

η(· − ai) vψi ,

where wΨ has been defined above. Obviously ∆vΨ is supported in Ω3σ/2 \ Ω2σ

and using classical regularity theory for the Laplacian, we find that ∆vΨ isbounded by a constant times ‖Ψ‖C2,α , in any Ck,α(Ωσ), k ≥ 0. Hence,

S − S : ΠNi=1C2,α(∂Bσ(ai)) −→ ΠN

i=1C1,α(∂Bσ(ai)),

is compact. We conclude that S is just a compact perturbation of S.

Step 2 : We now come to the definition of the ”interior Dirichlet to Neu-mann map”. Again, we let Ψ = (ψ1, . . . , ψN ) be a N -tuple of functions inΠN

i=1C2,α(∂Bσ(ai)). This time, we define wψi to be the unique solution of

∆wψi = 0 in Bσ(ai)

wψi = ψi on ∂Bσ(ai),(2.25)

Then the ”interior Dirichlet to Neumann map” is defined by

T (Ψ) := σ (∂r1wψ1 |∂Bσ(a1), . . . , ∂rNwψN

|∂Bσ(aN )). (2.26)

Again,T : ΠN

i=1C2,α(∂Bσ(ai)) −→ ΠNi=1C1,α(∂Bσ(ai)),

is a well defined linear bounded operator. And this operator is elliptic, withprincipal symbol |ξ|.

39

In this simple case, we even have some explicit representation of T . Indeed,assume for simplicity that ai = 0, if we decompose ψ over eigenfunctions

ψ =∑

j≥0

αj φj

Then, wψ is explicitly given by

wψ =∑

j≥0

αj

( r

σ

)γ+j

φj .

And thusσ ∂rwψ|∂Bσ

=∑

j≥0

αj γ+j φj .

Again, the fact that each coefficient of ψ is multiplied by γ+j ∼ λ

1/2j reflects the

fact that the operator T is elliptic of principal symbol |ξ|.Step 3 : We claim that

S − T : ΠNi=1C2,α(∂Bσ(ai)) −→ ΠN

i=1C1,α(∂Bσ(ai)),

is an isomorphism.To begin with, let us notice that S − T is clearly a isomorphism. Indeed,

this operator acts diagonally on N -tuples and, in terms of the eigenfunctionsφj , the ith component is determined by

∑

j≥0

αj φj −→∑

j≥0

(γ−j − γ+j )αj φj .

Notice that γ−i − γ+i 6= 0 and γ−i − γ+

i ∼ −2 λ1/2j . In particular, this operator

is Fredholm of index 0. Hence, S − T , which is just a compact perturbationof S − T , is also Fredholm of index 0. In order to prove that S − T is anisomorphism, we only need to prove that it is injective.

Assume that Ψ ∈ ΠNi=1C2,α(∂Bσ(ai)) satisfies

(S − T )(Ψ) = 0.

Now, we can solve ∆w = 0 with the set of boundary data given by ψi on each∂Bσ(ai), first in Ωσ (with exterior boundary data equal to 0), and then on eachBσ(ai). We obtain a function w which is globally defined in Ω, which satisfies∆w = 0 in Ωσ and in each Bσ(ai) and satisfies w = 0 on ∂Ω. Furthermore,the condition (S − T )(Ψ) = 0 implies that w is a C1 function. Thus, w is aweak solution of ∆w = 0 in all Ω. Necessarily, w = 0 which in turn implies thatΨ = 0. This ends the proof of the injectivity of S − T .

Step 4 : We can now complete the proof of Proposition 2.6. For all f ∈C0,α

ν−2(Ω \ Σ), we define wext to be the unique solution of

∆wext = f in Ωσ

wext = 0 on ∂Ωσ.(2.27)

40

And, for all i = 1, . . . , N , we also define wint,i to be the solution of

∆wint,i = f in Bσ(ai) \ aiwint,i = 0 on ∂Bσ(ai),

(2.28)

which is given by Proposition 2.5. Notice that wint,i ∈ C2,αν (Bσ(ai)) \ ai). In

addition, there exists some constant c > 0, such that

||wint,i||C2,αν

≤ c ||f ||C0,αν−2

,

||wext||C2,α ≤ c ||f ||C0,α ,

where all the norms are taken over the sets where the functions are defined.

To complete the construction, we now look for a solution of

∆wker = 0 in Ω \ ∪i∂Bσ(ai), (2.29)

which is continuous in Ω \ Σ and such that wker = 0 on ∂Ω. In addition, wewant to be able to choose wker in such a way that the function

w :=

wext + wker in Ωσ

wint,j + wker in Bσ(ai), ∀i = 1, . . . , N,(2.30)

is a C1 function in all Ω\Σ. Furthermore, we want the restriction of wker to anyBσ(ai) to belong to C2,α

ν (Bσ(ai)\ai). Assuming we have already constructedwker, we see that ∆w = f in Ω \ Σ with w = 0 on ∂Ω will be a solution of ourproblem. Hence, this will end the construction of a right inverse for ∆.

In order to build wker, we must check that we can find a solution of (2.29),which is continuous through every ∂Bσ(ai) and for which the discontinuity of∂riwker through ∂Bσ(ai) is equal to (∂riwext−∂riwint,i)|∂Bσ(ai). Since solutionsof (2.29) are parameterized by their values on ∪i∂Bσ(ai), this problem reducesto find Ψ solution of the equation

(S − T )(Ψ) = − ((∂r1wext − ∂r1wint,1)|∂Bσ(a1) , . . .

. . . , (∂rN wext − ∂rN wint,N )|∂Bσ(aN )

).

(2.31)

Since the existence of Ψ follows from Step 3, the properties of wker then followfrom the construction of T . The proof of the surjectivity of ∆ is thereforecomplete.

Step 5: It remains to show that the kernel of ∆ has dimension N j0. Let ηdenotes a cutoff function identically equal to 1 in Bσ and identically equal to 0outside B2σ. Given λ := (λi

j)i,j , we define

w =N∑

i=1

η(· − ai)j0−1∑

j=0

λij r

γ−ji φj(θi).

41

Obviously, w is harmonic in every Bσ(ai)\ai and has nonremovable singulari-ties at each ai. Moreover, the function f := ∆w, defined in Ω\Σ, has support inAσ. We extend this function by 0 at each ai and we define v to be the (unique)solution of

∆v = ∆w in Ω

v = 0 on ∂Ω.

Then, w − v belongs to C2,αν (Ω \ Σ) and is harmonic in Ω \ Σ. We have thus

obtained a N j0-dimensional family of elements of the kernel of ∆. Finally, it iseasy to check that any element of the kernel belongs to this family. This endsthe proof of the Proposition. 2

As a consequence we may now state the

Corollary 2.5 Assume that ν < 0 and that ν /∈ γ±j : j ∈ N. Let j0 ∈ N bethe least index for which ν > γ−j0 . Then,

∆ : Ck,αν,D(Ω \ Σ) −→ Ck−2,α

ν−2 (Ω \ Σ),

is Fredholm of indexIndex = N j0.

Notice that when n ≥ 3, if we choose ν ∈ (2− n, 0), then the above operator isan isomorphism.

Remark 2.5 When the weight parameter ν belongs to γ±j : j ∈ N, the oper-ator

∆ : Ck,αν,D(Ω \ Σ) −→ Ck−2,α

ν−2 (Ω \ Σ),

is not Fredholm since it does not even have closed range.

2.3 Applications to nonlinear problems

The previous framework turns out to be powerful in analyzing a certain numberof nonlinear problems. The first application of this weighted spaces approach ina nonlinear context probably appeared in the work of L.A. Caffarelli, R. Hardtand L. Simon [18], [30] where minimal surfaces and hypersurfaces with isolatedsingularities are studied.

Later on, N. Smale has also used these spaces in [92] and [93], to producenew examples of minimal hypersurfaces with singularities. More recently, C.C.Chan [19] has also used this framework together with tools from geometricmeasure theory to prove the existence of minimal hypersurfaces with prescribedasymptotics at infinity. We can also mention the work of L. Mou [67], G. Liaoand N. Smale [43], R. Hardt and L. Mou [33] for applications of this weightedspaces scheme to the study of harmonic maps with isolated singularities.

We now illustrate how the previous linear theory can by applied to nonlinearproblems. We will only consider three examples which, in our opinion, are rathertypical. All these examples are analyzed using the weighted spaces and in eachof them, the crucial point is the choice of the weight parameter ν.

42

2.3.1 Minimal surfaces with one catenoidal type end

As a first example, we explain how the previous framework, together with theimplicit function Theorem, allows to build an infinite dimensional space of min-imal surfaces with one catenoidal end (and with boundary), which are topolog-ically equivalent to [0, +∞)×S1. The material from this example is essentiallyborrowed from [54] and also from [56].

We recall that, up to a dilation and a rigid motion, a catenoid C can beparameterized by

X0 : (s, θ) ∈ R× S1 −→ (cosh s cos θ, cosh s sin θ, s) ∈ R3.

The normal vector field can be taken to be

N0 : (s, θ) ∈ R× S1 −→ 1cosh s

(cos θ, sin θ,− sinh s).

Now, all surfaces C close enough to C can be parameterized (at least locally)as normal graphs over C, namely

X = X0 + w N0 ∈ C,

for some scalar function w. In this case, the mean curvature at the point ofparameters (s, θ) is given by [54]

H(s, θ) = − 1cosh4 s

Lw +1

cosh sQ

(w

cosh s,∇w

cosh s,∇2w

cosh s

), (2.32)

where, by definition,

L := cosh2 s

(∂2

ss + ∂2θθ +

2cosh2 s

),

is the linearized mean curvature operator about C and where Q is a polynomialwithout any constant or first order terms. In addition, all the coefficients ofQ only depend on s and have derivatives with respect to s which are boundeduniformly in s. To enlighten the relation between this operator and the previousanalysis, let us notice that

e2s(∂2

ss + ∂2θθ

),

is just the Laplacian in R2, once the change of variables r = e−s, for s > 0, isperformed.

Some Jacobi fields (i.e. solutions of the homogeneous equation Lw = 0)can be explicitly determined since they correspond to one-parameter families ofminimal surfaces of which C is an element. To exhibit these, first note that anysmooth one-parameter family t → C(t) of minimal surfaces, with C(0) = C,will have differential at t = 0 which is a Jacobi field on C. For example, the oneparameter family of translations along the axis of C yields the Jacobi field

Ψ0,+(s, θ) = tanh s.

43

While, the one parameter family of dilations of C yields

Ψ0,−(s, θ) = s tanh s− 1.

Translating the axis of C in the orthogonal direction, we obtain two linearlyindependent Jacobi fields

Ψ1,+(s, θ) =1

cosh scos θ and Ψ−1,+(s, θ) =

1cosh s

sin θ,

and finally, rotation of the axis of C in the orthogonal direction, gives anothertwo linearly independent Jacobi fields

Ψ1,−(s, θ) =( s

cosh s+ sinh s

)cos θ,

andΨ−1,−(s, θ) =

( s

cosh s+ sinh s

)sin θ.

The indicial roots associated to L, at both +∞ and −∞, are given by

γ±j = ±j.

As usual, they provide all the possible asymptotic behaviors of solutions of thehomogeneous equation L = 0 at ±∞. Now, for all ν ∈ R and all S ∈ R, wedefine the weighted Holder space

Ck,αν ([S, +∞)× S1) :=

e−νsw : w ∈ Ck,α

loc ([S, +∞)× S1)

.

Notice that we recover the usual definition of Ck,αν (B1 \ 0), once we have

performed the change of variables r = e−s.

The mapping properties of L can be studied as before and one proves that

L : C2,αν,D([S, +∞)× S1) −→ C0,α

ν−2([S, +∞)× S1),

is Fredholm provided ν /∈ Z, is injective if ν > 0, and is surjective if ν < 0 andν /∈ Z.

We only sketch the proof of the injectivity property. Assume that ν > 0 andthat w ∈ C2,α

ν,D([S,+∞)× S1) solves Lw = 0. We decompose w in Fourier series

w =∑

j∈Zwj eijθ,

where the functions wk are complex valued.Now, w0 has to be a linear combination of both Ψ0,±, so cannot decay

exponentially unless w0 ≡ 0. Similarly w−1e−iθ + w1e

iθ has to be a linearcombination of the 4 independent Jacobi fields Ψ±1,±. Since w−1e

−iθ + w1eiθ

has to decay exponentially at +∞, we readily see that this function is in fact a

44

linear combination of Ψ±1,+. Furthermore, since w−1e−iθ + w1e

iθ is equal to 0for s = S, it is also identically 0.

Finally, for |j| ≥ 2, we multiply the equation Lw = 0 by cosh−2 swj e−ijθ

and integrate by parts over [S, +∞)× S1 to obtain∫ ∞

S

|∂swj |2 ds + j2

∫ ∞

S

|wj |2 ds−∫ ∞

S

2cosh2 s

|wj |2 ds = 0.

And, since |j| ≥ 2, we conclude that wj ≡ 0 also. This ends the proof of theinjectivity of L when ν < 0.

Paralleling Proposition 2.3, we prove :

Proposition 2.7 Assume that ν > 0 and that ν is not an indicial root, i.e.ν /∈ N. Let j0 ∈ N be chosen so that j0 − 1 < ν < j0. Then, there existsc > 0 such that, for all f ∈ C0,α

ν−2([S, +∞) × S1) there exists a unique w ∈C2,α

ν ([S, +∞)× S1) solution of

Lw = f in [S, +∞)× S1,

with boundary data given by w(S, ·) ∈ Spanφj : j = 0, . . . , j0 − 1 on S1. Inaddition, ‖w‖C2,α

ν≤ c ‖f‖C0,α

ν−2.

Given S ∈ R, we can use the above Proposition, with ν ∈ (0, 1) and j0 = 1,together with a straightforward application the implicit function Theorem, inorder to solve

Lw = cosh3 s Q

(w

cosh s,∇w

cosh s,∇2w

cosh s

)in [S, +∞)× S1

w − ψ ∈ R on S × S1,(2.33)

for all sufficiently small functions ψ ∈ C2,α(S1). Notice that Spanφ0 = R.Since w solves (2.33), we see that the surface parameterized by

X = X0 + w N0,

has mean curvature 0, see (2.32). In particular, we have produced an infinitedimensional family of minimal surfaces (parameterized by the boundary data),with one end which is asymptotic to a catenoidal end.

In the past years, weighted Holder spaces have been used to understand themoduli space of minimal surfaces in R3 (or constant mean curvature surfaces)[74], [57], [39] and also to build new examples of surfaces [54], [37], [99].

2.3.2 Semilinear elliptic equations with isolated singular-ities

We are interested in solutions of ∆u+up = 0 with isolated singularities in somedomain of Rn, n ≥ 3. To begin with, let us recall that, for all p ∈ ( n

n−2 , n+2n−2 ),

the functionu := C |x|− 2

p−1 ,

45

where Cp−1 := 2p−1

(n− 2 p

p−1

), is a weak solution of

∆u + up = 0 in Rn,

which singular at the origin (we refer to [17] and [25] for further results onthe asymptotic behavior of solutions of the above equation near an isolatedsingularity).

The linearized operator about u reads

L := ∆ + p up−1.

In polar coordinates, L takes the form

L := ∂2rr +

n− 1r

∂r +1r2

∆Sn−1 + pCp−1

r2,

As already discussed, the asymptotic behavior of solutions to Lw = 0 at 0 aredetermined by the indicial roots of this operator. These indicial roots are givenby [53]

γ±j :=2− n

2±

√(n− 2

2

)2

+ λj − 2p

p− 1

(n− 2p

p− 1

).

Observe that, for some values of the parameter p, γ±0 could belong to C. Since,p ∈ ( n

n−2 , n+2n−2 ), we have

. . . ≤ − 2p− 1

− 1 = γ−1 < − 2p− 1

< <γ−0 ≤ 2− n

2≤ <γ+

0 < γ−1 ≤ . . . (2.34)

Moreover, it is shown in [53] that

L : C2,αν,D(B1 \ 0) −→ C0,α

ν−2(B1 \ 0),

is Fredholm provided ν /∈ <γ±j : j ∈ N, is injective for all ν > <γ+0 , and is

surjective for all ν < <γ−0 and ν /∈ γ±j : j ∈ N. In addition, if we assumethat γ−1 < ν < <γ−0 , the dimension of the kernel of L is equal to 1.

In particular, we have the following :

Proposition 2.8 Assume that γ−1 < ν < <γ−0 . Then, there exists c > 0 andan operator

G : C0,αν−2(B1 \ 0) −→ C2,α

ν (B1 \ 0),such that, for all f ∈ C0,α

ν−2(B1 \ 0), the function w := G(f) is a solution of

Lεw = f, in B1 \ 0w = 0 on ∂B1.

In addition, ‖w‖C2,αν

≤ c ‖f‖C0,αν−2

.

46

If we choose ν ∈ (− 2p−1 ,<γ−0 ) (notice that this is always possible, thanks to

(2.34)), we can apply the result of Proposition 2.8, together with the implicitfunction Theorem, to show that, for all x0 close enough to 0, there exists a oneparameter family of positive weak solutions of

∆u + up = 0 in B1 \ x0

u = C on ∂B1,

which has a nonremovable singularity at x0. Notice that the restriction ν >− 2

p−1 is needed to ensure that, near x0, the solution of the above equation

looks like u = C |x− x0|−2

p−1 +O(|x− x0|ν), and thus will has a nonremovablesingularity at x0.

For subcritical exponents, further existence results of singular solutions havebeen obtained by R. Mazzeo and F. Pacard [53], [73]. While, for the criticalexponent, the existence results can be found in the work of R. Schoen [87], R.Mazzeo and F. Pacard [54], Y. Rebaı [77] and also in the work of R. Mazzeo,D. Pollack and K. Uhlenbeck [58] for special configuration of the singular set.

Again, weighted spaces have been used to study the moduli space of singularsolutions of ∆u + up = 0. This issue is discussed in [58] and also in [57].Finally, still using these weighted spaces, precise asymptotics, which improvethe previous results in [17], and some balancing formulæ have been establishedin [38].

2.3.3 Singular perturbations for Liouville equation

Here, we are interested in solutions of ∆u + %2 eu = 0 in some domain of R2.This equation has been extensively studied and we refer to [49], [68], [97] forprecise results and further references.

For all % ∈ (0, 2], we define ε > 0 to be the least solution of

%2 =8 ε2

(1 + ε2)2.

Then, we setuε := 2 log(1 + ε2)− 2 log(ε2 + |x|2).

and we check that this is a solution of ∆u + %2 eu = 0 in R2. The linearizedoperator about uε reads

Lεw := ∆w +8 ε2

(ε2 + r2)2w.

Again, some Jacobi fields (i.e. solutions of the homogeneous equation) can beexplicitly computed, since they correspond to one parameter families of solu-tions. For example, the Jacobi field

Ψ0ε(x) =

ε2 − r2

ε2 + r2,

47

corresponds to the one parameter family of solutions η −→ u(η x) − 2 log η.Similarly

Ψ1ε(x) =

2 ε2

ε2 + r2eiθ and Ψ−1

ε (x) =2 ε2

ε2 + r2e−iθ,

correspond to the invariance of the equation with respect to translations.

It is proved in [5] that, for all ε < 1, the operator

Lε : C2,α(B1) −→ C0,α(B1),

is an isomorphism. Unfortunately, the norm of its inverse tends to +∞ as εtends to 0. To see this last fact, it is enough to notice that Ψ±1

ε , which is aJacobi field, satisfies |Ψ±1

ε | ≤ 2 in B1 and |Ψ±1ε | ≤ 2 ε2 on ∂B1. Hence, the

inverse of Lε cannot be bounded uniformly as ε tends to 0.In order to understand more closely the way the inverse of Lε blows up as

ε tends to 0, it is again useful to work in weighted Holder spaces, this time,paralleling Proposition 2.3, we get

Proposition 2.9 [5] Assume that 1 < ν < 2. Then, there exist ε0 > 0, c > 0and for all ε ∈ (0, ε0), for all f ∈ C0,α

ν−2(B1 \ 0), there exists a unique w ∈C2,α

ν (B1 \ 0) solution of Lεw = f, in B1 \ 0,

w ∈ Spane−iθ, 1, eiθ on ∂B1.

In addition, ‖w‖C2,αν

≤ c ‖f‖C0,αν−2

.

The key point is that, in the above result, the constant c > 0 is independentof ε. In other words, we have obtained, in a suitable weighted space, a rightinverse for Lε, whose norm is bounded in dependently of ε.

By opposition to the previous example, this time, the solutions we are in-terested in are not singular anymore. Nevertheless, using the weighted Holderspaces, we are able to obtain a result which bears some features of the wellknown Liapunov-Schmidt decomposition. This proposition has been applied in[5] to produce multiple solutions of ∆u + %2eu = 0, when the parameter % issmall. Somehow, this can be understood as the counterpart of the theory ofcritical points at infinity of A. Bahri, Y. Li and O. Rey [4], [78] and [79] in anon variational framework.

48

Chapter 3

The Ginzburg-LandauEquation in C

In this Chapter, we begin the study of Ginzburg-Landau vortices. To beginwith, we define u1, the radially symmetric solution of the Ginzburg-Landauequation in all C, and also L1, the linearized Ginzburg-Landau operator aboutu1. Next, we cary out a careful study of all the possible asymptotic behaviors ofa solution of the homogeneous equation L1 w = 0 both near the origin and near∞. This yields a classification of all bounded solutions of L1w = 0 in C. Thisstudy provides a key for understanding the definition of the weighted spaces wewill work with in the subsequent Chapters.

3.1 Radially symmetric solution on CWe recall some well known facts about what we will call the ”radially symmetricsolution” of

∆u + u (1− |u|2) = 0. (3.1)

By a radially symmetric solution, we mean a solution of (3.1) which can bewritten as

u1(r, θ) := eiθ S(r). (3.2)

If such a solution exists, it is easy to see that the scalar function S has to be asolution of the following second order ordinary differential equation

d2S

dr2+

1r

dS

dr− 1

r2S + S (1− S2) = 0. (3.3)

Since we want S to be a non constant bounded solution of (3.3), it is necessaryto impose the following behavior at infinity

limr→+∞

S = 1.

49

As already mentioned, the proof of the existence of such a solution S, as wellas some qualitative properties of the function S itself, are available in [34]. Theresults we will need are collected in the :

Theorem 3.1 [34] There exists a unique, non constant solution of (3.3) whichis defined for all r ≥ 0 and whose limit at +∞ is 1. This solution S is strictlyincreasing, we even have

dS

dr> 0, for all r ≥ 0,

and 0 < S < 1, for all r > 0. Furthermore

S = 1− 12r2

+O(

1r4

),

for large r, and there exists some constant κ > 0 such that

S = κ r − κ

8r3 +O (r5),

for r close to 0.

In addition to all these informations, we will need the following result.

Lemma 3.1 Let us define

T :=dS

dr− S

r.

Then T < 0 in (0, +∞).

Proof : Granted the asymptotic behavior of S near 0 and ∞, one can checkdirectly that T < 0 for r > 0 close enough to 0 or for r large enough. Fur-thermore, since S is a solution of (3.3) and since 0 < S < 1, we have for allr > 0

dT

dr+ 2

T

r= −S (1− S2) < 0.

We conclude that the function T cannot achieve a positive maximum value in(0, +∞). Hence T < 0. 2

3.2 The linearized operator about the radiallysymmetric solution

3.2.1 Definition

For any complex valued function u, we define the nonlinear mapping

N (u) := ∆u + u (1− |u|2). (3.4)

50

The linearized operator about a function u is then defined, for any complexvalued function w, by

Lw := ∆w + (1− |u|2) w − 2 u (u · w), (3.5)

where, by definition2 u · w := u w + w u.

is the usual scalar product in C.

When u = u1, the radially symmetric solution defined above, we will denotethe corresponding linearized operator by L1. In order to study L1, it happensto be more convenient to write any complex valued function w as

w := (wr + i wi) eiθ,

where wr and wi are real valued functions. Granted this decomposition, it isnatural to define the conjugate linearized operator by

L1 := e−iθ L1 eiθ. (3.6)

It follows from a simple computation that

L1 (wr + i wi) =(

∆wr + (1− 3 S2)wr − 1r2

wr − 2r2

∂θwi

)

+ i

(∆wi + (1− S2)wi − 1

r2wi +

2r2

∂θwr

).

(3.7)

As we will see, the study of the conjugate operator L1 will be much easier thanthe study of the original linearized operator L1.

3.2.2 Explicit solutions of the homogeneous problem

By definition, the Jacobi fields are solutions of the homogeneous problem L1w =0. For the remaining of the analysis, it is very important to notice that we knowexplicitly some Jacobi fields. These Jacobi fields correspond to the invarianceof (3.1) under the action of some group. The first example corresponds to theinvariance of (3.1) under the one parameter group of transformation u −→ eiαu,for α ∈ R. This gives us the following Jacobi field

Φ01 := i e−iθ u1 = i S. (3.8)

Using the same procedure, we also find two more Jacobi fields

Φ+11 := e−iθ ∂xu1

=12

((dS

dr− S

r

)eiθ +

(dS

dr+

S

r

)e−iθ

)

=dS

drcos θ − i

S

rsin θ,

(3.9)

51

and

Φ−11 := e−iθ ∂yu1

=i

2

(−

(dS

dr− S

r

)eiθ +

(dS

dr+

S

r

)e−iθ

)

=dS

drsin θ + i

S

rcos θ,

(3.10)

which correspond to the invariance of (3.1) under the action of the group oftranslations u −→ u(· − b), for b ∈ C.

3.3 Asymptotic behavior of solutions of the ho-mogeneous problem

This entire subsection is devoted to the study of the asymptotic behavior of thesolutions of L1w = 0 both at the origin and at ∞. This study is a key point ofour analysis since it motivates the definition of the function spaces we will workwith.

3.3.1 Classification of all possible asymptotic behaviors at0

We give in the next Proposition, a complete classification of all the possibleasymptotic behaviors of the solutions of L1w = 0 which are bounded near theorigin :

Proposition 3.1 Let n ∈ N \ 0. Assume that

w = a einθ + b e−inθ,

is a solution ofL1w = 0,

where a and b are complex valued functions only depending on r > 0. Then :

- either |w| is unbounded at 0 (and blows up, as r tends to 0, at least liker1−n, if n ≥ 2 or at least like − log r, if n = 1),

- or w is a linear combination, with real valued coefficients, of the 4 inde-pendent solutions of L1w

jn = 0, for j = 1, . . . , 4, whose behavior near the origin

is given by

w1n = rn+1 (1 +O(r2)) einθ +O(rn+5) e−inθ,

w2n = i

(rn+1 (1 +O(r2)) einθ +O(rn+5) e−inθ

),

w3n = rn−1 (1 +O(r2)) e−inθ +O(rn+3) einθ,

w4n = i

(rn−1 (1 +O(r2)) e−inθ +O(rn+3) einθ

).

52

Notice that w2n 6= i w1

n and w4n 6= i w3

n.

In the case where n = 0, if we assume that w, is a solution of L1w = 0 whichonly depends on r > 0. Then :

- either |w| is unbounded near 0 (and blows up at least like r−1),

- or w is a linear combination, with real valued coefficients, of the 2 inde-pendent solutions of L1w

j0 = 0, for j = 1, 2, whose behavior near the origin is

given byw1

0 = r (1 +O(r2)) and w20 = i r (1 +O(r2)).

Notice that w20 6= i w1

0.

Proof : Assume that w = aeinθ + be−inθ is a solution of L1w = 0, with n ≥ 1.Then, a and b are solutions of the following system of ordinary differentialequations

d2a

dr2+

1r

da

dr− (n + 1)2

r2a = S2b− (1− 2S2)a

d2b

dr2+

1r

db

dr− (n− 1)2

r2b = S2a− (1− 2S2)b.

(3.11)

These ordinary differential equations are second order and the functions a andb are complex valued, hence the space of solutions of (3.11) is an 8-dimensionalreal vector space.

In order to prove the result it is sufficient to prove that there exists 8 in-dependent solutions of this system which have the behavior described in thestatement of the result, and 4 of which are bounded near 0. The proof of theexistence of these solutions follows closely the proof of the classical Cauchy-Lipschitz Theorem. For example, to get the existence of the 4 independentsolutions which are bounded near the origin, we rewrite the above system as

a = α rn+1 + rn+1

∫ r

0

s−2n−3

∫ s

0

tn+2(S2b− (1− 2S2)a) dt ds

b = β rn−1 + rn−1

∫ r

0

s−2n+1

∫ s

0

tn(S2a− (1− 2S2)b) dt ds,

(3.12)

where (α, β) ∈ C2. Obviously, any solution of (3.12) is a solution of (3.11).Now, (α, β) being fixed, we set

a := rn+1 (α + v), and b = rn−1 (β + w).

Then, the existence of (a, b) reduces to the existence of a fixed point (v, w)for some affine mapping. Provided r0 is chosen small enough, the existence of aunique fixed point (v, w) in the space C0(0, r0)×C0(0, r0) follows from a standardapplication of the fixed point Theorem for contraction mappings, so we skip thedetails. Finally, the choice of the coefficients (α, β) ∈ (1, 0), (i, 0), (0, 1), (0, i)yields the 4 solutions wi

n which are given in the statement of the result.

53

Similarly, for n ≥ 2, if we want to get the existence of the 4 independentsolutions which blow up at the origin, it is sufficient to solve

a = α r−1−n − r−1−n

∫ r

0

s2n+1

∫ r0

s

t−n(S2b− (1− 2S2)a) dt ds

b = β r1−n − r1−n

∫ r

0

s2n−3

∫ r0

s

t2−n(S2a− (1− 2S2)b) dt ds,

(3.13)

for (α, β) ∈ C2. Again, for r0 > 0 small enough, the existence of a solution of(3.13) which is defined in (0, r0], follows from a simple fixed point argument.Notice that, all the last 4 solutions do blow up at least like r1−n, as r tends to0. So the result is proven for all n ≥ 2. When n = 1, a similar analysis leads tothe existence of 4 independent solutions which blow up at least like − log r, asr tends to 0.

Finally, the case n = 0 can be treated similarly. We omit the details. 2

3.3.2 Classification of all possible asymptotic behaviors at∞

Our next task will be to prove a result similar to the above one but this timefor the behavior of solutions of L1w = 0 at ∞. Before stating our result, weprove a result concerning the asymptotic behavior of some Bessel functions at∞.

Lemma 3.2 For all n ≥ 0, there exist r0 > 0, C > c > 0 and a function J+n ,

solution of

d2J

dr2+

1r

dJ

dr− 1

r2(n2 − 2)J − 2J = 0, in (r0,+∞), (3.14)

such thatc r−1/2 e

√2r ≤ J+

n ≤ C r−1/2 e√

2r.

for all r > r0.

For all n ≥ 0 and all η > 0, there exist r0 > 0, Cη > cη > 0 and a functionJ−n solution of (3.14) in (r0, +∞), such that

cη r−1/2−η e−√

2r ≤ J−n ≤ Cη r−1/2+η e−√

2r,

for all r ≥ r0.

Proof : Many information about Bessel functions are available in the literature[98] and this result is a classical one. Nevertheless, we give here a short prooffor the sake of completeness.

Notice that the maximum principle holds for the operator

B :=d2

dr2+

1r

d

dr−

(2 +

1r2

(n2 − 2))

,

54

provided r ≥ 1. Hence, the existence and the estimates for J±n are easy toobtain using the method of sub- and supersolutions once an appropriate barrierfunction is constructed.

For all ξ ∈ R, we compute

B (rξ e−√

2r) = −(√

2 (1 + 2 ξ) + (n2 − 2− ξ2)1r

)rξ−1 e−

√2r.

Let η > 0 be fixed. If we choose ξ := −1/2+η, the previous computation showsthat the function

I+η := r−1/2+η e−

√2r,

is a supersolution in [r0, +∞), provided r0 > 1 is chosen large enough. While,if we take ξ := −1/2− η, we conclude that the function

I−η := r−1/2−η e−√

2r,

is a subsolution in [r0, +∞), provided r0 > 1 is chosen large enough. Thesebarrier functions yield the existence and estimates for J−n .

For all ξ ∈ R, we now compute

B (r−1/2 (1 + ξ/r) e√

2r) = −(

2 ξ√

2 + n2 − 2− 14

)r−5/2 e

√2r

+(

4− n2 +14

)ξ r−7/2 e

√2r.

If we take ξ such that 2√

2 ξ > 2− n2 + 14 , we conclude that the function

I+ξ := r−1/2 (1 + ξ/r) e

√2r,

is a supersolution in [r1, +∞), provided r1 > 1 is chosen large enough. While,if we take ξ such that 2

√2 ξ < 2− n2 + 1

4 , we can conclude that the function

I−ξ := r−1/2 (1 + ξ/r) e√

2r,

is a subsolution in [r1,+∞), provided r1 > 1 is chosen large enough. Theexistence and estimates for J+

n are now standard and left to the reader. 2

Granted the previous Lemma, we prove the :

Proposition 3.2 Let n ∈ N \ 0. Assume that

w = a einθ + b e−inθ,

is a solution ofL1w = 0,

where a and b are complex valued functions only depending on r. Then :

55

- either w is unbounded at +∞ (and blows up like rn or like J+n ),

- or w decays at +∞ (at least like r−n or like J−n ).

In the case where n = 0, if w is a solution of L1w = 0 only depending onr. If in addition w is a real valued function then, either w blows up at +∞ likeJ+

0 or decays at +∞ like J−0 . If w is an imaginary valued function then, eitherw blows up at +∞ like log r or is bounded at +∞.

Proof : We give the proof of the result when n 6= 0 and omit the proof of theresult when n = 0 since only slight changes are needed to deal with this latercase. We have already seen in the proof of the previous Proposition that thespace of solutions of L1(a einθ +b e−inθ) = 0 is a 8-dimensional real vector space.

In order to prove the result it is sufficient to prove that there exists 8 in-dependent solutions of this system which have the behavior described in thestatement of the result, and 4 of which decay at +∞. To this aim, we define

A := a + b and B := a− b.

Using (3.11), it is a simple exercise to see that A and B satisfy

d2A

dr2+

1r

dA

dr− 1

r2

((n2 + 1)A + 2nB

)+ (1− 2S2)A = S2A

d2B

dr2+

1r

dB

dr− 1

r2

((n2 + 1)B + 2nA

)+ (1− 2S2)B = −S2B.

(3.15)

If we take the real part of these two equations, we find that

x := <A, and y := <B,

are solutions of the system

d2x

dr2+

1r

dx

dr− 1

r2(n2 − 2)x− 2x =

2n

r2y − 3(1− S2 − 1

r2)x

d2y

dr2+

1r

dy

dr− n2

r2y =

2n

r2x− (1− S2 − 1

r2)y.

(3.16)

If we had taken the imaginary part of (3.15), we would have obtained thatx := =A and y := =B are solution of the same system with x replaced by y andy replaced by x.

In order to get the existence of 2 independent solutions of (3.16) which arebounded at +∞, we rewrite the above system as

x = αJ−n − J−n

∫ r

r0

(J−n )−2t−1

∫ +∞

t

s J−n

(2n

s2y − 3(1− S2 − 1

s2)x

)dsdt,

and

y = βr−n + r−n

∫ +∞

r

t2n−1

∫ +∞

t

s1−n

(2n

s2x− (1− S2 − 1

s2)y

)dsdt,

56

where, (α, β) ∈ C2. Using the fact that, for r large, we have from Theorem 3.1

1− S2 − 1r2

= O(

1r4

),

and using the estimates of Lemma 3.2 for J−n (with any 0 < η < 1/4), theexistence of a solution, which is defined in [r0, +∞) and decays at ∞, follows asbefore from a fixed point argument. In particular, for (α, β) = (1, 0) we obtainthe existence of a solution, whose behavior at ∞ is given by

x = J−n (1 +O(r−1)) and y = O(r−1)J−n ,

and, for (α, β) = (0, 1), we get another solution, whose behavior at ∞ is givenby

x = O(r−n−1) and y = r−n (1 +O(r−1)).

In order to get the existence of 2 independent solutions which blow up at+∞, it is enough to find solutions of the following systems

x = J+n − J+

n

∫ +∞

r

(J+n )−2t−1

∫ t

1

s J+n

(2n

s2y − 3(1− S2 − 1

s2)x

)ds dt

y = rn

∫ r

r0

t−2n−1

∫ t

r0

sn+1

(2n

s2x− (1− S2 − 1

s2)y

)ds dt,

and

x = −J+n

∫ +∞

r

(J+n )−2t−1

∫ t

1

s J+n

(2n

s2y − 3(1− S2 − 1

s2)x

)ds dt

y = rn − rn

∫ +∞

r

t−2n−1

∫ t

r0

sn+1

(2n

s2x− (1− S2 − 1

s2)y

)ds dt,

Again, provided the constant r0 is chosen large enough, we obtain a solutionwhose behavior, at ∞, is given by

x = J+n (1 +O(r−4)) and y = O(r−2)J+

n ,

and another solution whose behavior, at ∞, is given by

x = O(rn−2) and y = rn (1 +O(r−2)).

When n = 0, the equation for a := w reads

d2a

dr2+

1r

da

dr− 1

r2a = S2 a− (1− 2S2) a,

and the existence of the desired solutions follow from similar arguments. 2

As a simple consequence of the former proof we have the :

57

Corollary 3.1 Assume that n 6= 0 and that w = a einθ + b e−inθ is a solutionof L1w = 0 which blows up exponentially at +∞.

If both a and b are real valued functions only depending on r, then thereexists a constant c ∈ R such that

w = c J+n (cos(nθ) +O(r−2)),

for r large enough.

If both a and b are imaginary valued functions only depending on r, thenthere exists a constant c ∈ R such that

w = c J+n (sin(nθ) +O(r−2)),

for r large enough.

Proof : We keep the notations of the previous proof. Since

a =A + B

2, and b =

A−B

2,

we getw = A cos(nθ) + i B sin(nθ).

Now, if a and b are real valued, we have seen that

A = <A = c J+n (1 +O(r−4)) and B = <B = O(r−2) J+

n ,

for some constant c ∈ R. While, if a and b are complex valued we have

B = i=B = i c J+n (1 +O(r−4)) and A = i=A = iO(r−2)J+

n ,

for some constant c ∈ R. The result is now straightforward. 2

As another byproduct of the proof of the previous Proposition, we also havethe :

Corollary 3.2 Assume that w is a real valued solution of L1w = 0 which blowsup exponentially at +∞. Then, there exists a constant c > 0 such that

w = c J+0 (1 +O(r−4)),

for r large enough.

3.4 Bounded solution of the homogeneous prob-lem

The main result of this section is a key result of our construction. Part of theproof of the following Theorem relies on the stability result of P. Mironescu in[64] and was communicated to us by L. Lassoued [40] :

58

Theorem 3.2 [64], [40]. All solutions of L1 w = 0 which are defined on all Cand bounded are linear combinations of Φ0

1, Φ+11 and Φ−1

1 .

Proof : Let us assume that we have a function w solution of L1w = 0, whichis bounded and defined on all C. We expend it into Fourier series

w = a0 +∑

n≥1

(an einθ + bn e−inθ). (3.17)

Our aim is to prove that w is a linear combination of Φ01, Φ+1

1 and Φ−11 . This will

be achieved in 3 steps. In the first step, we deal with the first eigenfrequencyand prove that a0 is proportional to Φ0

1, next, in the second step, we study theeigenfrequency corresponding to e±iθ and prove that a1e

iθ + b1e−iθ is a linear

combination of Φ±11 . In the third step, we prove that all other eigencomponents

of w are equal to 0.

Step 1. Since w is bounded and solves L1w = 0, it is easy to see that a0 isa bounded solution of the ordinary differential equation which is given by

d2a0

dr2+

1r

da0

dr− 1

r2a0 + (1− 2S2) a0 − S2 a0 = 0. (3.18)

The space of solutions of (3.18) which are bounded at 0 is a 2 dimensionalreal vector space spanned by the functions w1

0 and w20 (which are defined in

Proposition 3.1). We also know that Φ01 := i S is an imaginary valued solution

of L1w = 0 and, thanks to Theorem 3.1, we know that the following expansionholds

i S = i (κ r +O(r3)),

near the origin. Therefore, we have the identity i S = κw20. We set

S∗ := κw10,

where w10 is the real valued function which has been defined in Proposition 3.1.

We know that the following expansion holds

S∗ := κw20 = κ (r +O(r3)). (3.19)

Thus, by definition, S∗ > 0 for r close to but not equal to 0. We now provethe :

Lemma 3.3 For all r > 0, S∗(r) > 0. Furthermore, S∗ blows up at +∞ likeJ+

0 .

Proof : In order to prove that S∗ > 0 for all r > 0, we argue by contradiction.If the result were not true, there would exist r0 > 0 such that S∗ > 0 for allr ∈ (0, r0),

S∗(r0) = 0, anddS∗

dr(r0) ≤ 0.

59

Both i S and S∗ are solutions of (3.18). Hence we obtain

dS∗

drS − dS

drS∗ =

2r

∫ r

0

S3 S∗ t dt. (3.20)

Evaluating this expression at r0 and using the fact that S > 0, we obtain

0 ≥ dS∗

dr(r0)S(r0) =

2r

∫ r0

0

S3 S∗ t dt > 0.

Which is the desired contradiction. The proof of the positivity of S∗ is thereforecomplete.

Now that we have proven that S∗ > 0, we may use once more (3.20) togetherwith the fact that S is increasing, to obtain the inequality

dS∗

drS ≥ 2

r

∫ r

0

S3 S∗ t dt.

Using the fact that limr→+∞ S = 1, we conclude that S∗ blows up faster thanany power of r at +∞. Hence, using the result of Proposition 3.2, we concludethat S∗ blows up like J+

0 at +∞. 2

The space of solutions of (3.18) which are bounded near the origin being2 dimensional, we have therefore proved that, up to a constant Φ0

1 is the onlysolution of (3.18) which is bounded on all C.

Step 2. Now we treat the case n = 1, we have already seen in (3.11) that a1

and b1 solve the following system of ordinary differential equations

d2a1

dr2+

1r

da1

dr− 4

r2a1 + (1− 2S2) a1 − S2 b1 = 0

d2b1

dr2+

1r

db1

dr+ (1− 2S2) b1 − S2 a1 = 0.

(3.21)

As in the previous case, we take advantage from the fact that we know explicitlysome bounded solutions of L1w = 0. The first one is given by Φ+1

1 , which wedecompose as

Φ+11 := α1 eiθ + β1 e−iθ.

We know from (3.5) that

α1 :=12

(dS

dr− S

r

)= −κ

8r2 (1 +O(r2))

β1 :=12

(dS

dr+

S

r

)= κ (1 +O(r2)),

for r close to 0. Hence, it follows from Proposition 3.1 that

Φ+11 = κw3

1 −κ

8w1

1.

60

Using again the result of Proposition 3.1, we now define

Ψ+11 := κw3

1 +κ

8w1

1,

which also satisfies L1Ψ+11 = 0. We decompose

Ψ+11 := α∗1 eiθ + β∗1 e−iθ,

and we see that (α∗1, β∗1) is another real valued (independent) solution of (3.21).

This time we have

α∗1 =κ

8r2(1 +O(r2)) and β∗1 = κ (1 +O(r2)), (3.22)

for r close to 0.

Similarly, we know that Φ−11 is a bounded solution of L1w = 0, which we

decompose asΦ−1

1 := i (α−1 eiθ + β−1 e−iθ).

As above, (i α−1, i β−1) is an imaginary valued solution of (3.21) which is boundedfor all r ≥ 0. In addition

α−1 := −12

(dS

dr− S

r

)=

κ

8r2 (1 +O(r2))

β−1 :=12

(dS

dr+

S

r

)= κ (1 +O(r2)),

for all r close to 0. Using once more the result of Proposition 3.1, we get theidentity

Φ−11 = κw4

1 +κ

8w2

1.

We define, as aboveΨ−1

1 : κw41 −

κ

8w2

1,

which, thanks to Proposition 3.1, solves L1 Ψ−11 = 0. We decompose

Ψ−11 := i (α∗−1 eiθ + β∗−1 e−iθ),

and we see that (i α∗−1, i β∗−1) is another imaginary valued (independent) solutionof (3.21). Moreover

α∗−1 = −κ

8r2 (1 +O(r2)), and β∗−1 = κ (1 +O(r2)), (3.23)

for r close to 0.

We now prove the :

Lemma 3.4 For all r > 0, α∗1 > 0 and β∗1 > 0. In addition both functions blowup at +∞ like J+

1 .

61

Proof : Using (3.21), we get, after having performed an integration

dα∗1dr

α−1 − dα−1

drα∗1 =

1r

∫ r

0

S2 (β−1 α∗1 + α−1 β∗−1) t dt

dβ∗1dr

β−1 − dβ−1

drβ∗−1 =

1r

∫ r

0

S2 (β−1 α∗1 + α−1 β∗−1) t dt.

(3.24)

We first prove that

α∗1 > 0, and β∗1 > 0 for r > 0.

Observe that, granted the construction of α∗1 and β∗1 , these inequalities arealready known to be true for r small. To prove the result, we argue by contra-diction and assume that the result is not true. Then, there would exist r0 > 0such that

α∗1 > 0, and β∗1 > 0 in (0, r0),

and, for example

α∗1(r0) = 0, anddβ∗1dr

(r0) ≤ 0.

Using the first equality in (3.24), the fact that β−1 > 0 for all r > 0 and theresult of Lemma 3.1, which states that α−1 > 0 for all r > 0, we readily find

0 ≥ dα∗1dr

(r0)α−1(r0) =1r0

∫ r0

0

S2 (β−1 α∗1 + α−1 β∗−1) t dt > 0.

Which is the desired contradiction. The other case, where

β∗1(r0) = 0, and β∗1(r0) ≤ 0,

can be treated similarly using this time the second equation of (3.24), we omitthe details. The proof of the positivity of both α∗1 and β∗1 is therefore complete.We now prove that these two functions blow up at +∞. To this aim we useonce more (3.24), in order to get

r (α−1)2d

dr

(α∗1α−1

)=

∫ r

0

S2 (β−1 α∗1 + α−1 β∗−1) t dt. (3.25)

Notice that the function on the right hand side is positive, increasing. Takingadvantage from the fact that

α−1 =12r

+O(

1r3

),

at ∞, we see that there exists c > 0 such that

d

dr

(α∗1α−1

)≥ c r,

62

for all r > 1. It is now easy to conclude that α∗1 blows up at ∞, and, byProposition 3.2, we even know that it blows up like J+

1 .Using similar arguments, together with the fact that

β−1 =12r

+O(

1r3

),

at ∞, we show that β∗1 also blows up exponentially at ∞. 2

The same analysis can be carried out for the imaginary part of the system(3.21). Things being almost identical to what we have already done, we willonly sketch the proof of the :

Lemma 3.5 For all r > 0, α∗−1 < 0 and β∗−1 > 0. In addition both functionsblow up at +∞ like J+

1 .

Proof : As above, using (3.21), we get, after having performed an integration,

dα∗−1

drα1 − dα1

drα∗−1 = − 1

r

∫ r

0

S2 (β∗−1 α1 + α∗−1 β1) t dt

dβ∗−1

drβ1 − dβ1

drβ∗−1 = − 1

r

∫ r

0

S2 (β∗−1 α1 + α∗−1 β1) t dt.

(3.26)

This time, we want to prove that

α∗−1 < 0, and β∗−1 > 0 for all r > 0.

Again, we already know that these inequalities are true for small r. We argue bycontradiction and assume that the result is not true. In this case, there wouldexist r0 > 0 such that

α∗−1 < 0, and β∗−1 > 0 in (0, r0),

and, for example,

α∗−1(r0) = 0, anddβ∗−1

dr(r0) ≥ 0.

Using the first equality in (3.26), the fact that β1 > 0 for all r > 0, as well asthe result of Lemma 3.1, which also states that α1 < 0 for all r > 0, we readilyfind

0 ≥ dα∗−1

drα1 = − 1

r

∫ r

0

S2 (β∗−1 α1 + α∗−1 β1) t dt > 0,

which is the desired contradiction. The other case, where

β∗−1(r0) = 0, anddβ∗−1

dr(r0) ≤ 0,

can be treated similarly using this time the second equation of (3.26), we omitthe details and consider the proof to be complete.

63

The proof of the fact that these functions blow up at ∞ is identical to whatwe have already done, therefore we omit it. 2

We have therefore proved that linear combinations of Φ+11 and Φ−1

1 are theonly bounded solutions of L1(a1 eiθ + b1 e−iθ) = 0.

Step 3. [64], [40]. It remains to study all eigenfrequencies corresponding ton ≥ 2. To this aim, we set

Q(w) := − 14π

∫

C

(wL1w + wL1w

)r dr dθ,

which is well defined for all w ∈ H1(C).

Let us now consider the special case where

w := a einθ + b e−inθ,

for n 6= 0, where a and b only depend on r. In this case, we simply have

Q(w) =∫ +∞

0

(∣∣∣∣da

dr

∣∣∣∣2

+∣∣∣∣db

dr

∣∣∣∣2

+(n + 1)2

r2|a|2 +

(n− 1)2

r2|b|2

)r dr

−∫ +∞

0

(1− S2) (|a|2 + |b|2) r dr +12

∫ +∞

0

S2 |a + b|2 r dr.

While, if n = 0 and w := a, we have

Q(w) =∫ +∞

0

(∣∣∣∣da

dr

∣∣∣∣2

+1r2|a|2

)r dr −

∫ +∞

0

(1− S2) |a|2r dr

+12

∫ +∞

0

S2|a + a|2 r dr.

Now, for any function

v := a e−inθ + b einθ,

we may define the auxiliary function

v := i (|a|2 + |b|2)1/2.

Using Cauchy-Schwartz inequality we obtain

(d

dr(|a|2 + |b|2)1/2

)2

≤∣∣∣∣da

dr

∣∣∣∣2

+∣∣∣∣db

dr

∣∣∣∣2

.

Hence∫ +∞

0

(d

dr(|a|2 + |b|2)1/2

)2

r dr ≤∫ +∞

0

(∣∣∣∣da

dr

∣∣∣∣2

+∣∣∣∣db

dr

∣∣∣∣2)

r dr.

64

This inequality yields

Q(v)−Q(v) ≥∫ +∞

0

(n (n + 2)

r2|a|2 +

n (n− 2)r2

|b|2)

r dr

+12

∫ +∞

0

(1− S2) |a + b|2 r dr.

(3.27)

Therefore, if n ≥ 2, we findQ(v) ≥ Q(v). (3.28)

Let us now assume that n ≥ 2 and that v := a einθ + b e−inθ is a boundedsolution of L1v = 0 in all C. It follows from Proposition 3.2 that v ∈ H1(C)and, since L1v = 0, we obtain

Q(v) = 0.

On the other hand, a result of P. Mironescu [64] asserts that

Q(v) ≥ 0.

This, together with (3.28), yields

Q(v) = Q(v) = 0.

In particular, it follows directly from the expression of Q(v)−Q(v) that

a = b = 0.

The proof of the Theorem is therefore complete. 2

As a byproduct of the previous proof, we have the :

Corollary 3.3 Let 0 ≤ r1 < r2 ≤ +∞ and assume that w is a bounded solutionof

L1w = 0 in Br2 \Br1 ,

with w = 0 on ∂Br1 if r1 > 0, and on ∂Br2 if r2 < +∞. Further assume thatw has no eigencomponent corresponding to the eigenfrequencies n = −1, 0, 1, inother words, w belongs to the space

Spanh±n e±inθ : n ≥ 2.

Then, w = 0.

Proof : To obtain this result, it suffices to extend the solution w by 0 outsideBr2 \Br1 and apply the arguments of Step 3 in the previous proof. 2

65

3.5 More solutions of the homogeneous equa-tion

We end this Chapter with some definitions. The functions Φ01 and Φ±1

1 havealready been defined in (3.8), (3.9) and (3.10). In the proof of Theorem 3.2, wehave defined many others solutions of the homogeneous problem L1w = 0.

In Step 1 of the proof of Theorem 3.2, we have defined

Ψ01 := S∗

:= κw20.

Let us recall that the properties of this function are stated in Lemma 3.3 aswell as in (3.18).

In Step 2 of the proof of Theorem 3.2, we have defined

Ψ+11 := α∗1 eiθ + β∗1 e−iθ

:= κw31 +

κ

8w1

1,

andΨ−1

1 := i (α∗−1 eiθ + β∗−1 e−iθ)

:= κw41 −

κ

8w2

1.

The properties of these functions are stated in Lemma 3.4 and in (3.23). Wewill also need the :

Lemma 3.6 At ∞, the following expansions hold

Φ+11 = −i

S

rsin θ +O

(1r3

)

Φ−11 = i

S

rcos θ +O

(1r3

).

(3.29)

In addition, there exists some constant c+1, c−1 ∈ R \ 0 such that

Ψ+11 = c+1 J+

1

(cos θ +O

(1r2

))

Ψ−11 = c−1 J+

1

(sin θ +O

(1r2

)).

(3.30)

Proof : The first expansions concerning Φ±11 follow at once from the explicit

expression for these two functions as well as from Theorem 3.1. While, thesecond expansions follow from Corollary 3.1. 2

66

Finally, we now define

Ψ+21 := w3

2

:= α∗2 e2iθ + β∗2 e−2iθ,(3.31)

andΨ−2

1 := w42

:= i (α∗−2 e2iθ + β∗−2 e−2iθ),(3.32)

which are respectively the unique solution of Lεw = 0 whose behavior near 0is given by

α∗2 = O(r5) and β∗2 = r (1 +O(r2)),

respectively by

α∗−2 = O(r5) and y∗−2(r, θ) = r (1 +O(r2)).

There is a qualitative difference between all the different solutions of thehomogeneous equation Lεw = 0 which have been defined so far. Indeed, all thesolutions Ψ±n

1 blow up at ∞, while all solutions Φ±n1 are bounded.

Remark 3.1 The inspection of Proposition 3.1 shows that all the above definedsolutions constitute the collection of (linearly independent) solutions of the ho-mogeneous equation L1w = 0 which are bounded at the origin, but which arenot bounded near 0 by any constant times rµ, when the parameter µ > 1.

3.6 Introduction of the scaling factor

For all ε > 0, we define the Ginzburg-Landau equation as

∆u +u

ε2(1− |u|2) = 0. (3.33)

Obviously,uε(z) := u1(z/ε) = eiθ S(r/ε), (3.34)

is a solution of (3.33). For the sake of simplicity in the notations, we define

Sε := S(·/ε).

We also introduce the nonlinear operator

Nε(u) := ∆u +u

ε2(1− |u|2), (3.35)

whose linearization about uε will be denoted by Lε. Again, we consider theconjugate operator

Lε := e−iθ Lε eiθ.

67

For the time being, let us just notice that, if wε and fε solve

Lε wε = fε,

thenw1 := wε(ε ·), and f1 := ε2 fε(ε ·),

solve L1 w1 = f1. Hence, the study of Lε reduces to the study of L1. Inparticular, all the results we have obtained for L1 can be translated for Lε.

In order to simplify the notations we set

Φjε := Φj

1(·/ε), and Ψkε := Ψk

1(·/ε),

for j = −1, 0,+1 and k = −2,−1, 0, +1,+2, which are for Lε the Jacobi fieldscorresponding to the Φj

1 and Ψk1 . For the same reason, we will also write

S∗ε := S∗(·/ε).

Finally, for all τ ∈ R, we may define

uε,τ := eiτ uε.

Observe that this is also a solution of (3.33). Now, the corresponding linearizedoperator about uε,τ is given by

Lε,τw := ∆w +w

ε2(1− |uε,τ |2)− 2

ε2uε,τ (uε,τ · w),

and clearly depends on τ . On the other hand the conjugate operator

Lε,τ := e−i(θ+τ) Lε,τ ei(θ+τ),

is equal to Lε and thus does not depend on τ . Therefore, for all τ ∈ R, thestudy of Lε,τ is equivalent to the study of Lε.

68

Chapter 4

Mapping Properties of Lε

This Chapter is devoted to the study of the problem Lεw = f in B1 \ 0

w = 0 on ∂B1,(4.1)

for all functions f defined in the punctured unit ball. We will first derive somesimple estimates which come from the fact that the operator Lε satisfies somemaximum principle type property, away from the origin. Then, we will definethe family of weighted spaces in which (4.1) will be studied. The main resultof this Chapter is the construction of some well behaved right inverse for Lε inthe unit ball.

4.1 Consequences of the maximum principle inweighted spaces

The study of equation (4.1) partially relies on the following Propositions whoseproof involves the maximum principle. The results are divided in two differentsections according to the Fourier decomposition of the functions w and f .

These results are of particular importance since they also help to understandthe motivations behind the definition of the function spaces we are going to workwith in the first part of the book.

4.1.1 Higher eigenfrequencies

We first consider the case where both functions w and f do not have anyeigencomponents corresponding to the eigenfrequencies n = −1, 0, 1, in theirFourier decompositions. We show that w, solution of L1 w = f , is controlled, inBr1 \Br0 , by f in the same set and w on the boundary, provided r0 and r1 arelarge enough.

69

Proposition 4.1 Assume that µ ∈ (1, 2). There exist some constants λ > 0and c > 0 such that, for all r0 ≥ λ and all r1 ≥ 2r0 and for all

w, f ∈ Spanh±n e±inθ : n ≥ 2,

solutions of L1w = f in Br1 \Br0

w = 0 on ∂Br1 ,

we have

supBr1\Br0

r−µ

(r2

r20

|wr|+ |wi|)≤ c

(sup∂Br0

r−µ |w|+ supBr1\Br0

r2−µ |f |)

,

where w := wr + iwi is the decomposition of w into its real and imaginary part.

The proof of this Proposition relies on the following two preliminary results.

Lemma 4.1 Assume that ν ∈ (0, 2). There exists a constant c1 > 0 such that,for all r0 > 0, r1 > 2r0 and for all

w, g, g ∈ Spanh±n e±inθ : n ≥ 2,

satisfying ∆w = g + ∂θ g in Br1 \Br0

w = 0 on ∂Br1 ,

we have

supBr1\Br0

r−ν |w| ≤ c1

(sup∂Br0

r−ν |w|+ supBr1\Br0

r2−ν (|g|+ |g|))

.

Before we proceed to the proof of the Lemma, let us notice that, in the state-ment, the constant c1 does not depend on r0 nor on r1.

Proof : To get the desired estimate, we first define v to be the solution of

∆v = 0 in Br1 \Br0

v = 0 on ∂Br1

v = w on ∂Br0 .

The maximum principle yields the estimate

supBr1\Br0

|v| ≤ sup∂Br0

|w|,

70

which, together with the fact that we have assumed ν > 0 implies that

supBr1\Br0

r−ν |v| ≤ sup∂Br0

r−ν |w|. (4.2)

Now, we define v to be the solution of

∆v = g in Br1 \Br0

v = 0 on ∂Br1 ∪ ∂Br0 .

We decompose both v and g into Fourier series and write

g :=∑

|n|≥2

gn einθ and v :=∑

|n|≥2

vn einθ.

For all |n| ≥ 2, the function vn is a solution of

d2vn

dr2+

1r

dvn

dr− n2

r2vn = gn.

in (r0, r1). In addition, we have the boundary conditions vn(r0) = vn(r1) = 0.We set

M := supBr1\Br0

r2−ν |g|,

so that |gn| ≤ M rν−2. Since we have assumed that ν ∈ (0, 2), it is easy to seethat, for all |n| ≥ 2, the function

r −→ M

n2 − ν2rν ,

can be used as a barrier function to obtain the estimate

|vn| ≤ M

n2 − ν2rν .

Summation over n yields

supBr1\Br0

r−ν |v| ≤ ∑

|n|≥2

1n2 − ν2

sup

Br1\Br0

r2−ν |g|. (4.3)

Finally, we define v to be the solution of

∆v = g in Br1 \Br0

v = 0 on ∂Br1 ∪ ∂Br0 .

And, thanks to what we have already proven in (4.3), we have the bound

supBr1\Br0

r−ν |v| ≤ c supBr1\Br0

r2−ν |g|.

71

Moreover, using rescaled Schauder’s estimates of Lemma 2.3 (as we have alreadydone in Corollary 2.1) together with Lemma 2.4, we conclude that

supBr1\Br0

r1−ν |∇v| ≤ c supBr1\Br0

r2−ν |g|.

In particularsup

Br1\Br0

r−ν |∂θv| ≤ c supBr1\Br0

r2−ν |g|.

This last estimate, together with (4.2) and (4.3), give the desired result since,by definition w = v + v + ∂θv. 2

Our second preliminary result reads as follows :

Lemma 4.2 Assume that ν ∈ R is fixed. There exist some constants λ1 > 0and c > 0 such that, for all r0 ≥ λ1 and r1 > 2r0, if

∆w − 1r2

w + (1− 3S2)w = g + ∂θg in Br1 \Br0

w = 0 on ∂Br1 .

Then

supBr1\Br0

r2−ν |w| ≤ c

(sup∂Br0

r2−ν |w|+ supBr1\Br0

r2−ν (|g|+ r |g|))

.

Proof : To obtain the desired estimate we look more closely at the operator

Λ := ∆− 1r2

+ (1− 3S2).

Since S tends to 1 as r tends to +∞, there exists λ1 > 0 such that the potentialin this operator satisfies

− 1r2

+ (1− 3S2) ≤ 0, in [λ1,+∞).

Thus, the maximum principle holds for Λ in C \Bλ1 .We now define v to be the solution of

Λv = g in Br1 \Br0

v = 0 on ∂Br1

v = w on ∂Br0 .

Let us compute

Λ rν−2 =(((ν − 2)2 − 1) r−2 + (1− 3S2)

)rν−2.

72

Increasing λ1, if this is necessary, this computation yields

Λ rν−2 ≤ −rν−2,

in C \Bλ1 . We set

M := sup∂Br0

r2−ν |w|+ supBr1\Br0

r2−ν |g|.

The previous computation shows that the function

z −→ M rν−2,

can be used as a barrier function for w in Br1 \Br0 . Hence, we obtain

supBr1\Br0

r2−ν |v| ≤ M := sup∂Br0

r2−ν |w|+ supBr1\Br0

r2−ν |g|. (4.4)

We finally define

Λv = g in Br1 \Br0

v = 0 on ∂Br1 ∪ ∂Br0 .

We make use of (4.4), with ν replaced by ν − 1, to obtain the bound

supBr1\Br0

r3−ν |v| ≤ c supBr1\Br0

r3−ν |g|.

Since, by assumption, r0 ≥ λ1 and r1 > 2r0, we may use Schauder’s estimates,as stated in Lemma 2.3 and in Lemma 2.4, to conclude that

supBr1\Br0

r3−ν |∇v| ≤ c supBr1\Br0

r3−ν |g|.

Hencesup

Br1\Br0

r2−ν |∂θv| ≤ c supBr1\Br0

r3−ν |g|.

The result follows from this last estimate together with (4.4), since by definitionw = v + ∂θv. 2

We are now in a position to prove Proposition 4.1, but before we do so, letus comment on the difference between the results of Lemma 4.1 and Lemma 4.2.In the first Lemma, where we are working with the operator ∆ but a similarresult would have been true for any operator of the form ∆ + c

r2 . The mainidea is that, when we solve ∆w = g, we gain a factor 2 in the weight scalesince the function g is bounded by rµ−2 while w is bounded by rµ. A fairlyreasonable way to understand this is to think that the operator ∇ (respectively∆) is somehow equivalent to multiplication by 1/r (respectively by 1/r2). Then,solving ∆w = f amounts to two integrations, or equivalently a gain of two inthe weight scale.

73

However, in the second Lemma, we are working with an operator which lookslike ∆−2 (since S ∼ 1 for r large) but a similar result would have been true forany operator of the form ∆− c2. This time, when we solve ∆w− 2w = g we donot gain any factor in the weight scale. A fairly reasonable way to understandthis is to think that the operator ∆ is now equivalent to multiplication by 1,and hence ∇ is equivalent to multiplication by 1. So, in this case, integrationsdo not produce any gain in the weight scale.

Proof of Proposition 4.1 : The equations being linear we may assume that

sup∂Br0

r−µ |w|+ supBr1\Br0

r2−µ |f | = 1. (4.5)

We shall prove that, if the parameter λ is chosen large enough, there exists someconstant c > 0 such that

supBr1\Br0

r−µ

(r2

r20

|wr|+ |wi|)≤ c.

Using the decomposition w = wr + i wi and f = fr + i fi where wr, wi, fr andfi are real valued functions, we see that wr solves

∆wr − 1r2

wr + (1− 3S2)wr =2r2

∂θwi + fr, (4.6)

and wi solves

∆wi = −(

1− S2 − 1r2

)wi − 2

r2∂θwr + fi. (4.7)

To begin with, let us prove that, if λ is chosen large enough, there existsc > 0 such that

supBr1\Br0

r−µ

(r

r0|wr|+ |wi|

)≤ c. (4.8)

Observe that this is a weaker estimate than the one we ultimately want to prove.

Let us denote by

Mr := supBr1\Br0

r1−µ |wr| and Mi := supBr1\Br0

r−µ |wi|.

Assuming that λ ≥ λ1 (where λ1 is the constant given in Lemma 4.2) andr0 ≥ λ, we may apply the result of Lemma 4.2 to (4.6), with ν = µ + 1, andobtain the inequality

supBr1\Br0

r1−µ |wr| ≤ c

(sup∂Br0

r1−µ |wr|+ supBr1\Br0

r1−µ

(|fr|+ 2

r|wi|

)).

This, together with (4.5), shows that

Mr ≤ c (r0 + Mi), (4.9)

74

for some constant c > 0.

Increasing the value of λ if necessary, we may always assume that∣∣∣∣1− S2 − 1

r2

∣∣∣∣ ≤1

4c1r−2,

in [λ, +∞), where c1 is the constant which appears in the statement of Lemma 4.1.Such a choice is always possible since, by Theorem 3.1

S = 1− 12r2

+O(

1r4

),

at ∞. We now apply the estimate of Lemma 4.1 to (4.7), with ν = µ, to getthe inequality

supBr1\Br0r−µ |wi| ≤ c1

(sup∂Br0

r−µ |wi|

+ supBr1\Br0

r2−µ

(|fi|+ 1

2c1|wi|+ 2

r2|wr|

)).

Hence, we have

Mi ≤ Mi

2+ c

(1 +

1r0

Mr

), (4.10)

for some constant c > 0.

Collecting (4.9) and (4.10), we conclude that

Mi ≤ c

(1 +

1r0

Mr

)and Mr ≤ c (r0 + Mi).

Then (4.8) follows at once by increasing the value of λ if this is necessary. Inparticular, we have obtained the inequalities

|wi| ≤ c rµ, and |wr| ≤ cr0

rrµ. (4.11)

We define wi to be the solution of

∆wi = − 2r2

∂θwr in Br1 \Br0

wi = 0 on ∂Br1 ∪ ∂Br0 .

(4.12)

Applying the result of Lemma 4.1 with ν = µ− 1 we obtain from (4.11) that

|wi| ≤ cr0

rrµ. (4.13)

Finally we defined wi to be the solution of

∆wi = −(

1− S2 − 1r2

)wi + fi in Br1 \Br0

wi = 0 on ∂Br1

wi = wi on ∂Br0 .

(4.14)

75

Since we have already proven that wi is bounded by a constant times rµ, wemay now apply Lemma 4.1 to obtain |wi| ≤ c rµ in Br1 \Br0 and then rescaledSchauder’s estimates to get

|∇wi| ≤ c rµ−1, (4.15)

in Br1 \B2r0 . Obviously, wi = wi + wi. We now write (4.6) as

∆wr − 1r2

wr + (1− 3S2)wr =2r2

∂θwi +(

2r2

∂θwi + fr

).

By virtue of (4.15) the last term in the right hand side is bounded by a constanttimes rµ−2. Applying once more the result of Lemma 4.2 with ν = µ and using(4.15), we now conclude that

|wr| ≤ c (1 + r0 + r20) rµ−2 ≤ c r2

0 rµ−2.

where the constant c only depends on λ. The proof of the result is thereforecomplete. 2

We now give a Corollary which is nothing but the result of Proposition 4.1,for Lε, in some special situation.

Corollary 4.1 Assume that µ ∈ (1, 2). There exist c > 0 and λ > 0 such that,for all ε ∈ (0, 1/(2λ)] and for all

w, f ∈ Spanh±n e±inθ : n ≥ 2,

solutions of Lεw = f in B1 \Bλε

w = 0 on ∂B1,

we have

supB1\Bλε

r−µ

(r2

ε2|wr|+ |wi|

)≤ c

(sup∂Bλε

r−µ |w|+ supB1\Bλε

r2−µ |f |)

,

where w := wr + iwi is the decomposition of w into its real and imaginary part.

Proof : It suffices to notice that whenever w solves Lεw = f , then v := w(ε·)and g := ε2 f(ε·) solve L1v = g and apply the former result to v. The constantλ is the one given in Proposition 4.1. 2

4.1.2 Lower eigenfrequencies

We now turn to the proof of a result which, in its spirit, is the counterpart ofthe previous Proposition, for lower eigenfrequencies.

76

Proposition 4.2 Assume that µ > 1. There exist some constants λ > 0 andc > 0 such that, for all r0 ≥ λ and r1 ≥ 2r0, and for all

w, f ∈ Spanh±1 e±iθ,solutions of L1w = f in Br1 \Br0

wr = 0 on ∂Br1 ,

we have

supBr1\Br0

r−µ

(r2

r20

|wr|+ |wi|)

≤ c

(sup∂Br0

r−µ(|w|+ r|∇wi|)

+ supBr1\Br0

r2−µ|f |)

,

where w = wr + i wi is the decomposition of w into its real and imaginary part.

Let us insist on the fact that on the right hand side of this estimate, a boundon |∇wi| on ∂Br0 is needed since we do not impose any boundary condition forwi on ∂Br1 . But we do not need any bound on |∇wr| on ∂Br0 , since we havewr = 0 on ∂Br1 .

The proof of this result is nearly identical to the proof of Corollary 4.1,though somehow less technical since we are only dealing with ordinary differen-tial equations instead of partial differential equations. Nevertheless, we have toreplace Lemma 4.1 by the following :

Lemma 4.3 Assume that µ > 1. There exists a constant c2 > 0 such that, forall r0 > 0 and r1 ≥ 2r0, and for all

w, g ∈ Spanh±1 e±iθ,solutions of

∆w = g in Br1 \Br0 ,

we have

supBr1\Br0

r−µ |w| ≤ c2

(sup∂Br0

r−µ (|w|+ r |∇w|) + supBr1\Br0

r2−µ |g|)

,

Observe that, since g and w only have eigencomponents over two eigenfrequen-cies, the equation ∆w = g reduces to two ordinary differential equations.Proof: To prove such an estimate, we first assume that w := v eiθ andg := f eiθ (the case were w := v e−iθ and g := f e−iθ can be treated similarly).Then, v and f solve

d2v

dr2+

1r

dv

dr− 1

r2v = f.

77

We setv0 := v(r0), and

dv

dr(r0 := v1.

We argue as in the proof of Lemma 4.1 and we define v to be the solution of

d2v

dr2+

1r

dv

dr− 1

r2v = 0 in [r0, r1]

v(r0) = v0

dv

dr(r0) = v1.

We have explicitly

v =12

((v0 + r0 v1)

r

r0+ (v0 − r0 v1)

r0

r

),

from which it follows that

sup[r0,r1]

r−µ |v| ≤ 2 r−µ0 (|v0|+ r0 |v1|) .

Finally, the variation of the constant formula yields

v − v = r

∫ r

r0

s−3

∫ s

r0

g t2 dt.

It is then easy to prove the estimate

sup[r0,r1]

r−µ |v − v| ≤ c sup[r0,r1]

r2−µ |g|.

And the result follows at once. 2

Proof of Proposition 4.2 : Again, the equations being linear we may assumethat

sup∂Br0

r−µ (|w|+ r |∇wi|) + supBr1\Br0

r2−µ |f | = 1.

We shall prove that, if the parameter λ is chosen large enough, there exists someconstant c > 0 such that

supBr1\Br0

r−µ

(r2

r20

|wr|+ |wi|)≤ c.

As in the proof of Proposition 4.1, we write

∆wr − 1r2

wr + (1− 3S2)wr =2r2

∂θwi + fr, (4.16)

and

∆wi = −(

1− S2 − 1r2

)wi − 2

r2∂θwr + fi. (4.17)

78

Let us denote by

Mr = supBr1\Br0

r2−µ |wr| and Mi = supBr1\Br0

r−µ |wi|.

We assume that λ is larger than λ1 (defined in Lemma 4.3) and apply the resultof Lemma 4.3 to (4.16). This gives us the inequality

Mr ≤ c (r20 + Mi), (4.18)

for some constant c > 0.

Let us further assume that λ is chosen large enough so that∣∣∣∣1− S2 − 1

r2

∣∣∣∣ ≤1

2c2r−2,

for all r ≥ λ, where in this inequality, c2 is the constant which appears in thestatement of Lemma 4.3. We now apply the estimate of Lemma 4.3 to (4.17)and obtain the inequality

Mi ≤ Mi

2+ c

(1 +

1r20

Mr

), (4.19)

for some constant c > 0.

Collecting (4.18) and (4.19), we conclude that

Mi ≤ c

(1 +

1r20

Mr

)and Mr ≤ c (r2

0 + Mi).

The result then follows at once taking λ large enough. 2

Again, we give a Corollary which is nothing but the result of Proposition 4.2,for Lε, in some special situation.

Corollary 4.2 Assume that µ > 1. There exist c > 0 and λ > 0 such that, forall ε ∈ (0, 1/(2λ)] and for all

w, f ∈ Spanh±1 e±iθ,solutions of Lεw = f in B1 \Bλε

wr = 0 on ∂B1,

we have

supB1\Bλε

r−µ

(r2

ε2|wr|+ |wi|

)≤ c

(sup∂Bλε

r−µ(|w|+ r|∇wi|)

+ supB1\Bλε

r2−µ|f |)

,

where w = wr + i wi is the decomposition of w into its real and imaginary part.

79

4.2 Function spaces

Let O ⊂ C be a regular open bounded subset of C and Σ := b1, . . . , bk a finiteset of points of O. We now want to define some function spaces which will helpus understand the mapping properties of Lε as ε tends to 0.

It should be clear, from Chapter 2, that the weighted Holder spaces Ck,αµ (O\

Σ) which have already been given in Chapter 2, are suited to understand themapping properties of the operator

w −→ ∆w +w

ε2(1− S2

ε )− 1r2

w,

which appears in the definition of Lε. Indeed, this operator behaves like theoperator ∆− 1

r2 near 0 and behaves like the operator ∆ at +∞, two operatorsfor which the above weighted spaces have already proved to be well adapted.However, these spaces will not be useful for understanding the operator

w −→ ∆w +w

ε2(1− 3S2

ε )− 1r2

w.

While this operator still behaves like the operator ∆− 1r2 near 0, it behaves like

the operator ∆− 2ε2 at +∞, operator for which we already know from Lemma 4.2

that they above spaces are not well adapted at all. This, together with the resultof Corollary 4.2, motivates the definition of the spaces Ck,α

µ,ε (O \ Σ), which willbe given below.

We chose σ > 0 in such a way that, if bi 6= bj ∈ Σ, then B2σ(bi) and B2σ(bj)are disjoint and both included in O. For all s ∈ (0, σ), we define a closed subsetof O

As := z ∈ O : dist(z, Σ) ∈ [s, 2s] .

We also setOσ := z ∈ O : dist(z, Σ) > σ .

For any function w ∈ Ck,α(O \Σ), we recall that we have already defined inChapter 2 the family of semi-norms indexed by s ∈ (0, σ).

[w]k,α,s :=k∑

j=0

sj supAs

|∇jw|+ sk+α supx,y∈As

|∇kw(x)−∇kw(y)||x− y|α . (4.20)

We also recall that :

Definition 4.3 Given k ∈ N, α ∈ [0, 1) and µ ∈ R, the space Ck,αµ (O \ Σ)

is defined as the set of real valued functions w ∈ Ck,αloc (O \ Σ) for which the

following norm is finite

‖w‖Ck,αµ (O\Σ) := ‖w‖Ck,α(Oσ) + sup

s∈(0,σ)

s−µ [w]k,α,s. (4.21)

80

The functions belonging to this space can roughly be described as the functionswhich are bounded by a constant times dist(z, Σ)µ and whose k-th derivativesare bounded by a constant times dist(z, Σ)µ−k.

Similarly, for any function w ∈ Ck,αloc (Ω \ Σ), we consider another family of

semi-norms indexed by s ∈ (0, σ).

[w]′k,α,s :=k∑

j=0

inf(ε, s)j supAs

|∇jw|

+ inf(ε, s)k+α supx,y∈As

|∇kw(x)−∇kw(y)||x− y|α .

(4.22)

And, we now define

Definition 4.4 Given k ∈ N, α ∈ [0, 1), µ ∈ R and ε > 0, the space Ck,αµ (O\Σ)

is defined as the set of real valued functions w ∈ Ck,αloc (O \ Σ) for which the

following norm is finite

‖w‖′Ck,αµ (O\Σ)

:= ‖w‖Ck,α(Oσ) + sups∈(0,σ)

inf(s, ε)−k sk−µ [w]′k,α,s. (4.23)

Observe that this space of functions depends on an additional parameter ε andthough this parameter appears in the definition of the norm and in the definitionof the space, it is not explicit in the notations. This time, functions belongingto this space can be described as functions which are bounded by a constanttimes inf(ε, dist(z, Σ))k dist(x, Σ)µ−k and whose k-th derivatives are boundedby a constant times dist(z, Σ)µ−k.

Finally, we define spaces of complex valued functions by

Ek,αµ (O \ Σ) := Ck,α

µ (O \ Σ)⊕ i Ck,αµ (O \ Σ), (4.24)

which is endowed with the norm

||wr + i wi||Ek,αµ

:= ||wr||′Ck,αµ (O\Σ)

+ ||wi||Ck,αµ (O\Σ). (4.25)

Granted the definition of Lε which is given in (3.7), it is an easy exercise tocheck that

Lε : E2,αµ (B1 \ 0) −→ E0,α

µ−2(B1 \ 0),is an operator whose norm is bounded independently of ε ∈ (0, 1).

4.3 A right inverse for Lε in B1 \ 0Now, we would like to obtain for Lε, a result similar to the results we haveobtained for the Laplacian in Chapter 2. More precisely, we shall prove the :

81

Theorem 4.1 For all µ ∈ (1, 2), there exists ε0 > 0 such that, for all ε ∈ (0, ε0)and for all f ∈ E0,α

µ−2(B1 \ 0), there exists a unique solution of Lεw = f in B1 \ 0

w = 0 on ∂B1,(4.26)

which can be decomposed as

w = v + d0 Φ0ε + ε−1 (d−1 Φ−1

ε + d+1 Φ+1ε )

+ εµ η(·/ε)(δ−2 Ψ−2

ε + δ0 Ψ0ε + δ+2 Ψ+2

ε

),

Where η is a cutoff function which is identically equal to 1 in B1 and equal to0 outside B2. In addition, if we set d := (d−1, d0, d+1) and δ := (δ−2, δ0, δ+2),the linear mapping

f ∈ E0,αµ−2 −→ (v, d, δ) ∈ E2,α

µ × R3 × R3,

is bounded independently of ε ∈ (0, ε0).

Before we proceed to the proof of this Theorem, let us comment on the reasonwhy we have chosen µ ∈ (1, 2). This choice is dictated by the fact that wewill use this linear result to perturb an approximate solution into a solution ofthe Ginzburg-Landau equation. Since our approximate solution will behave ateach vortex like uε which has been defined in (3.1), and therefore decays neareach vortex like r, it is natural to chose µ > 1 so that the perturbation willdecay faster. Unfortunately, with this choice, we are no longer in the range ofsurjectivity of the operator Lε. To overcome this lack of surjectivity we havethus introduced the 6 Jacobi fields Φj

ε and Ψjε.

To begin with, we decompose f and w in lower and higher eigenfrequencies.More precisely, we set

f =2∑

n=0

fn + g, and w =2∑

n=0

wn + v,

where both g and v belong to Spanh±n e±inθ : n ≥ 3 and where fn and wn

belong to Spanh±n e±inθ, for n = 0, 1, 2. The proof is now divided into twoparts. In the first part, we will deal with all eigenfrequencies corresponding to|n| ≥ 3 and finally we will handle the lower eigenfrequencies.

4.3.1 Higher eigenfrequencies

Obviously, the function v has to satisfy the equation Lεv = g in B1 \ 0

v = 0 on ∂B1.(4.27)

We prove the

82

Lemma 4.4 Assume that µ ∈ (1, 2). There exist some constants ε0 > 0 andc > 0 such that, for all ε ∈ (0, ε0) and for all

g ∈ E0,αµ−2(B1 \ 0) ∩ Spanh±n e±inθ : n ≥ 3,

there exists a unique

v ∈ E2,αµ (B1 \ 0) ∩ Spanh±n e±inθ : n ≥ 3,

solution of (4.27). In addition

||v||E2,αµ

≤ c ||g||E0,αµ−2

.

Proof : The operator Lε is clearly self adjoint and, thanks to the result ofCorollary 3.3, we know that it does not have any kernel in C2,α

D (B1 \ Bρ) ∩Spanh±n e±inθ : n ≥ 3, where the subscript D refers to the fact that thefunctions have 0 boundary data (the result is even true in C2,α

D (B1 \ Bρ) ∩Spanh±n e±inθ : n ≥ 2). Therefore, for any ρ ∈ (0, λ ε] (where λ is theconstant which is defined in the statement of Proposition 4.1), there exists aunique solution of

Lεv = g in B1 \Bρ

v = 0 on ∂B1 ∪ ∂Bρ.

We decompose v into its real and imaginary part, writing v := vr+i vi. Standardregularity theory tells us that there exists some constant cρ,ε > 0 (which a prioridepends in both ε and ρ), such that

supB1\Bρ

((inf(ε, r))−2r2−µ |vr|+ r−µ |vi|) ≤ cρ,ε supB1\Bρ

r2−µ |g|.

We claim that, since we have chosen µ ∈ (1, 2), the constants cρ,ε are boundedby some constant c > 0 which is independent of ρ ∈ (0, λ ε] and ε ∈ (0, 1

2λ ].Assuming that we have already proved this claim, we may pass to the limitρ → 0 and obtain the existence of v solution of (4.27) such that

supB1\0

((inf(ε, r))−2r2−µ |vr|+ r−µ |vi|) ≤ c supB1\0

|z|2−µ|g|,

where the constant c does not depend on g, neither on ε. As usual, we use bothSchauder’s estimates and rescaled Schauder’s estimates to obtain the existenceof some constant c > 0 independent of g and ε ∈ (0, ε0) such that

||v||E2,αµ

≤ c ||g||E0,αµ−2

.

To finish the proof of the Lemma, it suffices to apply Corollary 3.3 which ensuresthe uniqueness of the solution v.

83

It remains to prove the claim. The proof of this claim is very close to theproof of Proposition 2.1. Again, we argue by contradiction. If the result werenot true, there would exist sequences (εj)j≥0 ∈ (0, 1

2λ ], (ρj)j≥0 ∈ (0, λ εj ], asequence (gj)j≥0 for which

supB1\Bρj

r2−µ |gj | = 1,

and (vj)j≥0 the sequence of solutions of

Lεj vj = gj in B1 \Bρj

vj = 0 on ∂B1 ∪ ∂Bρj,

such thatAj := sup

B1\Bρj

((inf(εj , r))−2 r2−µ |vj,r|+ r−µ |vj,i|

), (4.28)

tends to +∞. We setCj := sup

Bλεj\Bρj

r−µ |vj |.

Obviously Aj ≥ (1 + λ2)Cj . Moreover, it follows from Corollary 4.1 that thereexists a constant c > 0, independent of j, such that

Aj ≤ c (1 + sup∂Bλεj

r−µ |vj |) ≤ c (1 + Cj).

Since, by assumption, Aj tends to +∞, so does Cj .

Now, we choose zj ∈ Bλεj \Bρj in such a way that

|zj |−µ |vj |(zj) = Cj .

We set rj := ρj/|zj |, rj := λεj/|zj | and rj := 1/|zj | and we define the sequenceof rescaled functions

wj := C−1j |zj |−µ vj(|zj | ·).

By definition,sup

Brj\Brj

r−µ|wj | ≤ 1.

Moreover, it follows from Corollary 4.1, that we also have

r−µ|wj | ≤ c in Brj \Brj ,

for some constant c > 0 independent of j. Finally, the function wj solves

Lεj/|zj |wj = gj in Brj \Brj

wj = 0 on ∂Brj ∪ ∂Brj ,

84

where we have setgj := C−1

j |zj |2−µ gj(|zj | ·).Notice that by construction

supBrj

\Brj

r2−µ |gj | = C−1j ,

which tends to 0 as j tends to +∞.

Up to a subsequence, we may always assume that (zj)j≥0 converges to z∞ ∈C and we set r∞ := 1/|z∞|. Similarly we may assume that (εj/|zj |)j≥0 convergesto ε∞ in [1/λ,+∞] and finally that (rj)j≥0 converges to r∞ ∈ [0, 1) (notice that,thanks to Lemma 2.4, we have r∞ < 1). We also know that gj tends to 0 inevery compact subset of z ∈ C : r∞ < |z| < r∞. We shall now distinguishtwo cases according to the values of ε∞.

1 - Assume that ε∞ < +∞. After extracting some subsequence, if this isnecessary, we may assume that the sequence (wj)j≥0 converges to some functionw which is a nontrivial solution of

Lε∞w = 0,

in the set z ∈ C : r∞ < |z| < r∞. Furthermore, w = 0 on ∂Br∞ , ifr∞ 6= 0 and also on ∂Br∞ if r∞ < +∞. In addition, we know that w ∈Spanh±n(r)e±inθ : n ≥ 3, w 6= 0 and also that, by construction, |w| ≤ c rµ.

Let us decompose w into Fourier series

w :=∑

n≥3

(an einθ + bn e−inθ),

then an einθ +bn e−inθ is also bounded by a constant times rµ. If r∞ = 0, we useProposition 3.1 and the fact that µ ∈ (1, 2) to conclude that an einθ +bn e−inθ isin fact bounded and even decays like rn−1 at 0. Similarly, if r∞ = +∞, we useProposition 3.2 and the fact that µ ∈ (1, 2), to conclude that an einθ + bn e−inθ

is in fact bounded and even decays like r−n at ∞.

Finally, we use Theorem 3.1, when r∞ = 0 and r∞ = +∞, or Corollary 3.3otherwise to conclude that w ≡ 0. Which is the desired contradiction.

2 - Assume that ε∞ = +∞ and hence rj = +∞. After extracting somesubsequence, if this is necessary, we may assume that the sequence (wj)j≥0

converges to some function w which is a nontrivial solution of

∆(eiθw) = 0,

in C \Br∞ . The presence of the extra eiθ is due to the fact that we are workingwith the conjugate linearized operator instead of the linearized operator itself.Furthermore, w = 0 on ∂Br∞ if r∞ > 0. In addition, we know that w ∈Spanh±n e±inθ : n ≥ 3, and also that, by construction, |w| ≤ c rµ.

85

Again, we decompose w into Fourier series

w :=∑

|n|≥3

an einθ,

then the function an is also bounded by a constant times rµ. Now, it is a simpleexercise to see that

an = αn rn+1 + βnr−n−1.

for some α, β ∈ R. Since |an| has to be bounded by a constant time rµ withµ ∈ (1, 2), and taking into account the behavior of an at ∞ we conclude that,necessarily αn = 0. Then, considering either the behavior of wn at 0 if r∞ = 0or simply the fact that wn(r∞) = 0 otherwise, we conclude that α = 0 too.Hence w ≡ 0 and we have also obtained the desired contradiction.

Since we have obtained a contradiction in each case, the proof of the claimis complete and so is the proof of the Lemma. 2

4.3.2 Lower eigenfrequencies

We are now going to prove some results similar to Lemma 4.4, for the lowereigenfrequencies. Again three cases are distinguished according to the value ofthe eigenfrequency.

Eigenspace corresponding to n = 0. This case is the easiest since we justwant to solve the ordinary differential equation

d2w0

dr2+

1r

dw0

dr− 1

r2w0 +

w0

ε2(1− 2S2

ε )− 1ε2

S2ε w0 = f0, (4.29)

in (0, 1), with w0(1) = 0. We prove the :

Lemma 4.5 Assume that 1 < µ. There exist some constants ε0 > 0 and c > 0such that, for all radially symmetric function f0 ∈ E0,α

µ−2(B1 \ 0) and for allε ∈ (0, ε0), there exists a unique w0, solution of (4.29) which can be decomposedas

w0 = v0 + d0 Φ0ε + εµ η(·/ε) δ0 Ψ0

ε,

where v0 ∈ E2,αµ (B1 \ 0) and d0, δ0 ∈ R. In addition, we have

||v0||E2,αµ

+ |d0|+ |δ0| ≤ c ||f0||E0,αµ−2

.

Proof : If we set w0 = α0 + iβ0 and f0 = g0 + ih0, the ordinary differentialequation (4.29) is equivalent to the system

d2α0

dr2+

1r

dα0

dr− 1

r2α0 +

α0

ε2(1− 3S2

ε ) = g0, (4.30)

86

d2β0

dr2+

1r

dβ0

dr− 1

r2β0 +

β0

ε2(1− S2

ε ) = h0. (4.31)

Since these two equations are decoupled and different in their nature, we aregoing to solve them independently.

Let us notice that Sε is a solution of the second equation when h0 ≡ 0.Since S > 0 for all r > 0, the variation of the constant formula yields an explicitsolution of (4.31). Namely

β0 = d0 Sε + Sε

∫ r

0

S−2ε s−1

∫ s

0

t Sε h0 dt ds, (4.32)

where the constant d0 is chosen so that β0(1) = 0. Using the expansions ofTheorem 3.1 for S together with the fact that µ > 1, it is an easy exercise tosee that there exists some constant c > 0 such that for all r ∈ (0, 1]

∣∣∣∣Sε

∫ r

0

S−2ε s−1

∫ s

0

t Sε h0 dt ds

∣∣∣∣ ≤ c

(sup(0,1]

s2−µ|h0|)

rµ.

And, it follows from the definition of d0 that

|d0| ≤ c ||h0||E0,αµ−2

.

Hence, we have solved the second ordinary differential equation and obtainedthe relevant estimates. Notice that the estimates for the derivatives of β0 nowfollow by direct estimation.

We turn to the study of the first ordinary differential equation. By construc-tion, the function S∗ε is a solution of (4.30), when g0 ≡ 0 and we have seen inthe proof of Lemma 3.3 that S∗ε > 0, for all r > 0. Again, we may apply thevariation of the constant formula and obtain explicitly

α0 = −S∗ε

∫ 1

r

(S∗ε )−2 s−1

∫ s

0

t S∗ε g0 dt ds. (4.33)

Now, the result of Corollary 3.2 implies that, there exists a constant c > 0such that

1c

(ε

r

)1/2

e√

2r/ε ≤ S∗ε ≤ c(ε

r

)1/2

e√

2r/ε,

for all r ≥ ε and1c

r

ε≤ S∗ε ≤ c

r

ε,

for all r ∈ (0, ε]. Using these estimates, it is again an easy exercise to show thatthere exists c > 0 such that

|α0| ≤ c ||g0||E0,αµ−2

ε2 rµ−2,

for all r ≥ ε. And also that there exists δ0 ∈ R such that

|α0 − εµ δ0 S∗ε | ≤ c ||g0||E0,αµ−2

rµ,

87

for all r ≤ ε, with|δ0| ≤ c ||g0||E0,α

µ−2.

Once we have derived these first estimates, the estimates for the derivativesfollow at once. This ends the proof of the Lemma. 2

Eigenspace corresponding to n = ±1. Next, we study the equation Lεw1 = f1 in B1 \ 0

w1 = 0 on ∂B1,(4.34)

when w1, f1 ∈ Spanh±1 e±iθ. This time, things are slightly more involvedsince we are dealing with a coupled system of ordinary differential equations.We shall take advantage from the fact that we already know four independentsolutions of the homogeneous related problem, all of which are bounded nearthe origin. These solutions are given by

Φ+1ε = α1(r/ε)eiθ + β1(r/ε)e−iθ, Φ−1

ε = i(α−1(r/ε)eiθ + β−1(r/ε)e−iθ),

and

Ψ+1ε = α∗1(r/ε)eiθ + β∗1(r/ε)e−iθ, Ψ−1

ε = i(α∗−1(r/ε)eiθ + β∗−1(r/ε)e−iθ).

Let us recall that both Φ±1ε decay at ∞ while both Ψ±ε blow up exponentially

at ∞. We prove the :

Lemma 4.6 Assume that µ ∈ (1, 2). There exist some constants ε0 > 0 and c >0 such that, for all ε ∈ (0, ε0) and for all f1 ∈ E0,α

µ−2(B1 \0)∩Spanh±1 e±iθ,there exists a unique w1 solution of (4.34) which can be decomposed as

w1 = v1 + ε−1(d−1 Φ−1

ε + d+1 Φ+1ε

),

where v1 ∈ E2,αµ (B1 \ 0) and d±1 ∈ R. In addition

|d−1|+ |d+1|+ ||v1||E2,αµ

≤ c ||f1||E0,αµ−2

.

Proof : The proof of the Lemma goes as follows, first we construct a solutionwhich has the correct behavior near the origin but is not equal to 0 on ∂B1.Then, we use the two Jacobi fields Ψ±1

ε to correct the boundary data of thissolution in such a way that its real part is equal to 0 on ∂B1. We obtain someuniform bound for this special solution and finally using all the Jacobi fieldsΦ±1

ε and Ψ±1ε , we obtain the desired solution.

Step 1. We first prove that we can find a solution of Lεw = f1 in B1 \ 0

<w = 0 on ∂B1,(4.35)

88

which can be decomposed as

w = v + δ−1Ψ−1ε + δ+1 Ψ+1

ε ,

where the function v is bounded by a constant times rµ at the origin.

If we set v := a eiθ + b e−iθ and f1 = α eiθ +β e−iθ, the existence of v followsfrom the existence of a solution for the following system of ordinary differentialequations

d2a

dr2+

1r

da

dr− 4

r2a = − 1

ε2(1− 2S2

ε )a +S2

ε

ε2b + α

d2b

dr2+

1r

db

dr= − 1

ε2(1− 2S2

ε )b +S2

ε

ε2a + β,

(4.36)

having the correct behavior at 0. For r close to 0, the existence of such a solutionfollows as usual from a fixed point Theorem, then, the solution can be extendedfor all r > 0 without difficulties. Indeed, we can rewrite the above system as

a = −r2

∫ ζε

r

s−5

∫ s

0

t3(

S2ε

ε2b− (1− S2

ε )ε2

a + α

)dt ds

b =∫ r

0

s−1

∫ s

0

t

(S2

ε

ε2a− (1− S2

ε )ε2

b + β

)dt ds

(4.37)

where the constant ζ > 0 is chosen small enough independently of ε. Theexistence of a solution of this system for r ≤ ζ ε follows without any difficultyusing the fact that Sε is bounded by 1. We now choose λ as in Proposition 4.2.Since v is a solution of some ordinary differential equation, it is easy to see that,for all ζ > λ, there exists a constant c > 0 (depending on ζ but independent ofε ∈ (0, 1

2ζ )) such that

supBζε\0

r−µ (|v|+ r |∇v|) ≤ cζ supB1\0

r2−µ |f1|. (4.38)

Now, we are going to chose δ±1 ∈ R in such a way that <w = 0 on ∂B1. Tosee that such a choice is possible, we observe that we have

<(δ+1 Ψ+1ε (eiθ) + δ−1 Ψ−1

ε (eiθ)) = δ+1(α∗1(1/ε) + β∗1(1/ε)) cos θ

− δ−1(α∗−1(1/ε)− β∗−1(1/ε)) sin θ.

Moreover on ∂B1, we also have <v ∈ Spancos θ, sin θ. Since the functions α∗1and β∗1 have the same sign and since the functions α∗−1 and β∗−1 have oppositesigns we can conclude that it is always possible to choose δ+1, δ−1 ∈ R in sucha way that

< (δ+1 Ψ+1

ε (eiθ) + δ−1 Ψ−1ε (eiθ)− v(eiθ)

)= 0.

This ends the proof of the existence of w.

Step 2. For the time being, we do not have any estimate of the norm ofw. This is the content of what follows. Let us recall that λ is chosen as in

89

Proposition 4.2 or Corollary 4.2. We are going to show that there exists aconstant c > 0 independent of f1 and ε such that the solution w constructedabove satisfies

supB1\Bλε

r−µ

(r2

ε2|wr|+ |wi|

)+ ε−µ |δ±1| ≤ c sup

B1\0r2−µ |f1|, (4.39)

where, as usual we have decomposed w := wr + i wi. We do not have a directargument to derive this estimate, this is the reason why the proof of this resultis by contradiction. If the above estimate were not true, there would existsequences εj , fj and wj a sequence of solutions of

Lεj wj = fj in B1 \ 0<(wj) = 0 on ∂B1,

which are constructed above, for which

supB1\0

r2−µ |fj | = 1.

and

Aj := supB1\Bλεj

(r−µ (

r2

ε2j

|wj,r|+ |wj,i|))

+ ε−µj |δ±1,j | (4.40)

tends to +∞. We claim that

Cj := ε−µj |δ±1,j |,

tends to +∞. To see this, let us decompose wj := vj + δ+1,j Ψ+1ε + δ−1,j Ψ−1

ε .Thanks to (4.38), we already know that the corresponding sequence of functionsvj is bounded by a constant times rµ and also that ∇vj is bounded by a constanttimes rµ−1 in Bλεj \ 0, all bounds being uniform with respect to j. Hence,applying the result of Corollary 4.2, we see that there exists c > 0, which doesnot depend on j, such that

Aj ≤ c (1 + Cj). (4.41)

Observe that we always have Cj ≤ Aj . This former inequality clearly showsthat Cj also tends to +∞.

We define the sequences of rescaled functions

wj := C−1j ε−µ

j wj(εj ·) and fj := C−1j ε2−µ

j fj(εj ·).

Then we have L1wj = fj . Moreover wj is bounded by a constant times r inBλ \ 0 and, thanks to (4.41), we see that, for all ζ > λ, wj is bounded bya constant times rµ in Bζ \ Bλ, all bounds being uniform with respect to j.Finally fj tends to 0 on every compact subset of C \ 0.

90

By construction, we have the decomposition of wj

wj := vj + δ±1,j Ψ±1εj

.

And we have the corresponding decomposition of wj

wj := vj + C−1j ε−µ

j δ±1,j Ψ±11 .

By virtue of Corollary 4.2, vj is uniformly bounded by a constant times rµ

on every B1/εj\ Bλ, and using (4.38) it is easy to see that vj tends to 0 on

every compact subset of C \ 0. Hence, up to a subsequence, we may assumethat wj converges to a nontrivial linear combination of Ψ+1

1 and Ψ−11 which,

by Corollary 4.2is bounded by a constant times rµ in C \ Bλ. But all nonzerolinear combinations of Ψ±1

1 blow up exponentially at ∞. This is the desiredcontradiction, so the proof of (4.39) is complete.

Step 3. Up to now, we have proven the existence of w solution of (4.35),which can be decomposed as

w = v + δ+1Ψ+1ε + δ−1Ψ−1

ε ,

in Bλε \ 0. Moreover, we know that

supBλε\0

r−µ|v| + supB1\Bλεr−µ

(r2

ε2|wr|+ |wi|

)

+ ε−µ|δ±1| ≤ c supB1\0

r2−µ|f1|,(4.42)

where as usual w := wr + i wi.

We now add to this solution a suitable linear combination of Φ±1ε and Ψ±1

ε

in order to ensure that we obtain a solution with 0 boundary data on ∂B1. So,we are looking for d±1, δ±1 ∈ R such that

w(eiθ) = d±1 Φ±1ε (eiθ) + δ±1 Ψ±1

ε (eiθ).

The existence of these parameters follows easily from Lemma 3.6, which yields

Φ+1ε = −i ε sin θ +O(ε3), Φ−1

ε = i ε cos θ +O(ε3),

and also

Ψ+1ε = c+1 J+

1 (1/ε) (cos θ +O(ε2)), Ψ−1ε = c−1 J+

1 (1/ε) (sin θ +O(ε2)),

on ∂B1. Furthermore, we also get the existence of a constant c > 0 such that

ε−2 J+1 (1/ε) |δ±1|+ ε |d±1| ≤ c sup

∂B1

|w|. (4.43)

Observe in the first part of this estimate, the factor ε−2 which comes from thefact that, on the one hand we already have <w = 0 on ∂B1 and on the otherhand we have <(Φ±1

ε ) = O(ε3) on ∂B1.

91

We setw1 := w − δ±1 Ψ±1

ε − d±1 Φ±1ε ,

which is the solution to our problem. To complete the proof, let us check thatthe relevant estimates hold for w. We set

d±1 := ε(δ±1 − δ±1 − d±1).

Obviously, (4.42) together with (4.43) yields

|d±1| ≤ c supB1\0

r2−µ |f1|. (4.44)

Now, we may decompose w in B1 \ 0 as

w1 := v1 +1ε

d±1 Φ±1ε .

Using once more (4.43), we get

∣∣∣δ±1 Ψ±1ε

∣∣∣ ≤ c ε2 J+(r)J+

1 (1/ε)≤ c ε2 rµ−2,

in B1 \Bλε. Moreover, using (4.42) as well as Lemma 3.6, we also get∣∣δ±1 Φ±1

ε

∣∣ ≤ c εµ+1 r−1 ≤ c ε2 rµ−2,

in B1 \Bλε. Similarly, using (4.43) we obtain∣∣∣δ±1 Φ±1

ε

∣∣∣ ≤ c ε2 rµ−2,

Collecting the last estimates, (4.42) together with the identity

v1 = w − δ±1 Ψ±1ε − (δ±1 − δ±1)Φ±1

ε ,

we conclude that

supB1\Bλε

r−µ

(r2

ε2|v1,r|+ |v1,i|

)≤ c sup

B1\0r2−µ |f1|. (4.45)

Now, we notice that, by construction, we have

|Ψ±1ε − Φ±1

ε | ≤ cr2

ε2,

in Bλε \ 0, for some constant independent of ε. Using, one last time, (4.42)and (4.43) together with the identity

w1 = v + (δ±1 − δ±1) (Ψ±1ε − Φ±1

ε ) +1ε

d±1 Φ±1ε ,

92

we getsup

Bλε\0r−µ |v1| ≤ c sup

B1\0r2−µ |f1|. (4.46)

This ensures the existence of w1 as well as the desired decomposition. Further-more the relevant estimates hold thanks to (4.44)-(4.46), since all the estimatesfor the derivatives follow from rescaled Schauder’s estimates. 2

Eigenspace corresponding to n = ±2. In this final step, we want to solve Lεw2 = f2 in B1 \ 0

w2 = 0 on ∂B1,(4.47)

when w2, f2 ∈ Spanh±2 e±2iθ. For fix ε, the existence of a solution is clearand, as the proof of Step 1, follows from Corollary 3.3, which states that, whenrestricted to the space of functions spanned by e±2iθ, the operator Lε is injective.We will therefore devote ourselves to the derivation of the correct estimates.

Lemma 4.7 Assume that µ ∈ (1, 2). There exist some constants ε0 > 0 andc > 0 such that, for all f2 ∈ E0,α

µ−2 ∩ Spanh±2 e±2iθ and for all ε ∈ (0, ε0),there exists a unique w2 solution of (4.47) which can be decomposed as

w2 = v2 + εµ η(·/ε) (δ−2 Ψ−2ε + δ+2 Ψ+2

ε ),

where v2 ∈ E2,αµ (B1 \ 0) and δ±2 ∈ R. In addition

|δ−2|+ |δ+2|+ ||v2||E2,αµ

≤ c ||f2||E0,αµ−2

.

Proof : We decompose w2 := wr + i wi. For the time being, let us show thatthere exists a constant c > 0 independent of f2 and ε such that

supB1\0

r2−µ(ε−2 |wr|+ inf(r−2, ε−2) |wi|

) ≤ c supB1\0

r2−µ |g2|. (4.48)

Observe that this result is weaker than the result we would like to prove. Theproof of these estimates is again by contradiction. Let us assume that the resultis not true. Since these estimates are easily seen to be true if ε stays boundedaway from 0, there would exist a sequence of solutions of Lεj wj = gj , withεj → 0, such that

Aj := supB1\0

r2−µ(ε−2

j |wj,r|+ inf(r−2, ε−2j ) |wj,i|

),

tends to +∞, whilesup

B1\0r2−µ |gj | = 1.

Furthermore, wj , gj ∈ Spanh±2 e±2iθ.We define

Cj := supBλεj

\0ε−2j r2−µ |wj | = +∞,

93

and we choose λ as in Proposition 4.1. Thanks to Corollary 4.1, we know that

Cj ≤ Aj ≤ c (1 + Cj),

where the constant c does not depend on j. In particular, this implies that Cj

tends to +∞. Let zj ∈ Bλεj\ 0 be a point where the above supremum is

achieved (observe that the supremum is always achieved since wj is bounded atthe origin and rµ−2 blows up at 0). We define

wj := A−1j |zj |2−µ ε−2

j wj(|zj |·),and

gj := A−1j |zj |4−µ ε−2

j gj(|zj |·).We also set rj := λ εj/|zj | and rj := 1/|zj |. Then Lεj/|zj |wj = gj . Moreoverwj is bounded by rµ−2 in Brj \ 0 and, thanks to Corollary 4.1, we also knowthat wj is bounded by a constant times rµ in Brj

\ Brj. Finally gj tends to 0

on every compact subset of C \ 0.Up to a subsequence we may assume that εj/|zj | converges to ε∞ ∈ [1/λ, +∞].

We now distinguish two cases according to the value of ε∞.

1 - If ε∞ = +∞, then after the extraction of subsequences (if this is neces-sary) we obtain a nontrivial solution of

∆(eiθw) = 0 in C \ 0,which is bounded by a constant times rµ−2 and which belongs toSpanh±2 e±2iθ. We decompose

w = a e2iθ + b e−2iθ.

Obviously, a is a linear combination of r3 and r−3 while b is a linear combinationof r and r−1. Inspection of the behavior at both the origin and ∞ shows thatw cannot be bounded by a constant times rµ−2 since µ− 2 ∈ (−1, 1).

2 - If ε∞ < +∞, then after the extraction of subsequences, if this is necessary,we obtain a nontrivial solution of

Lε∞w = 0 in C \ 0,which is bounded by a constant times rµ−2 if r ≤ ε∞ and by a constant timesrµ if r ≥ ε∞. Again, we decompose

w = a e2iθ + b e−2iθ.

Using the result of Proposition 3.1 and the fact that µ − 2 ∈ (−1, 1), we seethat w is bounded by a constant times r near 0. Finally using the result ofProposition 3.2 together with the fact that µ < 2, we see that w decays at leastlike r−2 near ∞. Therefore w is a nontrivial bounded solution of Lε∞w = 0 inC. But this would contradict the result of Theorem 3.1.

94

Since we have reached a contradiction in all the cases, the proof of (4.48)is complete. The next step consists in proving that one can choose δ±2 ∈ R insuch a way that

supBε\0

r−µ∣∣w − εµ δ±2 Ψ±2

ε

∣∣ ≤ c supB1\0

r2−µ |f2|, (4.49)

for some constant c > 0 independent of ε and f2. In order to prove such a result,we write w = a e2iθ + b e−2iθ and f2 = α e2iθ + β e−2iθ. We have already seenthat, there exists a0, b0 ∈ C such that the functions a and b are solutions of

a = a0 r3 − r3

∫ ε

r

s−7

∫ s

0

t4(

S2ε

ε2b− (1− 2S2

ε )ε2

a + α

)dt ds

b = b0 r + r

∫ r

0

s−3

∫ s

0

t2(

S2ε

ε2a− (1− 2S2

ε )ε2

b + β

)dt ds.

Going back to the definition of Ψ±2ε , we see that it suffices to chose

δ+2 + i δ−2 = ε1−µ b0.

Since we already know that

supBλε\0

ε−2 r2−µ |w| ≤ c supB1\0

r2−µ |f2|,

we easily find that

ε3−µ |a0|+ ε1−µ |b0| ≤ c supB1\0

r2−µ |f2|.

This proves the existence of δ±2 such that (4.49) holds. Moreover, we also have

|δ±2| ≤ c supB1\0

r2−µ |f2|.

In order to complete the proof of the Lemma, it remains to apply rescaledSchauder’s estimates. 2

The proof of the Proposition is immediate once we have collected the resultsof all the different Lemmas. 2

95

Chapter 5

Families of ApproximateSolutions with PrescribedZero set

In this Chapter, we construct finite dimensional families of approximate solu-tions of the Ginzburg-Landau equation, which have prescribed zero set. In orderfor our approximate solutions to be close enough to true solutions of (1.1), wecrucially will use the fact that the points a1, . . . , aN corresponding to the zeroset of our solutions have to be critical points of the renormalized energy Wg

which is defined in (1.10). Finally, we derive some estimates which, in somesense, measure how close our approximate solution is from a true solution.

5.1 The approximate solution u

To begin with, we give a list of notation we will keep in the next Chapters.Then, we define the approximate solutions near its zero set. We end up thissection by the definition of the approximate solution away from the zero set.

5.1.1 Notations

Assume we are giveng : ∂Ω −→ S1,

a C2,α function. We denote by d(g) the topological degree of g. Notice that wedo not necessarily assume that Ω is simply connected. In the case where ∂Ω hasmany connected components, the degree of g is defined by taking into accountthe orientation of the different pieces of ∂Ω.

Assume that we are given N points a1, . . . , aN ∈ Ω. Further assume that,

96

for all j = 1, . . . , N , we are given dj ∈ −1, 1, such thatN∑

j=1

dj = d(g). (5.1)

Finally, let us assume that (aj)j=1,...,N is a nondegenerate critical point of therenormalized energy Wg which has been defined in (1.10), namely we want∇2Wg to be a non degenerate bilinear form in Cn. Recall that Wg implicitlydepends on the choice of (dj)j=1,...,N .

Near aj , we may write the limit solution u?, which is defined in (1.6) as

u? :=(

z − aj

|z − aj |)dj

eiHj , (5.2)

where the function Hj is harmonic in some neighborhood of aj . As alreadymentioned in Chapter 1, it follows from the work of F. Bethuel, H. Brezis andF. Helein [11] that, since we have chosen (aj)j=1,...,N to be a critical point ofWg, we have

∇Hj(aj) = 0.

For all j = 1, . . . , N , it will be convenient to define

τ0j := Hj(aj). (5.3)

In all the subsequent chapters, we set

δ := ε1/2. (5.4)

It will be convenient to adopt polar coordinates about each aj , to this aim,we define rj and θj by

rj := |z − aj | and eiθj :=z − aj

|z − aj | (5.5)

Finally, we will denoteSj := Sε(| · −aj |). (5.6)

For all j = 1, . . . , N , Lε,j will denote the conjugate of the linearized operatorabout Sj ei(djθj+τ0

j ). Let us notice that this operator does not depend on τ0j

but does depend on dj . Thanks to the results of the previous Chapter, themapping properties of Lε,j are well known when dj = +1. Observe that, whendj = −1, it suffices to change θ into −θ to recover all results of Chapter 4 forthe corresponding operator.

Remark 5.1 In order to simplify the notations, we will assume from now onthat all

dj = +1,

and we will do it without further comment. The general case, when dj ∈−1,+1, requires very few modifications though the notations are more in-volved since, at each stage, we have to keep track of the sign of dj.

97

5.1.2 The approximate solution near the zeros

Near each aj , the approximate solution is defined to be a perturbation of thesolution

Sj ei(θj+τ0j ),

since we have assumed that all dj = +1. The perturbation wj will be easy todefine. However, good estimates of this function are quite difficult to derive.This will be the main content of this paragraph.

On each ball B4δ(aj), we define wj to be the solution of Lε,jwj = 0 in B4δ(aj) \ aj

wj = i (Hj − τ0j ) Sj on ∂B4δ(aj).

(5.7)

The existence of wj follows at once from the result of Theorem 4.1. We evenhave some estimates for wj , which are provided by the same Theorem. However,these estimates will not be precise enough for our purposes. Here, we are going totake advantage from the fact that the function Hj is not arbitrary. Indeed, thisfunction is harmonic, moreover we also know that, by assumption ∇Hj(aj) = 0.Hence we can state that

Hj − τ0j ∈ Spanrn

j e±inθj : n ≥ 2.

In particular there exists some constant c > 0 such that

|Hj − τ0j | ≤ c r2

j ,

in B2σ(aj). This information will help us to obtain sharp estimates for wj .In particular, we would like to prove that wj is close to i (Hj − τ0

j )Sj . Moreprecisely, we have the :

Proposition 5.1 There exists some constant c > 0 such that, if wj = wj,r +i wj,i is the decomposition of the solution of (5.7) into its real and imaginarypart, then

|wj,r|+ |wj,i − (Hj − τ0j )Sj | ≤ c

ε r if r ≤ ε

ε2 if ε ≤ r ≤ 4δ.(5.8)

In addition, for all k ∈ N, there exists a constant ck > 0 such that

|∇kwj,r|+ |∇k(wj,i − (Hj − τ0j )Sj)| ≤ ck ε2 r−k if ε ≤ r ≤ 2δ. (5.9)

Proof : For the sake of simplicity in the notations, we will assume that aj = 0and drop the index j. To begin with, we perform a dilation of magnitude 1/δ.We set

wδ := w(δ ·) and Hδ := H(δ ·).

98

Let us recall that, by definition δ2 = ε so we have

Sε/δ = Sδ := S(·/δ).

Now, we have to solve Lδwδ = 0 in B4 \ 0

wδ = i (Hδ − τ0)Sδ on ∂B4.(5.10)

As we have claimed, the existence of wδ (and hence the existence of w itself) fol-lows without difficulty from the result of Theorem 4.1. We therefore concentrateon the derivation of the relevant estimates.

Step 1. Here we construct an auxiliary function whose behavior is wellcontrolled and which is close to wδ. We define the imaginary part of the auxiliaryfunction by

wi := (Hδ − τ0)Sδ. (5.11)

Since S tends to 1 at ∞ we may always assume that λ > 0 is chosen sufficientlylarge so that

1δ2

(1− 3S2δ )− 1

r2≤ − 1

δ2,

in C \Bλδ. In particular, this choice implies that the operator

Λ := ∆ +1δ2

(1− 3S2δ )− 1

r2,

satisfies the maximum principle in B4 \Bλδ. Hence, we are able to solve

Λ wr =2r2

∂θwi in B4 \Bλδ

wr = 0 on ∂B4 ∪ ∂Bλδ.

(5.12)

Since the right hand side of (5.12) is bounded by a constant times δ2, we findthat the constant function c δ4 can be used as a barrier function, provided theconstant c > 0 is chosen large enough. Thus, we conclude that

supB4\Bλδ

|wr| ≤ c δ4. (5.13)

Now that we have proven this first estimate, it follows from (5.12) and the useof rescaled Schauder’s estimates that

supB4\Bλδ

|∇wr| ≤ c δ3. (5.14)

Now, observe that the function ∂θwr satisfies

Λ (∂θwr) =2r2

∂2θ wi in B4 \Bλδ

∂θwr = 0 on ∂B4 ∪ ∂Bλδ.

99

Again, the right hand side is bounded by a constant times δ2 and we conclude,as above, that

supB4\Bλδ

|∂θwr| ≤ c δ4. (5.15)

Using once more rescaled Schauder’s estimates, we also get

supB4\Bλδ

|∇(∂θwr)| ≤ c δ3. (5.16)

We want to improve (5.14). To this aim, we first notice that ∇wr satisfies

Λ (∇wr) = ∇(

2r2

∂θwi

)+∇

(3δ2

S2δ +

1r2

)wr,

in B4\Bλδ. Thanks to (5.13), we see that the second term on the right hand sideof this equation is bounded by a constant times δ4 r−3 and the first term on theright hand side of the same equation in bounded by δ2/r. In addition, thanksto (5.14), we know that ∇wr is bounded by a constant times δ3 on ∂B4 ∪ ∂Bλδ.Increasing λ if necessary, it is now easy to show that, provided the constantc > 0 is chosen large enough, the function

r −→ c (δ4 r−1 + δ3 e(r−4)/δ),

can be used as a barrier function. Therefore, we conclude that, for ε smallenough

supB3\Bλδ

r |∇wr| ≤ c δ4.

Using similar arguments, we prove by induction that, for all k ≥ 0

supB2\Bλδ

rk |∇kwr| ≤ c δ4, (5.17)

where the constant c depends on k. Let us notice that the last two estimatesdo not hold up to ∂B4.

We can now define the real part of auxiliary function by

wr := (1− ηλδ) wr (5.18)

where, η is a cutoff function identically equal to 1 in B1 and equal to 0 outsideB2 and where ηt := η(·/t). Finally, the auxiliary function itself is given by

w := wr + i wi.

Step 2. We now estimate Lδw in the space E0,αl−2, for l ∈ (1, 2). Naturally, in

the definition of E0,αl−2 the parameter ε, which was used in Definition 4.4 has to

be replaced by δ.

100

We have

Fr := ∆wr +1δ2

(1− 3S2δ ) wr − 1

r2wr − 2

r2∂θwi

= −wr ∆η − 2∇wr ∇η − η2r2

∂θwi.

Obviously, Fr is supported in B2λδ and, using (5.17) as well as the definition ofwi, we easily get

supB2λδ

r−1 (|Fr|+ r |∇Fr|) ≤ c δ.

Similarly, using the fact that Hδ is harmonic and also that S is a solution of(2.2) we compute

Fi := ∆wi +1ε2

(1− S2δ ) wi − 1

r2wi +

2r2

∂θwr

= 2∇Sδ ∇Hδ + (1− η)2r2

∂θwr.

This time, using (5.15) and (5.16), we get the bound

supB2λδ

r−1 (|Fi|+ r |∇Fi|) ≤ c δ

supB2\B2λδ

r2 (|Fi|+ r |∇Fi|) ≤ c δ4

supB4\B2

(|Fi|+ δ |∇Fi|) ≤ c δ4.

It is now a simple exercise to check that

‖Lδw‖E0,αl−2

≤ c δ4−l.

Hence we also have ‖Lδ(wδ − w)‖E0,αl−2

≤ c δ4−l. In addition, we know thatwδ − w = 0 on ∂B4. At this point, we apply the result of Theorem 4.1 with εreplaced by δ. Using the fact that both wδ and w do not have any componentcorresponding to the eigenfrequencies n = −1, 0, +1, we see that the coefficientsof Φj

δ, for j = −1, 0, +1 and of Ψ0δ are all equal to 0. Thus, we can already

conclude that

supBδ\0

(δr)−1

(2∑

k=0

rk |∇k(wδ − w)|)≤ c δ2,

while

supB4\Bδ

(δr)2−l

(2∑

k=0

δk |∇k(wδ,r − wr)|)

≤ c δ8−2l

supB4\Bδ

(δr)−l

(2∑

k=0

rk |∇k(wδ,i − wi)|)

≤ c δ4−2l.

101

Performing the scaling backward, and taking into account that ε = δ2, we obtain

supBε\0

r−1

(2∑

k=0

rk |∇k(w − w(·/δ))|)≤ c ε, (5.19)

while

supB4δ\Bε

r2−l

(2∑

k=0

εk |∇k(wr − wr(·/δ))|)

≤ c ε4−l

supB4δ\Bε

r−l

(2∑

k=0

rk |∇k(wi − wi(·/δ))|)

≤ c ε2−l.

(5.20)

Observe that the proof of (5.8) for wr is already complete.

Step 3. In this last step we derive (5.8) for wi as well as (5.9) when k = 1.It follows from (5.20), that

supB4δ\Bε

|∇2wi| ≤ c. (5.21)

Now, observe, as we have already done to derive (5.15) that ∂θwr satisfies

∆(∂θwr) +(

(1− 3S2ε )

ε2− 1

r2

)∂θwr =

2r2

∂2θwi. (5.22)

Moreover, ∂θwr = 0 on ∂B1 and, using (5.20) together with (5.15), we also have|∂θwr| ≤ c ε2 on ∂Bλε. Hence, we conclude that

supB4δ\Bλε

|∂θwr| ≤ c ε2. (5.23)

Moreover, using rescaled Schauder’s estimates together with (5.20), we also get

supB4δ\Bλε

|∇(∂θwr)| ≤ c ε. (5.24)

We are now going to improve (5.20). To this aim, we define

Wi := wi − wi(·/δ).

A simple computation yields

∆Wi +(

(1− S2ε )

ε2− 1

r2

)Wi = − 2

r2∂θwr − 2∇Sε · ∇H, (5.25)

and

∆wr +(

(1− 3S2ε )

ε2− 1

r2

)wr =

2r2

∂θWi − 2r2

∂θwi. (5.26)

Both functions wr and Wi vanish on ∂B1. Furthermore, (5.20) implies that|wr|+ |Wi| ≤ c ε2 on ∂Bλε.

102

Granted (5.23), we see that the right hand side of (5.25) is bounded by aconstant times ε2/r2 and furthermore belongs to

Spanh±n e±inθ : n ≥ 2.Now, the asymptotics of S as r tends to +∞, show that

∣∣∣∣(1− S2

ε )ε2

− 1r2

∣∣∣∣ ≤ cε2

r4≤ c λ−2r−2,

in B4δ \Bλε. Applying the result of Lemma 5.4 which is given at the end of thisChapter (with µ = 0 for example), we see that

supB4δ\Bλε

|Wi| ≤ c ε2,

provided we have chosen λ large enough. In particular this ends the proof of(5.8). Rescaled Schauder’s estimates also yield

supB4δ\Bλε

r |∇Wi| ≤ c ε2,

which proves (5.9) for wi, when k = 1. Finally, making use of (5.24) togetherwith rescaled Schauder’s estimates, we also get

supB4δ\Bλε

r |∇2Wi| ≤ c ε,

Using once more (5.26), we have

∆(∇wr) +(

(1− 3S2ε )

ε2− 1

r2

)∇wr = ∇

(2r2

∂θWi

)−∇

(2r2

∂θwi

)

− ∇(

(1− 3S2ε )

ε2− 1

r2

)wr.

(5.27)Moreover, |∇wr| ≤ c ε on ∂Bλε and |∇wr| ≤ c ε3−l on ∂B4δ. This time, theright hand side of (5.27) is bounded by a constant times r−1. As in the proofof (5.17) we see that

r −→ c (ε2 r−1 + ε3−l e(r−4δ)/ε),

can be used as a barrier function, provided the constant c is chosen large enough.Therefore, we conclude that, for ε small enough

supB3δ\Bλε

r |∇wr| ≤ c ε2.

Which ends the proof of (5.9) for wr, when k = 1.

The proof of (5.9) for general k can be done by induction using the samestrategy. 2

103

5.1.3 The approximate solution away from the zeros

We now devote ourselves to the construction of our approximate solution awayfrom the aj . To this aim, we will solve the semilinear (scalar) equation

∆ξ − |∇u?|2ξ +(1− ξ2)

ε2ξ = 0 in Ωδ/2,

ξ = Sε + wj,r on ∂Bδ/2(aj)

ξ = 1 on ∂Ω.

(5.28)

Though, ξ obviously depends on ε, we will not make this dependence explicitin the notation.

The existence of ξ is fairly easy. However, information about the behaviorof ξ as ε tends to 0 are harder to derive. This is the content of the followingLemma in which we prove that, as ε tends to 0, the behavior of the function ξis qualitatively the same as the behavior of Sε away from 0.

Lemma 5.1 There exists ε0 > 0 such that, for all ε ∈ (0, ε0), the above equationhas a solution which satisfies for all k ≥ 1

1− c ε2 ≤ ξ ≤ 1 in Ωσ

1− cε2

r2j

≤ ξ ≤ 1 in Bσ(aj) \Bδ/2(aj)

|∇kξ| ≤ ck ε2 r−2−kj in B2σ(aj) \Bδ(aj).

(5.29)

Proof : We set ξ := 1 + w. Then, we have to look for w solution of

∆w = |∇u?|2 (1 + w) +w(w + 1)(w + 2)

ε2in Ωδ/2

w = Sε + wj,r − 1 on ∂Bδ/2(aj)

w = 0 on ∂Ω.(5.30)

Given the asymptotics of S at ∞, we see that

−c ε ≤ w ≤ 0,

on each ∂Bδ/2(aj).

To begin with, let us observe that |∇u?|2 ≤ c ε−1 in Ωδ/2. Hence, thefunction

f(t) := |∇u?|2 (1 + t) +t(t + 1)(t + 2)

ε2,

is increasing in [−1/4, +∞), provided ε is chosen small enough. We modifythis function into a function f which is increasing in all R and is equal to f on[−1/4, +∞). We set

F (t) :=∫ t

0

f ds.

104

Let us define w to be the solution of

∆w = f(w) on Ωδ/2

w = Sε + wj,r − 1 on ∂Bδ/2(aj)

w = 0 on ∂Ω,

(5.31)

which is obtained by minimizing the functional

E(ξ) :=∫

Ωδ/2

|∇ξ|2 +∫

Ωδ/2

F (ξ),

among all functions ξ ∈ H1(Ωδ/2) satisfying the correct boundary conditions.

The maximum principle, together with the fact that f > 0 in [0, +∞), implythat w cannot achieve a positive maximum. Hence we already have

w ≤ 0.

Now, provided the constant c > 0 is chosen large enough

w0 := −c ε2N∑

j=1

r−2j ,

satisfies

∆w0 ≥ f(w0) in Ωδ/2

w0 ≤ w on ∂Ωδ/2.

Hence ∆(w − w0) ≤ f(w)− f(w0). The function f being increasing, the maxi-mum principle implies that w−w0 cannot achieve a negative minimum in Ωδ/2.Hence

w0 ≤ w ≤ 0,

in Ωδ/2. In particular w ≥ −1/4, for ε small enough, and therefore w is asolution of (5.30) which satisfies the first two inequalities of (5.29).

It remains to prove the estimates concerning the gradient of w in eachB2σ(aj) \ Bδ(aj). The proof of these is similar to what has already been donein the proof of Proposition 5.1. To simplify the notations, let us assume thataj = 0. Notice that w satisfies

∆w − 2w

ε2= |∇u?|2(1 + w) +

(3w2 + w3)ε2

.

Hence, it follows from rescaled Schauder’s estimates that |∇w| is bounded by aconstant times ε r−2 in B4σ \Bδ/2. Now, we may write

∆(∇w)− 2∇w

ε2= ∇

(|∇u?|2(1 + w) +

(3w2 + w3)ε2

).

105

Using the estimate we have already proved, it is easy to see that the right handside of this last equation is bounded by a constant times r−3 in B4σ \ Bδ/2. Itis then a simple exercise to prove that, provided ε is chosen small enough, thefunction

r −→ c (ε2 r−3 + e(δ/2−r)/ε + ε e(r−4σ)/ε),

can be used as a barrier function to prove that ∇w is bounded by ε2 r−3 inB3σ\B2δ/3. The estimates for higher derivatives are then obtained by induction,following the same strategy. 2

In addition to the above result, we will also need the following Lemma whichprovides an estimate which is not as good as the one we have just proved butwhich turns out to be valid up to the boundary of Ω.

Lemma 5.2 Let ξ be defined as above, then, for all k ≥ 1, there exists a con-stant ck > 0, independent of ε, such that

supΩσ

|∇kξ| ≤ ck ε2−k. (5.32)

Proof : The result has been obtained in the course of the proof of the previousLemma. 2

In the next Proposition, we compare the function ξ with Sε.

Proposition 5.2 For all k ≥ 0, there exists ck > 0, independent of ε, such that

|∇k (ξ − Sj)| ≤ ck ε2 r−kj , (5.33)

in B2σ(aj) \Bδ(aj).

Proof : As usual, we assume that aj = 0 and drop the indices j, just to simplifythe notations. Now, in B4σ \Bδ/2, we have

|∇u?| = |∇(θ + H)|.

Hence, the function ξ satisfies

∆ξ − |∇(θ + H)|2 ξ +ξ

ε2(1− ξ2) = 0,

while the function Sε solves

∆Sε − |∇θ|2 Sε +Sε

ε2(1− S2

ε ) = 0.

Therefore, we obtain for the difference D := ξ − Sε

∆D −(

ξ

ε2(Sε + ξ)− (1− S2

ε )ε2

+ |∇θ|2)

D = ξ∇(2θ + H) · ∇H. (5.34)

106

By definition of ξ, we have from (5.8)

|D| ≤ c ε2 on ∂Bδ/2.

Moreover, it follows from (5.29) that

|D| ≤ cε2 on ∂B4σ.

Finally, let us remark that the right hand side of (5.34) is bounded by someconstant c > 0 independent of ε and that the potential is bounded from belowby −1/ε2, provided ε is small enough. Then, the constant function c ε2 can beused as a barrier function in B4σ \Bδ/2, provided the constant c is chosen largeenough. This proves (5.33) for k = 0. Using rescaled Schauder’s estimates wealso find that

|∇D| ≤ c ε on ∂B3δ/4 ∪ ∂B3σ.

Now, ∇D satisfies an equation similar to (5.34) where, this time, the righthand side is bounded by a constant times r−1. It is then easy to see that, for εsmall enough, the function

r −→ c

(ε2

r+ ε e(r−3σ)/ε + ε e(3δ−4r)/(4ε)

),

can be used as a barrier function to prove that

supB5σ/2\B4δ/5

r |∇D| ≤ c ε2.

Hence, the proof of (5.33) is complete in the case where k = 1. The general casecan then be obtained by induction. 2

In our last Lemma, we want to show that the function

wj := (ξ − Sj) + i (Hj − τ0j )Sj ,

is almost a solution of the homogeneous problem Lεw = 0 in the region whereit is defined. More precisely we have

Lemma 5.3 For all k ≥ 0, there exists some constant ck > 0 such that

|∇k(Lεwj)| ≤ ck r2−kj ,

in B2σ(aj) \Bδ(aj). Where, by definition

wj := ((ξ − Sj) + i (Hj − τ0j )Sj).

Proof : As usual, let us assume that aj = 0 and drop the index j. We set

F := Lε((ξ − Sε) + i (H − τ0) Sε).

107

Going back to the definition of Lε and using the equations satisfied by ξ andSε, we find that

F = |∇(θ + H)|2 ξ − ξ

ε2(1− ξ2)− |∇θ|2 Sε +

Sε

ε2(1− S2

ε )

+(1− 3S2

ε )ε2

(ξ − Sε)− 1r2

(ξ − Sε) + i2r2

∂θ(ξ − Sε)

+ i

(∆((H − τ0)Sε) +

Sε

ε2

(1− S2

ε −ε2

r2

)(H − τ0)

)− 2

r2Sε∂θH.

Using the fact that H is harmonic, and hence ∆H = 0, we can rearrange thisexpression into

F = |∇H|2 ξ +1ε2

(2Sε + ξ) (ξ − Sε)2 +2r2

(ξ − Sε) ∂θH

+ i

(2∇H∇Sε + (H − τ0)∆Sε +

2r2

∂θ(ξ − Sε))

+ iSε

ε2

(1− S2

ε −ε2

r2

)(H − τ0).

Now, the estimates follow at once from the estimates of Lemma 5.33 and fromthe expansions of Theorem 3.1. 2

5.2 A 3N dimensional family of approximate so-lutions

Thanks to the above work, we are now in a position to define our 3N dimensionalfamily approximate solutions. Moreover, we will derive estimates which measurehow far u is from an exact solution.

5.2.1 Definition of the family of approximate solutions

As usual, η denotes a regular cutoff function such that η = 1 in B1 and η = 0in C \B2. For all t > 0, we set

ηt := η(·/t).

andηt,j := η((· − aj)/t).

First we define u in each B2σ(aj). To simplify the notations, we will assumethat aj = 0 and drop the index j.

1 - In B2ε, we set

u :=(ηε (Sε + w) + (1− ηε) (Sε + wr) ei

wiSε

)ei(θ+τ0), (5.35)

108

where the function w is the one defined in (5.7).

2 - Next, the function u is defined in B2σ \B2ε by the formula

u := (ηδ (Sε + wr) + (1− ηδ) ξ) ei (θ+ηδ (wiSε

+τ0)+(1−ηδ) H). (5.36)

Finally, in Ω2σ, we simply define

u := ξ u?. (5.37)

Let us recall that, by definition, u? := ei(θ+H) in B2σ.

Frequently, it will be convenient to write

u := |u| eiφ,

where the real valued function φ is locally well defined away from the points aj .

Once u is defined, we have to define a 3N dimensional family of approximatesolutions, which are obtained from u by changing the position of the zeros of u orslightly changing the phase of u near each aj . More precisely, given d±1 ∈ RN ,we define a family of diffeomorphism

ϕd±1 : Ω −→ Ω,

by

ϕd±1 := z +N∑

j=1

ησ,j (d+1,j + i d−1,j). (5.38)

Similarly, given d0 ∈ RN , we define a regular family of real valued functions ψd0

by

ψd0 :=N∑

j=1

ησ,j d0,j . (5.39)

The 3N dimensional family of approximate solutions is then given by

ud0,d±1 := eiψd0 u ϕd±1 . (5.40)

When d0 = 0 and d±1 = 0, we will simply write u instead of u0,0.

Observe that, by construction, the zeros of ud0,d±1 are precisely the pointsaj − (d+1,j + i d−1,j).

5.3 Estimates

We have already defined

Nε(u) := ∆u +u

ε2(1− |u|2).

109

Here our aim will be to derive a precise estimates of Nε(u) which, in somesense, measures how far u is from an exact solution of our problem.

Let Lε denote the linearized Ginzburg-Landau operator about u, namely

Lε w := ∆w +w

ε2(1− |u|2)− 2u (w · u).

We will also provide a precise estimate of the norm of Lε (Dd0 u) and ofLε

(Dd±1 u

). We will need these estimates because, in the next Chapters, both

Dd±1 u and Dd0 u will play, for the operator Lε, the role played by the functionsΦj

ε for the operator Lε.

Proposition 5.3 Assume that µ ∈ (1, 2). There exists some constant c > 0independent of ε such that

∥∥∥e−iφNε(u)∥∥∥

E0,αµ−2

≤ c δ2−2α. (5.41)

Moreover, when the norms are restricted to B2σ(aj), we also have∥∥∥e−iφNε(u)

∥∥∥E0,α

µ−2

+ δ∥∥∥∇(e−iφNε(u))

∥∥∥E0,α

µ−2

≤ c δ4−µ. (5.42)

and

δ∥∥∥e−iφ Lε(Dd±1 u)

∥∥∥E0,α

µ−2

+∥∥∥e−iφ Lε(Dd0 u)

∥∥∥E0,α

µ−2

≤ c δ4−µ. (5.43)

Proof : We devote ourselves to the derivation of (5.41) the proof of the otherestimates being essentially the same.

Step 1. Derivation of the estimate in B2ε(aj). For the sake of simplicity weassume that aj = 0 and drop the indices j in the notations. We have defined

u =(ηε (Sε + w) + (1− ηε) (Sε + wr) ei

wiSε

)ei(θ+τ0),

We claim that we can also write this identity as

u = (Sε + w) ei(θ+τ0),

in all B2ε, where the difference w− w is supported in B2ε \Bε and satisfies, forall k ≥ 1

|∇k(w − w)| ≤ ck ε4−k.

Indeed, to obtain this result is suffices to expend in B2ε \Bε

(Sε + wr) eiwiSε = Sε + wr + i wi +O(|w|2),

and the claim follows from (5.8) and (5.9).

110

With this notation, we find that

e−i (θ+τ0)Nε(u) = Lεw − w + 2Sε

ε2|w|2 − Sε

ε2w2,

in B2ε. Here, we have used the fact Sε is a solution of (2.2). Observe that, sincew solves (5.7) we find, Lεw = 0 in Bε, while, in B2ε \Bε, we have

Lεw = Lε(w − w),

which can be estimated by a constant times ε2 and whose k-th derivative isbounded by a constant times ε2−k.

Using once more (5.8) and (5.9), we conclude

supB2ε\0

r−3 (|Nε(u)|+ r |∇Nε(u)|) ≤ c ε−1. (5.44)

Step 2. We now turn to the evaluation of Nε(u) in B2δ \B2ε. To this aim itwill be convenient to write u in the form

u = (Sε + wr) ei (θ+τ0+wiSε

),

wherewr := ηδ wr + (1− ηδ) (ξ − Sε),

and wherewi

Sε:= ηδ

wi

Sε+ (1− ηδ) (H − τ0).

Observe that the function w − w has compact support in B2δ \Bδ.

An easy computation leads to

e−iφNε(u) = ∆wr +wr

ε2(1− 3S2

ε )− 1r2

wr − 2r2

∂θwi

+ i

(Sε ∆

(wi

Sε

)+ 2∇

(wi

Sε

)∇Sε +

2r2

∂θwr

)

+ 2 i∇wr∇(

wi

Sε

)+ i ∆

(wi

Sε

)wr − 2

r2

wr

Sε∂θwi

−∣∣∣∣∇

(wi

Sε

)∣∣∣∣2

(wr + Sε)− w2r

ε2(3Sε + wr).

The second line can be simplified since

Sε ∆(

wi

Sε

)+ 2∇

(wi

Sε

)∇Sε = ∆wi − ∆Sε

Sεwi.

Finally, using the fact that S solves (3.3), we conclude that

e−iφNε(u) = Lε w + 2 i∇wr∇(

wi

Sε

)+ i ∆

(wi

Sε

)wr − 2

r2

wr

Sε∂θwi

−∣∣∣∣∇

(wi

Sε

)∣∣∣∣2

(wr + Sε)− w2r

ε2(3Sε + wr).

111

By constructionLεw = 0,

in B4δ and w = w in Bδ \ B2ε. Hence we also have Lεw = 0 in this set. Using(5.8), (5.9) as well as (5.33), we can evaluate the nonlinear terms in the previouscomputation and we conclude that

supBδ\B2ε

r−2 (|Nε(u)|+ r |∇(Nε(u))|) ≤ c.

We setw := (ξ − Sε) + i (H − τ0) Sε.

Since w = η w + (1− η) w, we may now write

Lεw = (w − w)∆η + 2∇(w − w)∇η + (1− η)Lε(w − w).

Moreover, we have, for all k ≥ 0, the estimate

|∇k(w − w)| ≤ ck δ4−k,

in B2δ \Bδ, which follows from (5.8), (5.9) and from (5.33). This, together withthe result of Lemma 5.3, proves that

supB2δ\Bδ

δ−2 (|Lεw|+ δ |∇(Lεw)|) ≤ c.

Hence, we have

supB2δ\B2ε

r−2 (|Nε(u)|+ r |∇(Nε(u))|) ≤ c. (5.45)

Step 3. In Ω2δ, we have defined

u := ξ u?.

Since ξ is a solution of

∆ξ − |∇u?|2 ξ +ξ

ε2(1− ξ2) = 0,

and, in addition, since we also know that

∆u? = −|∇u?|2u?,

we obtainNε(u) = 2∇u?∇ξ. (5.46)

Observe that we have the identity

∇u? = i u?∇(θ + H).

112

We now take advantage from the fact that

∇θ · ∇Sε = 0,

in B2σ \ 0. Hence, we may write

e−iφNε(u) = 2 i∇(θ + H)∇ξ = i (∇θ∇(ξ − Sε) +∇H∇ξ).

Making use of (5.29) and (5.33), we can estimate

supB2σ\B2δ

r2 (|Nε(u)|+ r |∇(Nε(u)|) ≤ c ε2.

And also from Lemma 5.2

supΩ2σ

(|Nε(u)|+ ε |∇(Nε(u)|) ≤ c ε.

Collecting the last two estimates together with (5.44) and (5.45), it followsat once that

|Nε(u)|E0,αµ−2

≤ c δ2−2α.

This ends the proof of the estimate. 2

5.4 Appendix

Here, we prove the Lemma which is needed in the thirst Step of the proof ofProposition 5.1.

Lemma 5.4 Assume that µ ∈ (0, 2). There exists a constant c > 0 such that,for all r1 > r0 > 0 and for all

w, g ∈ Spanh±n e±inθ : n ≥ 2,

satisfying ∆w + V w = g in Br1 \Br0

w = 0 on ∂Br1 ,

we have

supBr1\Br0

r−µ |w| ≤ c

(sup∂Br0

r−µ |w|+ supBr1\Br0

r2−µ |g|)

,

provided the potential V ∈ R only depends on r and satisfies

sup[r0,r1]

r2 |V | ≤ 4− µ2.

113

Proof : As usual we define w to be the solution of

∆w = 0 in Br1 \Br0

w = 0 on ∂Br1

w = w on ∂Br0 ,

The maximum principle provides the estimate

supBr1\Br0

|w| ≤ sup∂Br0

|w|,

from which it follows that

supBr1\Br0

r−µ |w| ≤ sup∂Br0

r−µ |w|,

since we have assumed µ ≥ 0.

Now, we decompose w − w and g − V w into Fourier series

g − V w =∑

|n|≥2

hn einθ, and w − w =∑

|n|≥2

wn einθ.

Since∆(w − w) + V (w − w) = g − V w,

and w − w = 0 on ∂Br1 ∪ ∂Br0 , we see that, for all |n| ≥ 2, the function wn isa solution of

d2wn

dr2+

1r

dwn

dr− n2

r2wn + V wn = hn,

with wn(r1) = wn(r0) = 0. Observe that by assumption r2 |V | ≤ 4 and hence,the maximum principle holds for the operator

d2

dr2+

1r

d

dr− n2

r2+ V.

Furthermore|hn| ≤ Mrµ−2,

where we have set

M := supBr1\Br0

r2−µ (|g|+ 4r−2 |w|).

Notice that we have used once more the fact that r2 |V | ≤ 4. It is now easy tosee that the function

r −→ M

n2 − µ2 − sup (r2|V |) rµ,

114

can be used as a barrier function to prove that

|wn| ≤ M

n2 − µ2 − sup (r2 |V |) rµ.

Finally, summation over n yields

|w − w| ≤ c rµ.

And the result follows at once. 2

115

Chapter 6

Existence ofGinzburg-Landau Vortices

In this Chapter, we collect the result of the previous Chapter to derive theexistence of Ginzburg-Landau vortices. More precisely, given a nondegeneratecritical point of the renormalized energy, we prove the existence of a branchof solutions of the Ginzburg-Landau equation which is parameterized by ε andwhich is close to the approximate solution u which has been constructed inChapter 5. To obtain this branch of solutions, we rephrase our problem as afixed point problem. As a simple Corollary of our construction, we will obtaina local uniqueness result for the solution of the Ginzburg-Landau equation.

6.1 Statement of the result

The aim of this Chapter is to show that, there exists ε0 > 0 such that, for allε ∈ (0, ε0), we can prove the existence of a unique solution of

Nε(u) = 0,

which is close to the approximate solution u, we have constructed in Chapter 5.

We choose µ ∈ (1, 2). It will be convenient to define the space

E := E2,αµ (Ω \ Σ)⊕N

j=1 Spanεµ ηε,j Ψ0

ε,j , εµ ηε,j Ψ±2

ε,j

,

where as usual η is a cutoff function equal to 1 in B1 and equal to 0 outsideB2, and where we have set

ηε,j := η((· − aj)/ε),

andΨk

ε,j := Ψkε(· − aj).

116

Naturally, the space E is endowed with the norm

‖v + εµ∑

j

ηε,j(δ0,j Ψ0ε,j + δ±2,j Ψ±2

ε,j )‖E := ‖v‖E2,αµ

+ |δ0|+ |δ±2|.

We also renameF := E0,α

µ−2(Ω \ Σ).

We define

χε :=N∑

j=1

ηε,j .

Recall that we have decomposed the approximate solution u as

u := eiφ |u|.

Now, for all d0 ∈ RN , d±1 ∈ RN and w ∈ E , we define

u(d0, d±1, w) := eiψd0

(eiφ

((|u|+ wr) ei(1−χε)

wi|u| + i χε wi

)) ϕd±1 . (6.1)

Where we recall that we have defined in (5.38) and in (5.39)

ψd0 :=N∑

j=1

ησ,j d0,j .

and

ϕd±1 := z +N∑

j=1

ησ,j (d+1 + i d−1).

Observe that, when w = 0, we simply have

u(d0, d±1, 0) = eiψd0(u ϕd±1

)= ud0,d±1 .

Let us briefly comment on the structure of u(d0, d±1, w). Notice that, to firstorder, that is for d0, d±1 and w small, we have

u(d0, d±1, w) ∼ u + eiφ

w +

∑

j

(d0 Φ0ε,j +

1ε

d±1,j Φ±1ε,j )

. (6.2)

The presence of eiφ is justified by the fact that, when linearizing the nonlinearGinzburg-Landau operator about u, we want to find an operator close to Lε,which is the conjugate linearized operator. Now, the analysis of the nonlinearterms which appear in Nε, shows that the expression of u(d0, d±1, w) given by(6.1) is more adapted than the simpler expression given by (6.2). To be moreprecise, the structure of u(d0, d±1, w) is intimately linked to or analysis of thelinearized operator as well as the definition of the spaces we are working with.

117

Finally we define the nonlinear mapping

M(d0, d±1, w) := e−iφ e−i(1−χε)wi|u|

(e−iψd0 Nε (u(d0, d±1, w))

) ϕ−1d±1

.

Again, let us briefly try to justify the rather complicated expression of thisnonlinear mapping. As above, the presence of e−iφ is justified by the fact that,when linearizing the nonlinear Ginzburg-Landau operator about u, we want tofind an operator close to Lε, which is the conjugate linearized operator. Finally,it turns out that the presence of e−i(1−χε)

wi|u| is needed to improve the estimates

of some nonlinear terms.

For the time being, let us observe that M is well defined, and even C1, froma neighborhood of (0, 0, 0) in RN × (RN × RN )× E into F .

Theorem 6.1 Assume that µ ∈ (1, 2). There exist R > 0 and ε0 > 0, suchthat, for all ε ∈ (0, ε0), there exists a unique (d0, d±1, w) solution of

M(d0, d±1, w) = 0,

in the ball of radius R of RN × (RN × RN ) × E. Moreover, there exists c > 0,such that for all ε ∈ (0, ε0), the unique solution (d0, d±1, w) satisfies

|d0|RN + |d±1|RN×RN + ‖w‖E ≤ c ε1−α.

Before we proceed to the proof of the Theorem, let us rephrase the result as :

Theorem 6.2 Assume that dj = ±1, for all j = 1, . . . , N . Let (ai)i=1,...N

be a nondegenerate critical point of Wg. Then, there exists ε0 > 0 and forall ε ∈ (0, ε0) there exists a solution uε of (1.5) with exactly N isolated zerosa1(ε), . . . , aN (ε) such that :

limε→0

aj(ε) = aj ,

andlimε→0

uε = u? in C2,αloc (Ω \ Σ).

Where Σ := a1, . . . , aN and where u? is given by (1.6).

Hence, the proof of Theorem 1.9 is complete. As a byproduct of our construc-tion, we obtain some information about the functions uε. For example, we havethe :

Proposition 6.1 For all α′ > 0 there exists a constant cα′ > 0 independent ofε ∈ (0, ε0) such that

|aj(ε)− aj | ≤ cα′ εα′ . (6.3)

118

Further properties of uε will be proven in the last section of this Chapter.

For the time being let us write Taylor’s formula. It turns out to be easier forthe estimates we will have to perform, not to use the usual formula but rather

M(d0, d±1, w) = M(0, 0, 0) + DM(0,0,0)(d0, d±1, w)

+∫ 1

0

(DM(td0,td±1,0) −DM(0,0,0)

)(d0, d±1, 0) dt

+(DM(d0,d±1,0) −DM(0,0,0)

)(0, 0, w)

+∫ 1

0

(DM(d0,d±1,tw) −DM(d0,d±1,0)

)(0, 0, w) dt.

Observe thatM(0, 0, 0) = e−iφNε(u).

For the sake of simplicity, we will denote the nonlinear terms by

Q1(d0, d±1) :=∫ 1

0

(DM(td0,td±1,0) −DM(0,0,0)

)(d0, d±1, 0) dt,

Q2(d0, d±1, w) :=(DM(d0,d±1,0) −DM(0,0,0)

)(0, 0, w),

and

Q3(d0, d±1, w) :=∫ 1

0

(DM(d0,d±1,tw) −DM(d0,d±1,0)

)(0, 0, w) dt.

The reason why we have performed this rather unusual Taylor’s expansion isthat we want to avoid mixed second order differential of the form Dd0DwM orDd±1DwM evaluated at some point (d0, d±1, w) with w 6= 0, since these are noteasy to estimate.

Clearly, in order to prove Theorem 6.1, it is sufficient to find a fixed pointfor the mapping

(d0, d±1, w) −→ − (DM(0,0,0)

)−1 (M(0, 0, 0) + Q1(d0, d±1)

+Q2(d0, d±1, w) + Q3(d0, d±1, w)).

Now, the strategy of the proof of the different results of this Chapter is as follows: First, we show that DM(0,0,0) is close to Lε and thus, it is an isomorphism fromRN×(RN×RN )×E into F . Next we prove that the three nonlinear mappings Qk

are contraction mappings when restricted to a suitable neighborhood of (0, 0, 0)in RN × (RN × RN )× E .

119

6.2 The linear mapping DM(0,0,0)

We have an explicite expression for DM(0,0,0), namely

DM(0,0,0)(d0, d±1, w) = Lε

w +

N∑

j=1

(d0,j Φ0ε,j + d±1,j Φ±1

ε,j )

− e−iφ∇Nε(u)∇(ϕd±1 − z)

− i e−iφNε(u)(

ψd0 + (1− χε)wi

|u|)

.

(6.4)

We can now prove the :

Proposition 6.2 There exists ε0 > 0 such that, for all ε ∈ (0, ε0), linear oper-ator

DM(0,0,0)(β, b, w) : Rn × (RN × RN )× E −→ F ,

is an isomorphism whose inverse is bounded independently of ε ∈ (0, ε0).

Proof : First, we use (5.41), in Proposition 5.3, to get∥∥∥∥i e−iφNε(u) (1− χε)

wi

|u|

∥∥∥∥F≤ c δ2−2α ‖w‖E .

While, using (5.42), we obtain

‖i e−iφNε(u) ψd0‖F ≤ c δ4−µ ‖d0‖RN ,

and‖e−iφ∇Nε(u)∇(ϕd±1 − z)‖F ≤ c δ3−µ ‖d±1‖RN×RN .

Then, the result follows from a perturbation argument, using the result of The-orem ??. 2

Finally, let us notice that (5.41) also yields the estimate

‖M(0, 0, 0)‖F ≤ c δ2−2α. (6.5)

6.3 Estimates of the nonlinear terms

Here, we estimate in turn the nonlinear terms Qk. We will frequently use thefollowing basic computation for some real or complex valued functions f, h andsome complex valued function g, defined in some domains of C

(e−ih ∆

((f g) eih

))= f g

(i ∆h− |∇h|2) + 2 iDfg(∇g∇h)

+ D2fg(∇g,∇g) + Dfg(∆g).

WhereDfg(∇g∇h) := Dfg(∂xg ∂xh + ∂yg ∂yh),

120

and where

D2fg(∇g,∇g) := D2fg(∂xg, ∂xg) + D2fg(∂yg, ∂yg).

We apply this computation to g := ϕd±1 and h := ψd0 . Recall that, inBσ(aj), we simply have

ϕd±1 = z + (d+1,j + i d−1,j) ησ,j and h := ψd0 = d0,j ησ,j .

For the sake of simplicity in the notations, let us assume that aj = 0, drop allindices and write d1 instead of d+1 + id−1. We obtain

(e−iψ ∆

(eiψ f ϕ

)) ϕ−1 = ∆f + |∇η|2 ϕ−1 D2f (d1, d1)

+ 2 D2f (d1,∇η ϕ−1)

+ 2 i d0 Df (∇η ϕ−1)

+(∆η + 2 i d0 |∇η|2) ϕ−1 Df (d1)

+(i d0 ∆η − d2

0 |∇η|2) ϕ−1 f.

(6.6)

6.3.1 Estimates of Q1

The nonlinear term Q1 is probably the easiest to estimate. This is the contentof the following :

Lemma 6.1 There exists c > 0 such that, for all ε ∈ (0, ε0), we have

‖Q1(d0, d±1)−Q1(d0, d±1)‖F ≤ c l1 (‖d0 − d0‖RN

+‖d±1 − d±1‖RN×RN ),

provided l1 ≤ 1. Here, we have defined

l1 := ‖d0‖RN + ‖d±1‖RN×RN + ‖d0‖RN + ‖d±1‖RN×RN .

Proof : First of all let us notice that M(d0, d±1, 0) does not depend on d0 noron d±1 in Ω2σ ∪N

j=1 Bσ(aj). Therefore, we have

Q1(d0, d±1)−Q1(d0, d±1) = 0,

there. Finally, in B2σ(aj) \Bσ(aj), we have

M(d0, d±1, 0) = e−iφ(e−iψd0 ∆

(eiψd0 u ϕd±1

)) ϕ−1d±1

+|u|ε2

(1− |u|2).

121

Recall that, in B2σ(aj) \Bσ(aj), we have

u := ξ u?.

Then, using (6.6), it is easy to see that all partial differential (to any order) ofM(d0, d±1, 0), with respect to d0 and d±1 are bounded independently of ε. Theresult then follows at once. 2

6.3.2 Estimates of Q2

We now turn to the estimate of Q2. This is slightly more complicated.

Lemma 6.2 There exists c > 0 such that, for all ε ∈ (0, ε0), we have

‖Q2(d0, d±1, w) − Q2(d0, d±1, w)‖F ≤ c l1 ‖w − w‖E+ c l2 (‖d0 − d0‖RN + ‖d±1 − d±1‖RN×RN ),

provided l1 ≤ 1. Here, we have defined

l2 := ‖w‖E2,αµ (Ωσ\Ω2σ) + ‖w‖E2,α

µ (Ωσ\Ω2σ).

Proof : Again, we observe that M(d0, d±1, w) does not depend on d0 nor ond±1 in Ω2σ ∪j Bσ(aj). Therefore, Q2 is identically equal to 0 in this set. Itremains to evaluate Q2 in B2σ(aj) \Bσ(aj). But, in this set, we have

M(d0, d±1, w) = e−iφ e−iwi|u|

(e−iψd0 ∆

(eiψd0 f ϕd±1

)) ϕ−1d±1

+|u|+ wr

ε2(1− (|u|+ wr)2),

where we have definedf := ei(φ+

wi|u| ) (|u|+ wr).

In particular, the partial differential of M with respect to w, computed at(d0, d±1, 0) reads

DwM(d0,d±1,0)(0, 0, w) = e−iφ(e−iψd0 ∆

(eiψd0 (eiφw) ϕd±1

)) ϕ−1

d±1

− iwi

|u| e−iφ

(e−iψd0 ∆

(eiψd0 u ϕd±1

)) ϕ−1d±1

+wr

ε2(1− 3|u|2),

Thus, we obtain

Q2(d0, d±1, w) = e−iφ(e−iψd0 ∆

(eiψd0 (eiφw) ϕd±1

)) ϕ−1

d±1

−e−iφ ∆(eiφw)

− iwi

|u| e−iφ

(e−iψd0 ∆

(eiψd0 u ϕd±1

)) ϕ−1d±1

−iwi

|u| e−iφ ∆u.

122

The result is now an easy consequence of (6.6), if we decompose

Q2(d0, d±1, w)−Q2(d0, d±1, w) = Q2(d0, d±1, w)−Q2(d0, d±1, w)

+ Q2(d0, d±1, w)−Q2(d0, d±1, w).

At first glance, one would expect a term ε−α in front of ‖w‖E , in the state-ment of the result, since we only have

‖∇2wr‖C0,αµ−2

≤ c ε−α ‖wr‖C2,αµ

. (6.7)

But this is not the case. The key observation is that, in the expression of Q2,all terms involving second order partial derivatives of w are of the form

|∇η|2 ϕ−1 D2w (d1, d1) and D2w (d1,∇η ϕ−1).

Hence, the imaginary part of Q2 only involves second partial derivatives of wi

and does not involve any second order partial derivatives of wr. In particularwe do not have to use the estimate (6.7). 2

6.3.3 Estimates of Q3

The derivation of the estimates for Q3 are far more involved than the estimatesof the previous nonlinear quantities. We prove here the Lemma :

Lemma 6.3 There exists c > 0 such that, for all ε ∈ (0, ε0), we have

‖Q3(d0, d±1, w) − Q3(d0, d±1, w)‖F ≤ c l3 ‖w − w‖E

+ c l2

(‖d0 − d0‖RN + ‖d±1 − d±1‖RN×RN

),

(6.8)

provided l1 ≤ 1, l2 ≤ 1 and l3 ≤ 1. Where, by definition

l3 := (‖wi‖C2,α0

+ ‖wr‖C1,α0

) + (‖wi‖C2,α0

+ ‖wr‖C1,α0

).

Proof : The proof of the estimate being rather technical, we have divided it inseveral parts.

Step 1. Preliminary computation. In order to simplify the notations, we willwrite for short

u(w) := (|u|+ wr) + i χε wi e−i(1−χε)wi|u| .

It will also be convenient to decompose M into

M(d0, d±1, w) := MI(d0, d±1, w) +MII(w),

where we have defined

MI(d0, d±1, w) := e−iφ e−i(1−χε)wi|u|

×(e−iψd0 ∆

((eiψd0 eiφei(1−χε)

wi|u|u(w)) ϕd±1

)) ϕ−1

d±1,

(6.9)

123

and

MII(w) :=u(w)ε2

(1− |u(w)|2

). (6.10)

Already observe that MII does not depend on d0 nor on d±1.

Step 2. Study of MII . To begin with, let us simplify the expression of MII

in Bε(aj). In this set χε ≡ 1, hence, the above expressions yield

MII(w) =|u|+ w

ε2

(1− (|u|+ w)2

).

So, we can write

MII(w) = terms independent of w + terms linear in w

− 1ε2

(|u| |w|2 + 2|u|w wr + w |w|2) .

Since the terms which do not depend on w and the terms which are linear in ware irrelevant for the computation of (DMII,tw −DMII,0) (w), we find easily

(DMII,tw −DMII,0) (w) = − 1ε2

(2 t |u| |w|2 + 4 t |u|w wr + 3 t2 w |w|2) .

(6.11)In B2ε(aj) \Bε(aj), the expression for MII is much more involved but does

not introduce any additional difficulty. More precisely, we have

MII(w) = terms independent of w + terms linear in w

−|u|ε2

(w2

r + χ2ε w2

i + 2(|u|+ wr) χε wi sin(

(1− χε)wi

|u|))

− 1ε2

(wr + iχεwie

−i(1−χε)wi|u|

)

×(2|u|wr + w2

r + χ2ε w2

i + 2(|u|+ wr)χε wi sin((1− χε) wi

|u|))

+i

ε2χε wi(1− |u|2) (e−i(1−χε)

wi|u| − 1).

Since the terms which do not depend on w and the terms which are linear in ware irrelevant for the computation of (DMII,tw −DMII,0) (w), we find easilythat

(DMII,tw −DMII,0) (w) =t

ε2Qtw(wr, wi),

where Qtw(·, ·) is a quadratic form whose coefficients are functions bounded inC1,α0 (B2ε \Bε), provided the norm of w is bounded, say by 1, in C1,α

0 (B2ε \Bε).

We finally turn to the expression of MII in Ω2ε. This time χε ≡ 0 and wefind

MII(w) :=|u|+ wr

ε2

(1− (|u|+ wr)2

).

124

So, we can write

MII(w) = terms independent of w + terms linear in w

− 1ε2

(3 |u|w2

r + w3r

).

Since the terms which do not depend on w and the terms which are linear in ware irrelevant for the computation of (DMII,tw −DMII,0) (w), we find easily

(DMII,tw −DMII,0) (w) = − 1ε2

(6 t |u|w2

r + 3 t2 w3r

). (6.12)

At this stage, the key observation is that this quantity only involves wr anddoes not involve any wi. This is important since, in Ωσ, we can bound |wr| ≤c ε2 ‖w‖E , while we only have |wi| ≤ c ‖w‖E . Hence this expressions remainsbounded uniformly in ε if ‖w‖E does.

Step 3. Study of MI . Once again we use the fact that in Ω2σ ∪j Bσ(aj),the mapping MI does not depend on d0 nor on d±1. Hence, in this set, (6.9)reduces to

MI(d0, d±1, w) = e−iφe−i(1−χε)wi|u|∆

(eiφ((|u + wr)e

i(1−χε)wi|u| + iχεwi)

),

Hence, we have

MI(d0, d±1, w) = u(w)(

i∆(φ + (1− χε)wi

|u| )− |∇(φ + (1− χε)wi

|u| )|2

)

+2 i∇u(w)∇(φ + (1− χε)wi

|u| ) + ∆u(w).

With little work, we find

MI(d0, d±1, w) = terms independent of w + terms linear in w

+ 2(i∇wr − wr∇φ

)∇((1− χε)

wi

|u| )

+ iwr ∆((1− χε)wi

|u| )− (|u|+ wr)∣∣∣∣∇((1− χε)

wi

|u| )∣∣∣∣2

+ i e−iφ ∆(eiφ χε wi) (e−i(1−χε)wi|u| − 1).

125

Hence

(DMI,(d0,d±1,tw) − DMI,(d0,d±1,0))(0, 0, w)

= 2 t(i∇wr − wr ∇φ

)∇((1− χε)

wi

|u| )

+ t wr

(i ∆((1− χε)

wi

|u| )−∣∣∣∣∇((1− χε)

wi

|u| )∣∣∣∣2)

− t (|u|+ t wr) |∇((1− χε)wi

|u| )|2

+ i e−iφ ∆(eiφ χε wi) (e−i(1−χε) twi|u| − 1)

+ t (1− χε)wi

|u| e−iφ ∆(eiφ χε wi) e−i(1−χε) t

wi|u| .

(6.13)

At this stage, the key observation is that this quantity does not involve anysecond order partial derivatives of wr. This is important since, in Ωσ, we canbound |∇2wi|C0,α ≤ c ‖w‖E , while we only have |∇2wr|C0,α ≤ c ε−α ‖w‖E . Hencethis expressions remains bounded uniformly in ε if ‖w‖E does.

The expression of MI in B2σ(aj) \Bσ(aj) is much more involved since thistime it depends on d0 and d±1 and we have to use extensively (6.6). However,it does not introduce any new difficulty in the estimate. In particular, the keypoint is that (DMI,(d0,d±1,tw) − DMI,(d0,d±1,0))(0, 0, w) does not contain anysecond order partial derivatives of wr.

Step 4. The result then follows at once from the expressions given in (6.11),(6.12) and (6.13). 2

6.4 The fixed point argument

Clearly, collecting the results of Proposition 6.2, Lemma 6.1, Lemma 6.2 andLemma 6.3, one can show that there exists R > 0 such that for all ε ∈ (0, ε0),the operator

(d0, d±1, w) −→ − (DM(0,0,0)

)−1 (M(0, 0, 0) + Q1(d0, d±1)

+Q2(d0, d±1, w) + Q3(d0, d±1, w)).

is a contraction from the ball of radius R in RN × (RN × RN ) × E into itself.Application of a standard fixed point Theorem for contraction mapping alreadygives the local uniqueness result of Theorem 6.1. Using in addition (6.5), weconclude that, the unique fixed point (d0, d±1, w) satisfies the estimate

|d0|RN + |d±1|RN×RN + ‖w‖E ≤ c ε1−α.

where the constant c is independent of ε ∈ (0, ε0).

126

6.5 Further information about the branch of so-lutions

We finally derive some informations about the branch of solutions we haveconstructed.

To begin with, let us define what we call the density of area of a vectorvalued function u defined in some domain in R2.

Definition 6.5 Assume that u : Ω ⊂ R2 → R2. We set

Area (u) := det(Du). (6.14)

We now give some alternate definitions. For example, if we identify R2 with C,the definition of the area becomes

Area (u) := − i

2(∂xu ∂yu− ∂xu ∂yu) . (6.15)

Using polar coordinates, we obtain

Area (u) := − i

2r(∂ru ∂θu− ∂ru ∂θu) . (6.16)

As a first example, we may compute the density of area corresponding tothe radially symmetric solution of the Ginzburg-Landau equation

uε := Sε eiθ,

which is a model for any solution of the equation near a zero (of degree one).Using (6.16), we find

Area (uε) =1r

Sε∂rSε.

Since ∂rSε > 0, we conclude that

Area (uε) > 0,

in all C. What is more, given the asymptotic behavior of Sε at 0 and at ∞, wesee that there exists a constant c > 0 such that

|Area (u)| ≥ cε2

sup(ε4, r4), (6.17)

in C.

Our purpose is now to study to what extend the previous inequality holdsfor the solutions we have constructed. More precisely, we prove the

127

Proposition 6.3 There exist c > 0, σ0 > 0 and ε0 > 0 such that, for allε ∈ (0, ε0), if uε denotes the solution of Theorem 6.2, then

|Area (uε)| ≥ cε2

sup(ε4, r4),

in each Bσ0(aj), wherer := |z − aj(ε)|,

is the distance from the zero of uε in Bσ(aj).

Proof : For the sake of simplicity, let us assume that aj(ε) = 0, dj = +1 andlet us drop the j indices.

Up to a translation, the solution uε is a perturbation of u in B2σ. In partic-ular, we have

uε = eid0 ∇u +O(ε1−α rµ−1). (6.18)

We may always assume that d0 = 0 since we have the identity

Area (eid0u) = Area (u).

Now, observe that, not only do we have (6.17) but, decreasing the constant c ifnecessary, we also have

cε2

sup(ε4, r4)≤ Area (Sε eiθ) ≤ c−1 ε2

sup(ε4, r4), (6.19)

Step 1. In B2ε, we have using (5.8) and (5.9),

∇u = ∇(Sε ei(θ+τ0)) +O(ε).

Combining this estimate together with (6.18), we find that

Area (uε) = Area (Sε eiθ) +O(1 + ε−α rµ−1)

≥ c ε−2.

Step 2. In B2δ \B2ε, we have using (5.8), (5.9) and (5.33),

∇u = ∇(Sε ei(θ+H)) +O(ε2 r−1).

Combining this estimate together with (6.18), we find that

Area (uε) = Area (Sε ei(θ+H)) +O(ε2 r−2 + ε1−α rµ−2).

Moreover, using (6.16), we can compute explicitely

Area (Sε ei(θ+H)) =1r

(1 + ∂θH)Sε ∂rSε.

128

Hence, we conclude using (6.19) that

Area (uε) = Area (Sε eiθ) +O(ε2 r−2 + ε1−α rµ−2)

≥ ε2 r−4.

Observe that this estimate still holds in B4δ \B2δ.

Step 3. In Bσ \B2δ, we have explicitely

u = ξ u? = ξ ei(θ+H),

anduε = (ξ + wr) ei(θ+H+

wiξ ),

where w is the solution given by the fixed point argument. Hence, we find using(6.16)

Area (uε) =1r

(1 + ∂θH + ξ−1 ∂θwi) (ξ + wr) ∂r(ξ + wr) .

For the time being, we only know that

|wr|+ ε |∂rwr| ≤ c ε1−α rµ,

in B2σ \ B2δ. And this estimate is not sufficient for our purpose. However, wecan improve this estimate using the strategy we have already used in the proofof Proposition 5.2. More precisely, the function uε is a solution of

∆uε +uε

ε2(1− |uε|2) = 0.

Taking the real part of this identity, we obtain

∆wr − |∇(θ + H +wi

ξ)|2 (ξ + wr) +

wr + ξ

ε2(1− |ξ + wr|2) = 0.

Making use of the fact that ξ is a solution of (5.28), we conclude that

∆wr −(|∇(θ + H +

wi

ξ)|2 +

1ε2

(3 ξ2 + 3ξ wr + w2r − 1)

)wr

= (|∇(θ + H +wi

ξ)|2 − |∇θ|2) ξ.

The right hand side of this equation is easily seen to be bounded by a constant,independently of ε. Moreover |wr| ≤ c ε1−α δµ on ∂B2δ and |wr| ≤ c ε1−α on∂B2σ.

If ε is small enough, we can assume that the potential in the above equationsatisfies

(|∇(θ + H +

wi

ξ)|2 +

1ε2

(3 ξ2 + 3ξ wr + w2r − 1)

)≥ 1

ε2.

129

The maximum principle shows that the function

r −→ c (ε2 + ε1−α δµ e(2δ−r)/ε + ε1−α e(r−2σ)/ε),

can be used as a barrier function to prove that wr is bounded by a constanttimes ε2 in B3σ/2 \B3δ, provided ε is chosen small enough. Similarly, we provethat ∂rwr is bounded by a constant times ε2 r−1 in Bσ \ B4δ, provided ε ischosen small enough.

Using (5.33) together with the fact that

|wr|+ r |∂wr| ≤ c ε2 and |∂θwi| ≤ c ε1−α rµ,

we get

Area(uε) =1r(1 + ∂θH +O(ε1−αrµ))(Sε +O(ε2))(∂rSε +O(ε2r−1))

= Area (Sε eiθ) +O(ε2 r−2 + ε3−α rµ−4).

Therefore, using (6.19), we conclude that Area (uε) ≥ c ε2 r−4 in Bσ0 \ B4δ

provided σ0 is chosen small enough (but fixed). This completes the proof of theProposition. 2

130

Chapter 7

Elliptic Operators inWeighted Sobolev Spaces.

In this Chapter, we prove some estimates for solutions of elliptic problems inweighted Sobolev spaces. Again, the aim of this Chapter is not to providea thorough description of the theory of elliptic operators in weighted Sobolevspaces but rather to provide simple proofs of some results which are needed inthe subsequent Chapters. Further results can be found in the references alreadygiven in Chapter 2.

7.1 General overview

To begin with, let us explain what are the weighted Sobolev spaces we have inmind. Let B1 ⊂ Rn be the unit ball of Rn, n ≥ 2. We define the weightedLebesgue spaces as follows :

Definition 7.6 Given ν ∈ R, the space L2ν(B1 \ 0) is defined to be the set of

functions w ∈ L2loc(B1 \ 0) for which the following norm is finite

‖w‖L2ν

:=(∫

B1

|w|2 r−2ν−2 dx

)1/2

. (7.1)

In other words, we simply have

L2ν(B1 \ 0) := rν L2(B1, r

−2 dx).

Granted this definition, we may now define the weighted Sobolev spaces by :

Definition 7.7 Given k ∈ N and given ν ∈ R, we the space Hkν (B1 \ 0) is

defined to be the set of functions w ∈ L2loc(B1 \ 0) for which the following

131

norm is finite

‖w‖Hkν

:=k∑

j=0

‖∇jw‖L2ν−j

. (7.2)

The relation between weighted Sobolev spaces and weighted Holder spaces isgiven by the following simple observation :

C0,αµ (B1 \ 0) → w ∈ L2

ν(B1 \ 0),for all

ν < µ +n− 2

2.

Obviously, for all k ≥ 2

∆ : Hkν (B1 \ 0) −→ Hk−2

ν−2 (B1 \ 0),is well defined and bounded and the same is true for any operator of the form∆ + c r−2.

Paralleling what we have done for weighted Holder spaces, we can also inves-tigate the mapping properties of the Laplacian when defined between weightedSobolev spaces. The results are essentially the same and show that, again, theindicial roots play a crucial role in understanding the mapping properties of theLaplacian. In particular, if we set

δ±j = ±√(

n− 22

)2

+ λj , (7.3)

we can prove the :

Proposition 7.1 Assume that ν > 2−n2 and that ν /∈ δ±j : j ∈ N. Let j0 ∈ N

be the least index for which ν < δ+j0

. Then

∆ : H2ν,D(B1 \ 0) −→ L2

ν−2(B1 \ 0),injective and the dimension of its cokernel is given by j0. Here the subscript Drefers to the fact that the functions are equal to 0 on ∂B1.

Proof : The proof is essentially identical to the proof of the correspondingresult (see Proposition 2.4) in weighted Holder spaces. 2

We can also prove the :

Proposition 7.2 Assume that ν < n−22 and that ν /∈ δ±j : j ∈ N. Let j0 ∈ N

be the least index for which ν > δ−j0 . Then,

∆ : H2ν,D(B1 \ 0) −→ L2

ν−2(B1 \ 0),is surjective and the dimension of its kernel is given by j0.

132

Proof : This time the proof is close to the proof of Proposition 2.6. 2

We will not develop any further the theory of elliptic operators in weightedSobolev spaces but we will rather concentrate our attention on some specificestimates which will be needed in the subsequent Chapters.

7.2 Estimates for the Laplacian

Our first result is related to the study of the Laplacian in the unit puncturedball of R2, when the weight parameter ν is in the surjectivity range.

Lemma 7.1 Let B1 be the unit ball of R2 and assume that ν ∈ (−1, 0). Iff ∈ L2

ν−2(B1 \ 0 and if w ∈ H1ν (B1 \ 0) is a weak solution of

∆w = f in B1 \ 0

w = 0 on ∂B1.(7.4)

Then, there exists c0 ∈ R such that ∆w = f + c0 δ0 in B1 and

‖∇w‖2L2ν−1

≤ 2ν2 (1− ν)2

‖f‖2L2ν−2

+1

2πν2

(c0 +

∫

B1

f

)2

, (7.5)

provided all integrals are well defined.

Proof : We begin by the following simple observation. A priori w does notextend to a weak solution of ∆w = f in B1. However, since we have assumedthat ∫

B1

r−2−2ν |w|2,

is finite with ν ∈ (−1, 0), necessarily there exists c0 ∈ R such that

∆w = f + c0 δ0,

in B1. In addition, without loss of generality, we may assume that f is supportedaway from the origin since the general case can be obtained using a simpledensity argument.

We now, decompose both w and f as

w := v0 + v and f := g0 + g,

where both v and g belong to Spanhn einθ : n 6= 0, while v0 and g0 onlydepend on r.

Step 1. Notice that v is a weak solution of

∆v = g in B1

v = 0 on ∂B1.

133

Further observe that v(r, ·) ∈ Spaneinθ : n 6= 0, hence, we always have∫

∂Br

|v|2 dθ ≤∫

∂Br

|∂θv|2 dθ.

Multiplying this inequality by r−1−2ν and integrating over (0, 1), we obtain∫

B1

r−2ν−2 |v|2 ≤∫

B1

r−2ν−2 |∂θv|2. (7.6)

Now, we multiply the equation by − r−2ν v and integrate by parts. Weobtain

−∫

B1

r−2ν v g =∫

B1

r−2ν |∇v|2 +12

∫

B1

∇r−2ν ∇|v|2.

Since12

∫

B1

∇r−2ν ∇|v|2 = −2 ν2

∫

B1

r−2ν−2 |v|2,

we get

−∫

B1

r−2ν v g =∫

B1

r−2ν |∇v|2 − 2 ν2

∫

B1

r−2ν |v|2.

Which after the use of Cauchy-Schwarz inequality becomes∫

B1

r−2ν |∇v|2 − 2 ν2

∫

B1

r−2ν−2 |v|2

≤(∫

B1

r2−2ν |g|2)1/2 (∫

B1

r−2ν−2 |v|2)1/2

.

Finally, we use (7.6) to conclude that∫

B1

r−2ν |∇v|2 − 2 ν2

∫

B1

r−2ν−2 |∂θv|2

≤(∫

B1

r2−2ν |g|2)1/2 (∫

B1

r−2ν−2 |∂θv|2)1/2

.

(7.7)

Notice that all integrations are valid since g is supported away from the originand hence v is bounded by a constant times r near 0.

Now, we multiply the equation by r1−2ν ∂rv and integrate by parts. Thistime, we get

∫

B1

r1−2ν g ∂rv = ν

∫

B1

r−2ν(|∂rv|2 − r−2 |∂θv|2

)+

12

∫

∂B1

|∂rv|2. (7.8)

In particular, we have the inequality∫

B1

r1−2ν g ∂rv ≥ ν

∫

B1

r−2ν |∂rv|2 − ν

∫

B1

r−2−2ν |∂θv|2,

134

which becomes, after the application of Cauchy-Schwarz inequality

ν

∫

B1

r−2ν |∂rv|2 − ν

∫

B1

r−2−2ν |∂θv|2

≤(∫

B1

r2−2ν |g|2)1/2 (∫

B1

r−2ν |∂rv|2)1/2

.

(7.9)

In order to simplify the discussion, we set

X 2 :=∫

B1

r−2ν |∂rv|2 Y2 :=∫

B1

r−2−2ν |∂θv|2

andG2 :=

∫

B1

r2−2ν |g|2.

The inequalities (7.7) and (7.9) are then translated into

ν (X 2 − Y2) ≤ G XX 2 + (1− 2ν2)Y2 ≤ G Y

We can multiply the first inequality by −ν and add it to the second inequalityto get

(1− ν2) (X 2 + Y2) ≤ G (Y − ν X ).

And, since ν ∈ (−1, 0), we conclude

X 2 + Y2 ≤ 2(1− ν2)2

G2.

That is ∫

B1

r−2ν |∇v|2 ≤ 2(1− ν2)2

∫

B1

r2−2ν |g|2. (7.10)

Step 2. It remains to prove a similar estimate for v0. We use once more(7.8) which, since all functions are now radial functions, reads

−ν

∫

B1

r−2ν |∂rv0|2 = −∫

B1

r1−2ν g0 ∂rv0 + π |∂rv0(1)|2.

We use Cauchy-Schwarz inequality to get

−ν

∫

B1

r−2ν |∂rv0|2 ≤(∫

B1

r2−2ν |g0|2)1/2 (∫

B1

r−2ν |∂rv0|2)1/2

+ π |∂rv0(1)|2.It is then an easy exercise to see that

ν2

∫

B1

r−2ν |∂rv0|2 ≤∫

B1

r2−2ν |g0|2 + 2 π |∂rv0(1)|2.

135

Now, ∆v0 = g0 + c0 δ0, which after integration over B1 yields

2 π ∂rv0(1) =∫

B1

g0 + c0.

Hence

ν2

∫

B1

r−2ν |∂rv0|2 ≤∫

B1

r2−2ν |g0|2 +12π

(∫

B1

g + c0

)2

. (7.11)

The result of the Lemma is obtained by summing (7.10) with (7.11). 2

We now derive a similar result in more general domains. As usual, Ω is aregular open subset of R2 and Σ := a1, . . . , aN is a finite set of points in Ω.We chose σ > 0 in such a way that, if ai 6= aj ∈ Σ, then B2σ(ai) and B2σ(aj)are disjoint and both included in Ω. We define weighted Lebesgue spaces onΩ \ Σ by :

Definition 7.8 Given ν ∈ R, the space L2ν(Ω \ Σ) is defined to be the set of

functions w ∈ L2loc(Ω \ Σ) for which the following norm is finite

‖w‖L2ν

:=(∫

Ω

|w|2 dist(x, Σ)−2ν−2 dx

)1/2

. (7.12)

Our main result reads :

Proposition 7.3 Assume that ν ∈ (−1, 0). There exists a constant c > 0 and,for all w ∈ L2

ν(Ω \ Σ) weak solution of

∆w = f in Ω \ Σ

w = 0 on ∂Ω,(7.13)

there exists c1, . . . , cN ∈ R such that

∆w = f +N∑

j=1

cj δaj ,

in Ω and

‖∇w‖2L2ν−1

≤ c

‖f‖2L2

ν−2+

N∑

j=1

(cj +

∫

Bσ(aj)

f

)2 , (7.14)

provided all integrals are well defined.

136

Proof : The proof of the result relies on Lemma 7.1 and on the domain decom-position method we have already used in Chapter 2 and in Chapter 6.

Applying Lemma 7.1 to each Bσ(aj), we already see that there exists cj ∈ Rsuch that

∆w = f + cj δaj,

in Bσ(aj).

Step 1. Given φ := (φ1, . . . , φN ) ∈ ΠjH3/2(∂Bσ(aj)), we define vext to be

the harmonic extension of the φj in Ωσ which takes the value 0 on ∂Ω. Namely

∆vext = 0 in Ωσ

vext = φj on ∂Bσ(aj)

vext = 0 on ∂Ω,

and we may also define vint to be the harmonic extension of the φj in eachBσ(aj). Namely

∆vint = 0 in each Bσ(aj)

vint = φj on each ∂Bσ(aj)

As we have already done in Chapter 2 and in Chapter 6, we define the Dirichletto Neumann mapping by

DN(φ) :=((∂r1vext − ∂r1vint)|∂Bσ(a1), . . . , (∂rN vext − ∂rN vint)|∂Bσ(aN )

).

It is clear that

DN : ΠjH3/2(∂Bσ(aj)) −→ ΠjH

1/2(∂Bσ(aj)),

is a well defined bounded linear operator. Moreover, DN is a first order ellipticoperator whose principal symbol is −2|ξ|. Hence, in order to prove that DN isan isomorphism, it is enough to show that DN is injective.

Assume that DN(φ) = 0. Then the function v which is equal to vext in Ωσ

and equal to vint in ∪j Bσ(aj) is harmonic in all Ω and has 0 boundary valueson ∂Ω. Thus v = 0 and hence φ = 0. This proves the injectivity of DN .

Step 2. Now, we define wint and wext to be the solutions of the followingequations

∆wint = f + cj δaj in each Bσ(ai)

wint = 0 on each ∂Bσ(ai),(7.15)

and ∆wext = f in Ωσ

wext = 0 on ∂Ωσ.(7.16)

137

Finally, we set

φ := DN−1((∂r1wext − ∂r1wint)|∂Bσ(a1), . . .

. . . , (∂rN wext − ∂rN wint)|∂Bσ(aN )

).

(7.17)

We will denote φ := (φ1, . . . , φN ). We claim that there exists c > 0 such that,for all j = 1, . . . , N , we have

‖φj‖2H3/2(∂Bσ(ai))≤ c

‖f‖2L2

ν−2(Ω\Σ) +

(cj +

∫

Bσ(aj)

f

)2 . (7.18)

For the sake of simplicity in the notations, let us assume that aj = 0 and let usdrop the j indices. First, we use standard trace embedding results to get

‖∂rwint‖H1/2(∂Bσ) ≤ c ‖∇wint‖2H1(Bσ\Bσ/2) .

Now, using (7.15), we can state that there exists c > 0 such that

‖∇wint‖2H1(Bσ\Bσ/2) ≤ c(‖∇wint‖2L2(Bσ\Bσ/3)

+ ‖f‖2L2(Bσ\Bσ/3)

). (7.19)

Using once more the fact that wint satisfies (7.15) together with the result ofLemma 7.1, we conclude that

‖∇wint‖2L2(Bσ\Bσ/3)≤ c

(‖f‖2L2

ν−2(Bσ) +(

cj +∫

Bσ

f

)2)

.

Therefore

‖∂rwint‖2H1/2(∂Bσ) ≤ c

(‖f‖2L2

ν−2(Bσ) +(

cj +∫

Bσ

f

)2)

. (7.20)

Similarly, we get

‖∂rwext‖2H1/2(∂Bσ) ≤ c ‖f‖2L2(Ωσ). (7.21)

The claim follows at once from (7.20) and (7.21).

Step 3. Let φ be given by (7.17). We define vext to be the harmonic extensionof the φj in Ωσ which takes the value 0 on ∂Ω. And we define vint to be theharmonic extension of the φj in each Bσ(aj). By construction of φ, it is clearthat the following identity holds

w = wint − vint in ∪N

j=1 Bσ(aj)

w = wext − vext in Ωσ.(7.22)

138

In Ωσ, we make use of the following standard estimates

‖vext‖H1(Ωσ) ≤ c

N∑

j=1

‖φj‖H3/2(∂Bσ(aj)), (7.23)

and‖wext‖H1(Ωσ) ≤ c ‖f‖L2(Ωσ). (7.24)

In each Bσ(aj), standard regularity results yield

‖∇vint‖2L2ν−1(Bσ(aj))

≤ c ‖φj‖2H3/2(∂Bσ(aj)). (7.25)

Moreover, since wint satisfies (7.15), Lemma 7.1 yields

‖∇wint‖2L2ν−1(Bσ(aj))

≤ c

‖f‖2L2

ν−2(Bσ(aj))+

(cj +

∫

Bσ(aj)

f

)2 . (7.26)

Combining (7.23)-(7.26) with (7.18) leads to the desired estimate. The proofof the Proposition is therefore complete. 2

Following the strategy we have used to prove Lemma 7.1, we can prove the :

Proposition 7.4 Assume that ν ∈ (0, 1). There exists c > 0 such that, for allf ∈ L2

ν−2(R2 \ 0), if

u :=12π

log r ∗ f,

then‖u− u(0)‖H2

ν≤ c ‖f‖L2

ν−2.

Proof : We definew := u− u(0),

and we decompose both w and f as

w := v0 + v and f := g0 + g,

where both v and g belong to Spanhn einθ : n 6= 0, while v0 and g0 onlydepend on r.

Step 1. To begin with, let us assume that g has compact support in R2 \0.There is no loss of generality in doing so since, the general case can be obtainedby a classical density argument. Under this assumption, since v is a weaksolution of

∆v = g,

then v is smooth at the origin and v(0) = 0. Hence, there exists a constantc > 0 (depending on g) such that

|v|+ r |∇v| ≤ c r,

139

in B1. Furthermore, it is classical to see that there also exists a constant c > 0(depending on g) such that

|v| ≤ c r−1 and |∇v| ≤ c r−2, (7.27)

in R2 \B1.

Further observe that v(r, ·) ∈ Spaneinθ : n 6= 0, hence∫

∂Br

|v|2 dθ ≤∫

∂Br

|∂θv|2 dθ.

Multiplying this inequality by r−1−2ν and integrating over (0, +∞), we obtain∫

R2r−2ν−2 |v|2 ≤

∫

R2r−2ν−2 |∂θv|2. (7.28)

Now, we multiply the equation ∆v = g by r−2ν v and integrate by parts overBR. We obtain

∫

BR

r−2ν v g = −∫

BR

r−2ν |∇v|2 + 2 ν2

∫

BR

r−2−2ν |v|2

+∫

∂BR

r−2ν v ∂rv + ν

∫

∂BR

r−1−2ν |v|2

We let R tend to +∞ and obtain thanks to (7.27)∫

R2r−2ν v g = −

∫

R2r−2ν |∇v|2 + 2 ν2

∫

R2r−2−2ν |v|2. (7.29)

Multiplying the equation by r1−2ν ∂rv and integrating by parts over BR, we alsoget ∫

BR

r1−2ν g ∂rv = ν

∫

BR

r−2ν(|∂rv|2 − r−2 |∂θv|2

)

+12

∫

∂BR

r1−2ν (|∂rv|2 − r−2 |∂θv|2).

Again, we let R tend to +∞ and obtain∫

R2r1−2ν g ∂rv = ν

∫

R2r−2ν

(|∂rv|2 − r−2 |∂θv|2). (7.30)

Using this identity together with (7.29) yields

(1− ν2)∫

R2r−2ν |∂rv|2 + (1 + ν2)

∫

R2r−2−2ν |∂θv|2

−2 ν2

∫

R2r−2−2ν |v|2 = −

∫

R2r−2ν v g − ν

∫

R2r1−2ν g ∂rv.

Making use of (7.28), we get

(1− ν2)∫

R2r−2ν |∇v|2 ≤ −

∫

R2r−2ν v g − ν

∫

R2r1−2ν g ∂rv.

140

Now, we make use of Cauchy-Schwarz inequality and once more of (7.28),to conclude that

(1− ν2)2∫

R2r−2ν |∇v|2 ≤

∫

R2r2−2ν |g|2

It still follows from (7.28) that we also get

(1− ν2)2∫

R2r−2−2ν |v|2 ≤

∫

R2r2−2ν |g|2

Then, using classical regularity results for solutions of elliptic equations we get∫

R2r2−2ν |∇2v|2 ≤ c

∫

R2r2−2ν |g|2

for some constant c which does not depend on g.

Step 2. It remains to prove the relevant estimate for v0. Observe that (7.30)still holds and gives us

∫

R2r1−2ν g0 ∂rv0 = ν

∫

R2r−2ν |∂rv0|2.

and henceν2

∫

R2r−2ν |∂rv0|2 ≤

∫

R2r2−2ν |g0|2. (7.31)

Finally, notice that, since ν > 0, we have (provided all integrals are well defined)∫ ∞

0

r−1−2ν h2 = − 12ν

∫ ∞

0

∂r(r−2ν) h2

=1ν

∫ ∞

0

r−2ν h ∂rh

≤ 1ν

(∫ ∞

0

r−1−2ν h2

)1/2 (∫ ∞

0

r1−2ν |∂rh|2)1/2

.

Hence ∫ ∞

0

r−1−2ν h2 ≤ 1ν2

∫ ∞

0

r1−2ν |∂rh|2.

Applying this inequality to v0 and using (7.31) we conclude that∫

R2r−2ν |∂rv0|2 +

∫

R2r−2−2ν |v0|2 ≤

(1ν4

+1ν2

) ∫

R2r2−2ν |g0|2.

This ends the proof of the Proposition. 2

141

7.3 Estimates for some elliptic operator in di-vergence form

We now derive, for the operator

w −→ div(ρ−2∇w),

results which are similar in their spirit to those we have obtained in the previoussection for the Laplacian. Since, the assumptions we will make on the functionρ are very weak, we cannot obtain results as sharp as the ones we have obtainedfor the Laplacian. However, the results we obtain will be sufficient for ourpurposes.

The counterpart of Lemma 7.1 is given by the following Lemma in whichthe function ρ is assumed to satisfy the following properties :

1. The function ρ is smooth in B1 \ 0.2. The function ρ only depends on r and is increasing.

3. There exists a constant c0 > 0 such that

ρ ≥ c0 r and r∂rρ

ρ≤ 1

c0in B1.

Under these assumptions, we have :

Lemma 7.2 Assume that ν ∈ (1− 2−1/2, 1). Let w be a weak solution of

−div (ρ−2∇w) = ρ−2 f + div g in B1 \ 0w = 0 on ∂B1,

(7.32)

and further assume that∫

B1

r2−2ν ρ−2 |∇w|2 < +∞.

Then∫

B1

r2−2ν ρ−2 |∇w|2 ≤ c

(∫

B1

r4−2ν ρ−2 |f |2 +∫

B1

r2−2ν ρ2 |g|2

+(∫

∂B1

ρ−2 ∂rw

)2

+(∫

∂B1

g · x)2

),

(7.33)

where the constant c only depends on ν and c0.

142

Before we proceed to the proof of this result, let us briefly comment on thereason why some information on w is needed on the right hand side of theestimate. To simplify the discussion, let us assume that ρ is given by

ρ := rα,

where α ∈ (0, 1). In this particular case, the function w0 = 1− r2α is a solutionto the homogeneous problem

−div (ρ−2∇w0) = 0,

in B1 \0. Clearly, because of this solution of the homogeneous problem exists,one cannot hope to estimate w, the solution of (7.33) just in term of f and g.Yet another way to understand this extra term is to notice that a priori wsatisfies (7.33) away from the origin but, just as in Lemma 7.1, the function wis not a weak solution of the equation in all B1. Indeed, one can see that thereexists c0 ∈ R such that

−div (ρ−2∇w0) = ρ−2 f + div g + c0 δ0,

in B1. Somehow, the extra term is just here to evaluate the constant c0.

Proof : As usual, we decompose w, f and g as

w := w0 + w1, and f := f0 + f1,

where w1 and f1 belong to Spanhn einθ : n 6= 0, while w0 and f0 only dependon r.

Moreover, we decompose g := g0 + g1 in such a way that div g1 belongs toSpanhn einθ : n 6= 0, while div g0 only depend on r. If we write

g := (gx, gy),

we have

div g = (cos θ ∂rgx − 1

rsin θ ∂θg

x) + (sin θ ∂rgy +

1r

cos θ ∂θgy).

And, if we define the function

h0 :=12π

∫ 2π

0

(cos θ gx + sin θ gy) dθ,

we get explicitely

div g0 =1

2πr

∫

∂Br

div g =1r

h0 + ∂r h0.

Step 1. First, let us observe that, since w1 ∈ Spanhn einθ : n 6= 0, wehave ∫

∂Br

|w1|2 ≤∫

∂Br

|∂θw1|2,

143

which, once multiplied by r−2ν ρ−2 and integrated over (0, 1), yields∫

B1

r−2ν ρ−2 |w1|2 ≤∫

B1

r−2ν ρ−2 |∂θw1|2. (7.34)

This inequality is in particular useful to justify all the integrations by parts.

Step 2. Multiplying the equation satisfied by w1 by r2−2ν w1 and integratingover B1, we get

∫

B1

r2−2ν w1

(ρ−2 f1 + div g1

)=

∫

B1

r2−2ν |∇w1|2ρ2

+12

∫

B1

ρ−2∇r2−2ν ∇|w1|2.

Moreover, we have

12

∫

B1

ρ−2∇r2−2ν ∇|w1|2 = −2 (1− ν)2∫

B1

r−2ν ρ−2 |w1|2

+ 2 (1− ν)∫

B1

r1−2ν ∂rρ|w1|2ρ3

.

The function ρ being increasing and ν being less than 1, we conclude that∫

B1

r2−2ν ρ−2 |∇w1|2 − 2 (1− ν)2∫

B1

r−2ν ρ−2 |w1|2

≤∫

B1

r2−2ν ρ−2 w1 f1 −∫

B1

g1∇(r2−2ν w1

Using (7.34) together with Cauchy-Schwarz inequality, we conclude that∫

B1

r2−2ν ρ−2 |∂rw1|2 + (1− 2 (1− ν)2)∫

B1

r−2ν ρ−2 |∂θw1|2

≤(∫

B1

r4−2νρ−2|f1|2)1/2 (∫

B1

r−2νρ−2|∂θw1|2)1/2

+cν

(∫

B1

r2−2νρ2|g1|2)1/2 (∫

B1

r2−2νρ−2|∇w1|2)1/2

.

for some constant cν only depending on ν. Since we have assumed that ν >1− 2−1/2, this implies that

∫

B1

r2−2ν ρ−2 |∇w1|2 ≤ c

(∫

B1

r4−2ν ρ−2 |f1|2 +∫

B1

r2−2ν ρ2 |g1|2)

. (7.35)

Step 3. We now consider the proof of the estimate for w0. The equationsatisfied by w0 reads

−∂r

(ρ−2 r ∂rw0

)= ρ−2 r f0 + h0 + r ∂rh0. (7.36)

144

Multiplying this equation by r3−2ν∂rw0 and integrating between 0 and 1, weget

(1− ν)∫

B1

r2−2ν ρ−2 |∂rw0|2 +∫

B1

r3−2ν ∂rρ ρ−3 |∂rw0|2

= π ρ−2(1) |∂rw0|2(1) +∫

B1

r3−2ν (ρ−2 f0 + r−1 h0 + ∂rh0) ∂rw0.

(7.37)

We focus on the term∫

B1

r3−2ν ∂rh0 ∂rw0,

which appears in the last integral on the right hand side of (7.37). We use (7.36)together with an integration by parts to obtain

∫

B1

r3−2ν∂rh0 ∂rw0 =∫

B1

r3−2ν h0 (f0 + r−1 ρ2 h0 + ρ2 ∂rh0)

+ 2 π h0(1) ∂rw0(1)− (3− 2ν)∫

B1

r2−2ν ∂rw0 h0

− 2∫

B1

r3−2ν ∂rρ ρ−1 ∂rw0 h0.

(7.38)Finally, we evaluate the term

∫

B1

r3−2ν ρ2 h0 ∂rh0,

which appears in the first integral on the right hand side of (7.38). We write∫

B1

r3−2νρ2h0∂rh0 =12(ρ2|h0|2)(1)−

∫

B1

r2−2ν(2− ν + r−1ρ∂rρ

)ρ2|h0|2.

(7.39)

Observe that for all κ > 0, we have

a b ≤ κ a2 +14κ

b2 (7.40)

Hence, using Cauchy-Schwarz inequality, we obtain∫

B1

r3−2ν ρ−2 f0 ∂rw0 ≤ κ

∫

B1

r2−2ν ρ−2 |∂rw0|2 +14κ

∫

B1

r4−2ν ρ−2 |f0|2.

Similarly we estimate∫

B1

r2−2ν h0 ∂rw0 ≤ κ

∫

B1

r2−2ν ρ−2 |∂rw0|2 +14κ

∫

B1

r2−2ν ρ2 |h0|2.

145

Finally, we simply bound

h0(1) ∂rw0(1) ≤ 12

(|h0|2(1) + |∂rw0|2(1)),

and ∫

B1

r3−2ν h0 f0 ≤ 12

∫

B1

r4−2ν ρ−2 |f0|2 +12

∫

B1

r2−2ν ρ2 |h0|2.

Combining (7.37), (7.38) and (7.39), with the above inequalities, we concludethat

∫

B1

r2−2νρ−2|∂rw0|2 ≤ c (ρ2(1) |h0|2(1) + ρ−2(1) |∂rw0|2(1))

+ c

(∫

B1

r4−2νρ−2 |f0|2 +∫

B1

r2−2ν ρ2 |h0|2)

.

(7.41)where the constant c only depends on ν and on c0. To obtain the result in thestatement of the Lemma, it is enough to notice that

∂rw0(1) =12π

∫

∂B1

∂rw,

and, by definition

h0(1) =12π

∫

∂B1

g x.

The result of the Lemma is then a consequence of (7.35) and (7.41). 2

We now derive a similar result in more general domains. We keep the nota-tions of the previous section and assume that ρ satisfies the following properties :

1. The function ρ is smooth in Ω \ Σ.

2. In each Bσ(aj), the function ρ only depends on rj := |x − aj | and isincreasing.

3. There exists a constant c0 > 0 such that

ρ ≥ c0 rj and rj

∂rj ρ

ρ≤ 1

c0,

in each Bσ(aj) andρ ≥ c0,

in Ωσ.

Under these assumptions, we have the :

146

Proposition 7.5 Assume that ν ∈ (1− 2−1/2, 1). For all w weak solution of −div (ρ−2∇w) = ρ−2 f + div g in Ω \ Σ

∂νw = 0 on ∂Ω,(7.42)

if we further assume that∫

Ω

r2−2ν ρ−2 |∇w|2 < +∞.

Then∫

Ω

r2−2ν ρ−2 |∇w|2 ≤ c

(∫

Ω

r4−2ν ρ−2 |f |2 +∫

Ω

r2−2ν ρ2 |g|2)

+ c

N∑

j=1

(∫

∂Bσ(aj)

ρ−2 ∂rj w

)2

+ c

N∑

j=1

(∫

∂Bσ(aj)

g · (x− aj)

)2

,

(7.43)

where r := dist(x, Σ) and where the constant c > 0 only depends on ν and onthe constant c0.

Proof : We proceed exactly like in the proof of Proposition 7.1 using Lemma 7.2instead of Lemma 7.1. In particular, the operator

DN : ΠjH3/2(∂Bσ(aj)) −→ H1/2(∂Bσ(aj)),

can be defined as before to be the difference of the two Dirichlet to Neumannoperators in Ωσ and in ∪jBσ(aj), corresponding to the operator Λ : w −→−div(ρ−2∇w) with 0 Neumann data on ∂Ω. However, this time Λ has one di-mensional Kernel and one dimensional Cokernel. This induces a one dimensionalKernel and a one dimensional Cokernel for DN . More precisely, we have

KerDN = Span(1, . . . , 1),and

RangeDN = (φ1, . . . , φN ) ∈ ΠjH1/2(∂Bσ(aj)) |

∑

j

∫

∂Bσ(aj)

φj = 0.

Beside this remark, the proof follows the proof of Proposition 7.1, so we omitthe details. 2

We end this Chapter with a result which is close to the result of Propo-sition 7.5. The result involves a function ρ which is assumed to satisfy thefollowing set of assumptions :

1. The function ρ is smooth in R2 \ 0, only depends on r and is increasing.

147

2. There exists a constant c0 > 0 such that

ρ ≥ c0 r,

in each B1 \ 0,c0 ≤ ρ ≤ 1

c0,

in R \B1 and

r∂rρ

ρ≤ 1

c0,

in R2 \ 0.Under these assumptions, our result reads :

Proposition 7.6 Assume that ν ∈ (1 − 2−1/2, 1). There exists c > 0 andς ∈ (0, σ), such that, for all w weak solution of

−div (ρ−2∇w) + w = ρ−2 f + div g,

in R2 \ 0, if we assume that∫

R2r2−2ν ρ−2 |∇w|2 < +∞,

and if lim∞ |w| = 0, then∫

R2r2−2ν ρ−2 |∇w|2 ≤ c

(∫

R2r4−2ν ρ−2 |f |2 +

∫

R2r2−2ν ρ2 |g|2

+(∫

∂Bς

1ρ2

∂rw −∫

Bς

w

)2

+(∫

∂Bς

g · x)2

),

(7.44)provided ρ(ς) > 1/2.

Proof : As in the proof of Lemma 7.2, we decompose w, f and g as

w := w0 + w1, and f := f0 + f1,

where w1 and f1 belong to Spanhn einθ : n 6= 0, while w0 and f0 only dependon r. As in the proof of Lemma 7.2, we will write

div g0 =1r

h0 + ∂rh0.

Moreover, we decompose g := g0 + g1 in such a way that div g1 belongs toSpanhn einθ : n 6= 0, while div g0 only depend on r.

Step 1. Following Step 1 and Step 2 in the proof of Lemma 7.2, we alreadyobtain

∫

R2r2−2ν ρ−2 |∇w1|2 ≤ c

(∫

R2r4−2ν ρ−2 |f1|2 +

∫

R2r2−2ν ρ2 |g1|2

). (7.45)

148

Step 2. We now would like to derive the relevant estimate for w0. Keepingthe notations of Step 3 in the proof of Lemma 7.2, we see that w0 is a solutionof

−∂r

(ρ−2 r ∂rw0

)+ r w0 = ρ−2 r f0 + h0 + r ∂rh0. (7.46)

1 - Estimate on ∂Bς . By definition of h0, we have

h0(ς) =12π

ς−2

∫

∂Bς

g · x. (7.47)

Now, since we have assumed that lim∞ w = 0, we can write

w0 = −∫ +∞

r

∂rw0 dr.

Hence, applying Cauchy-Schwarz inequality, we get the pointwise bound

r2−2ν |w0|2 ≤ c

∫

R2\Br

r2−2ν |∂rw0|2 ≤ c

∫

R2\Br

r2−2ν ρ−2 |∂rw0|2, (7.48)

for some constant c depending on ν and c0. Observe that we have used theassumption ρ ≥ c0 to obtain the last inequality. Integration of this inequalityover Bς yields

∫

Bς

r2−2ν |w0|2 ≤ c ς2

∫

R2r2−2ν ρ−2 |∂rw0|2. (7.49)

Observe that∫

Bς

w =∫

Bς

w0 and∫

∂Bς

∂rw =∫

∂Bς

∂rw0.

Hence, we can estimate |∂rw0(ς)|2 in the following way

ς2

ρ4(ς)|∂rw0(ς)|2 =

(12π

∫

∂Bς

1ρ2

∂rw0

)2

≤ c

(∫

∂Bς

1ρ2

∂rw −∫

Bς

w

)2

+ c ς2ν

∫

Bς

r2−2ν |w0|2

≤ c

(∫

∂Bς

1ρ2

∂rw −∫

Bς

w

)2

+c ς2ν+2

∫

Bς

r2−2νρ−2|∂rw0|2.(7.50)

We have used Cauchy-Schwarz inequality to obtain the second estimate and(7.49) to obtain the last inequality.

149

2 - Estimate over Bς . Multiplying (7.46) by r3−2ν ∂rw0 and integratingbetween 0 and ς, we get

∫

Bς

r3−2ν(ρ−2f0 + r−1h0 + ∂rh0)∂rw0 = (1− ν)∫

Bς

r2−2ν ρ−2 |∂rw0|2

+∫

Bς

r3−2ν ρ−3 ∂rρ |∂rw0|2

− (2− ν)∫

Bς

r2−2ν |w0|2

− πς4−2ν(ρ−2|∂rw0|2 − |w0|2)(ς).

The function ρ being increasing, we obtain the inequality

(1− ν)∫

Bς

r2−2νρ−2|∂rw0|2 ≤∫

Bς

r3−2ν(ρ−2f0 + r−1h0 + ∂rh0)∂rw0

+(2− ν)∫

Bς

r2−2ν |w0|2

+π ς4−2ν ρ−2(ς) |∂rw0|2(ς).

Arguing as in Step 3 of the proof of Lemma 7.2, we conclude that∫

Bς

r2−2ν ρ−2 |∂rw0|2 ≤ c ς4−2ν(ρ2(ς) |h0|2(ς) + ρ−2(ς) |∂rw0|2(ς)

)

+ c

(∫

Bς

r4−2ν ρ−2 |f0|2 +∫

Bς

r2−2ν ρ2 |h0|2)

+ c

∫

Bς

r2−2ν |w0|2,(7.51)

where the constant c only depends on ν and on c0.

Using (7.50) and (7.51), we conclude that∫

Bς

r2−2ν ρ−2 |∂rw0|2 ≤ c

(∫

R2r4−2ν ρ−2 |f |2 +

∫

R2r2−2ν ρ2 |g|2

+(∫

∂Bς

1ρ2

∂rw −∫

Bς

w

)2

+(∫

∂Bς

g · x)2

)

+ c (ς2 + ς4 ρ2(ς))∫

R2r2−2ν ρ−2 |∂rw0|2,

(7.52)for some constant c only depending on ν and c0. The key point here is that theconstant c does not depend on ς.

3 - Estimate on R2 \ Bς . On R2 \ Bς , we multiply (7.46) by r2−2ν w0 and

150

integrate by parts. We obtain∫

R2\Bς

r2−2νρ−2|∂rw0|2 +∫

R2\Bς

r2−2ν |w0|2 =∫

R2\Bς

r2−2νρ−2f0 w0

−∫

R2\Bς

r2−2ν h0 ∂rw0

− 2 (1− ν)∫

R2\Bς

r1−2ν h0 w0

− 2 (1− ν)∫

R2\Bς

r1−2ν ρ−2 w0 ∂rw0

− 2π ς3−2ν(ρ−2 w0 ∂rw0 + h0 w0

)(ς).

(7.53)We now evaluate each term which appears on the right hand side of this in-equality. Let τ1, τ2 two positive parameters, to be fixed later on. We use theinequality

a b ≤ κ a2 +14κ

b2, (7.54)

to bound

−∫

R2\Bς

r2−2νh0 ∂rw0 ≤ τ1

∫

R2r2−2ν ρ−2 |∂rw0|2

+1

4τ1

∫

R2r2−2ν ρ2 |h0|2.

Similarly, we have∫

R2\Bς

r2−2ν f0 w0 ≤ τ2

∫

R2\Bς

r−2ν ρ−2 |w0|2

+1

4τ2

∫

R2r4−2ν ρ2 |f0|2.

and− (1− ν)

∫

R2\Bς

r1−2ν h0 w0 ≤ τ2

∫

R2\Bς

r−2ν ρ−2 |w0|2

+1− ν

4τ2

∫

R2r2−2ν ρ2 |h0|2.

We use (7.47) and (7.48) to get the bounds

2 π ς3−2ν (h0 w0)(ς) ≤ τ1

∫

R2\Bς

r2−2ν ρ−2 |∂rw0|2

+c

τ1ς4−2ν

(∫

∂Bς

g · x)2

.

151

Similarly, we use (7.48) and (7.50) to get

2 π ς3−2ν (ρ−2 w0 ∂rw0)(ς) ≤ 2π

(τ1(ς1−νw0)2 +

14τ1

(ς2−νρ−2∂rw0)2)

≤ c

τ1ς2−2ν

(∫

∂Bς

1ρ2

∂rw −∫

Bς

w

)2

+c

(τ1 +

1τ1

ς4

) ∫

R2r2−2ν ρ−2 |∂rw0|2.

It remains to estimate

−2 (1− ν)∫

R2\Bς

r1−2ν ρ−2 w0 ∂rw0.

We decompose

2∫

R2\Bς

r1−2ν ρ−2 w0 ∂rw0 = 2∫

R2\B1

r1−2ν ρ−2 w0 ∂rw0

+ 2∫

B1\Bς

r1−2ν ρ−2 w0 ∂rw0.

Using once more (7.54), we see that the first integral is bounded by

2∫

R2\B1

r1−2ν ρ−2 w0 ∂rw0 ≤∫

R2r2−2ν ρ−2 |∂rw0|2

+∫

R2\B1

r−2ν ρ−2 |w0|2.

Since ρ is increasing and since we integrate over r ≥ 1, we get

2∫

R2\B1

r1−2ν ρ−2 w0 ∂rw0 ≤∫

R2r2−2ν ρ−2 |∂rw0|2

+ ρ−2(1)∫

R2\B1

r2−2ν |w0|2.

Finally, for the second integral, we can write

2∫

B1\Bς

r1−2ν ρ−2 w0 ∂rw0 ≤∫

B1\Bς

ρ−2 |∂rw0|2

+∫

B1\Bς

r2−4ν ρ−2 |w0|2.(7.55)

We use on last time (7.46), multiply it by w0 and integrate to get∫

R2\Bς

ρ−2 |∂rw0|2 +∫

R2\Bς

|w0|2 =∫

R2\Bς

ρ−2 f0 w0

+∫

R2\Bς

h0∂rw0 − 2π ς (h0 − ρ−2∂rw0)(ς) w0(ς).

152

Using (7.54), we can state∫

R2\Bς

ρ−2 |∂rw0|2 +∫

R2\Bς

|w0|2 ≤ c

∫

R2\Bς

ρ−4 |f0|2

+∫

R2\Bς

ρ2 |h0|2 − 4π ς (h0 − ρ−2∂rw0)(ς) w0(ς).

This, together with (7.55) implies that

2∫

B1\Bς

r1−2νρ−2w0∂rw0 ≤ cς

∫

R2r4−2νρ−2|f0|2 + cς

∫

R2r2−2νρ2|h0|2

+4 π ς3−4ν(ρ−4(|h0|+ ρ−2|∂rw0|)|w0|

)(ς),

where the constant cς depends on ς. Notice that we have implicitly used thefact that 1− 2ν < 0. Now we use (7.47), (7.48) and (7.50) to get

2∫

B1\Bς

r1−2ν ρ−2 w0 ∂rw0 ≤ cς

∫

R2r4−2ν ρ−2 |f0|2

+ cς

∫

R2r2−2ν ρ2 |h0|2

+ cς

(∫

∂Bς

g · x)2

+ cς

(∫

∂Bς

1ρ2

∂rw −∫

Bς

w

)2

+ (τ1 +c

τ1ς4−4νρ−4(ς))

∫

R2r2−2νρ−2|∂rw0|2.

4 - Final argument. Collecting all above estimates in (7.53) together with(7.52), we obtain

(ν − cτ1 − c

τ1ς4−4νρ−4(ς))

∫

R2r2−2ν ρ−2 |∂rw0|2

+(1− (1− ν)ρ−2(1)− 2τ2)∫

R2\Bς

r2−2ν |w0|2

≤ cς

∫

R2r4−2ν ρ−2 |f0|2 + cς

∫

R2r2−2ν ρ2 |h0|2

+cς

(∫

∂Bς

g · x)2

+ cς

(∫

∂Bς

1ρ2

∂rw −∫

Bς

w

)2

.

Now, let us assume thatρ4(ς) ≥ 1/2.

In particular, this implies that

1− (1− ν)ρ−2(1) > 0,

since the function ρ is increasing and since we have assumed that 1−2−1/2 < ν.The result then follows at once by first choosing τ1 and τ2 to be small enoughbut fixed and then letting ς tend to 0. 2

153

Chapter 8

Generalized PohozaevFormula for ρ−ConformalFields

In this Chapter, we derive a Pohozaev type formula in order to compare twosolutions of the Ginzburg-Landau equation. This formula is a modification of thestandard Pohozaev formula which is commonly used in the study of semilinearelliptic equations.

In the beginning of the Chapter we briefly recall the use of Pohozaev identityin the classical framework. Then, we derive a Pohozaev type identity for theGinzburg-Landau equation, first in the particular case of radially symmetricsolutions in the unit ball and then in the general case.

8.1 The Pohozaev formula in the classical frame-work

To shed light on the developments of this Chapter, we slightly digress and brieflyrecall the classical use of Pohozaev identity and its variational interpretation.We refer the reader to the work of R. Schoen [87] for geometric consequences ofthis identity.

Let Ω be an open subset of Rn and p > 1, we are interested in positivesolutions of the equation

∆u + up = 0 in Ω

u = 0 on ∂Ω.(8.1)

Recall that a conformal Killing vector field X := (X1, . . . , Xn) in Rn is a vectorfield which corresponds to the infinitesimal action of a conformal transformation.

154

It satisfies∂xi

Xj + ∂xjXi = c δi,j ,

where the function x ∈ Rn −→ c(x) ∈ R depends on X and where δi,j is theusual Kronecker’s symbol. In Rn, n ≥ 3 the space of conformal Killing vectorfields has dimension (n+1)(n+2)

2 and is generated by

X1 = b, ∀ b ∈ Rn,

X2 = x,

X3 = (b · x) c− (c · x) b, ∀ b, c ∈ Rn,

X4 = (b · x)x− 12 |x|2 b, ∀ b, c ∈ Rn.

Notice that, in dimension n = 2, the conformal Killing vector fields are generatedby holomorphic function and thus the space if in this case infinite dimensional.

Now, assume that u is a solution of ∆u + up = 0. Multiplying this equationby

∂Xu :=n∑

i=1

Xi ∂xiu,

it is an easy exercise to see that

div(

(∂Xu)∇u−(

12|∇u|2 +

1p + 1

up+1

)X

)

+c

2

(n− 2

2|∇u|2 − n

p + 1up+1

)= 0,

(8.2)

where c(x) is the function which appears in the definition of the conformalKilling vector field.

Multiplying (8.1) by u we also find

div (u∇u) = |∇u|2 − up+1.

Hence we conclude that

div(

∂Xu∇u−(

12|∇u|2 − 1

p + 1up+1

)X +

n− 28

c∇u2 − u2∇c

)

=c

2

(n

p + 1− n− 2

2

)up+1.

(8.3)

The main feature of this kind of identity is that it involves on the left hand sidea quantity which is in divergence form and on the right hand side, a quantitywhose sign is controlled.

Assuming that u is regular enough (say in H1(Ω)∩Lp+1(Ω)), we see that usolution of ∆u + up = 0 is a critical point of the functional

E(u) :=12

∫

Ω

|∇u|2 − 1p + 1

∫

Ω

|u|p+1.

155

NamelydE

dt(u + tw)|t=0 = 0,

for all w ∈ H1(Ω) ∩ Lp+1(Ω) with compact support in Ω.

It is interesting to observe that the Pohozaev identity may also be derivedfrom the assumption that u is a critical point of E with respect to variationson the parameterization of the domain (these critical points are usually calledstationary critical point). Indeed, let

φ : Ω −→ Rn,

be a vector field in Ω, with compact support in Ω. We define, for t small enough

ut := u(·+ tφ).

By definition, u is said to be a critical point of E with respect to variations onthe parameterization of the domain, if

dE

dt(ut)|t=0 = 0, (8.4)

for all sufficiently smooth vector field φ having compact support in Ω. Obviously,assuming that u is regular enough, we have the expansion

ut = u + t ∂Xu +O(t2),

and, using this, we find explicitely

dE

dt(ut)|t=0 =

∫

Ω

∇u∇(∂φu)−∫

Ω

up ∂φu.

Integrating by parts, we see that (8.4) becomes

∑

i,j

∫

Ω

∂xiu ∂xj u (∂xiφj + ∂xj φ

i)−∫

Ω

(12|∇u|2 − 1

p + 1up+1

)divφ = 0. (8.5)

The Pohozaev identity then follows from this identity using some appropriatevector fields. More precisely, we define for all ε > 0

χε := 1− η

(dist(x, ∂Ω)

ε

),

where η is a cutoff function identically equal to 1 in [0, 1] and equal to 0 in[2,+∞). Given any conformal Killing vector field X, we set

φε := χε X,

156

and we apply (8.5) to φε. We find∫

Ω

(∂Xu) (∇χε∇u)−∫

Ω

(12|∇u|2 +

1p + 1

up+1

)∂Xχε

+∫

Ω

c

2χε

(n− 2

2|∇u|2 − 1

p + 1up+1

)= 0,

The Pohozaev formula (8.2), integrated over Ω, follows at once by letting ε tendto 0. We refer for example to [24] for the details.

A classical application of this Pohozaev identity is the following :

Theorem 8.1 Assume that Ω is a starshaped domain. Then, there are nononzero positive smooth solutions of (8.1), if p ≥ n+2

n−2 .

Proof : Assume that Ω is starshaped with respect to the origin and apply (8.3)to the vector field X = x. We find

div(

∂Xu∇u +(

12|∇u|2 − 1

p + 1up+1

)X +

n− 24

∇u2

)

=(

n

p + 1− n− 2

2

)up+1.

Integrating this identity over Ω and using the fact that u = 0 on ∂Ω, we concludethat ∫

∂Ω

∂Xu ∂νu− 12|∇u|2(x · ν) =

(n

p + 1− n− 2

2

) ∫

Ω

up+1,

where ν denotes the outward pointing normal to ∂Ω. Hence

12

∫

∂Ω

|∇u|2 (x · ν) =(

n

p + 1− n− 2

2

) ∫

Ω

up+1,

since Ω is starshaped we have (x · ν) > 0 on ∂Ω. This implies that ∇u = 0 on∂Ω and we conclude that u is identically equal to 0 too, using for example HopfLemma [26]. 2

8.2 Comparing Ginzburg-Landau solutions throughPohozaev’s argument

Before, we begin our study of the Pohozaev identity in the framework of theGinzburg-Landau equation, let us list the notations we will use in the subsequentChapters.

8.2.1 Notations

Let us recall that we have already defined the scalar product in C by

z · z′ = xx′ + y y′.

157

We will adopt the same notation for the scalar product in R2.

In the subsequent Chapters, it will be convenient to use the following nota-tion

z ∧ z′ := (iz) · z′ := x y′ − y x′.

Observe that the following identities hold

z ∧ z = 0, and z ∧ z′ = −z′ ∧ z.

Given a vector field X = (Xx, Xy), we will adopt the notation

∂X := Xx ∂x + Xy ∂y.

In the particular case where the vector field ν := (νx, νy) is the outward normalvector field on the boundary of a subset ω, we will write

∂ν := νx ∂x + νy ∂y,

just to be more consistent with usual notations.

Finally, given a vector field X = (Xx, Xy), we set

X⊥ := (Xy, Xx),

which corresponds to a rotation of angle −π/2. Similarly, we define

∇⊥ := (∂y ,−∂x ).

In the particular case where the vector field ν := (νx, νy) is the outward normalvector field on the boundary of a subset ω, we will set

τ := ν⊥,

and∂τ := νy ∂x − νx ∂y.

Observe that, with all these definitions,we have

∇⊥w · ν = −∂τw.

8.2.2 The comparison argument in the case of radiallysymmetric data

We show how to adapt the Pohozaev identity in order to compare two solutionsof the Ginzburg-Landau equation in a simple setting. To begin with, let usconsider the Ginzburg-Landau equation in the unit ball, with boundary datasimply given by eiθ. Namely

∆u +u

ε2(1− |u|2) = 0 in B1

u = eiθ on ∂B1.(8.6)

We recall the following simple result :

158

Theorem 8.2 [29], [34] There exists a unique, non constant solution of (8.6)which is of the form

uε := Sε eiθ.

This solution Sε satisfies

dSε

dr> 0, for all r ≥ 0,

and 0 < Sε < 1, for all r > 0.

Now, let u be any other solution of (8.6) which satisfies

u(0) = 0.

Following P. Mironescu [65], we set

w :=u

uε.

It is an easy exercise to check that w is a solution of

∆w + 2∂rSε

Sε

∂rw + 2 i1r2

∂θw + w S2ε

(1− |w|2)ε2

= 0, (8.7)

in B1 \ 0. Furthermore, w is a critical point of the functional

E(w) :=∫

B1

S2ε |∇w|2 + 2

∫

B1

S2ε

r2w ∧ ∂θw +

12ε2

∫

B1

S4ε (1− |w|2)2. (8.8)

In order to get the Pohozaev identity we are interest in, we consider thevector field X defined by

X := (x, y),

and we take the scalar product of equation (8.7) with

∂X w := x ∂xw + y ∂yw = r ∂rw.

It is an easy exercise to prove that

∂Xw ·∆w = ∂x

(∂Xw · ∂xw − x

2|∇w|2

)+ ∂y

(∂Xw · ∂yw − y

2|∇w|2

).

We will write for short

∂Xw ·∆w = div(

∂Xw · ∇w − X

2|∇w|2

). (8.9)

Similarly, we have

2 ∂Xw ·(

∂rSε

Sε

∂rw

)=

2r

∂rSε

Sε

|∂Xw|2. (8.10)

159

We also obtain

2 ∂Xw ·(

i

r2∂θw

)= − ∂x(w ∧ ∂yw) + ∂y(w ∧ ∂xw).

We recall that we have defined

∇⊥X := (Xy,−Xx).

Hence, the previous formula can be written for short

2 ∂Xw ·(

i

r2∂θw

)= −div (w ∧∇⊥w). (8.11)

Finally, we have

∂Xw ·(

S2ε

ε2w (1− |w|2)

)=

14ε2

div(S2

ε (1− |w|2)2 X)

+1

4ε2(1− |w|2)2 div (S2

ε X).

(8.12)

Combining (8.7)-(8.12), together with the fact that w is a solution of (8.6),we conclude that

div

(∂Xw · ∇w − 1

2|∇w|2 X − w ∧∇⊥w − S2

ε

4ε2(1− |w|2)2 X

)

= −2r

∂rSε

Sε

|∂Xw|2 − 14ε2

(1− |w|2)2 div (S2ε X).

(8.13)

This is, in a particular case, the Pohozaev identity for Ginzburg-Landau equa-tion we are interested in. For the time being, let us observe that on the lefthand side we have a quantity which is in divergence form and on the right handside, we have a negative quantity.

We can also give another form of the Pohozaev identity by integrating theprevious identity over any ω which has compact support in B1 \ 0. We find

∫

∂ω

((∂Xw) · ∂νw − 1

2|∇w|2 X · ν + w ∧ ∂τw − S2

ε

4ε2(1− |w|2)2 X · ν

)

= −∫

ω

2r

∂rSε

Sε

|∂Xw|2 − 14ε2

∫

ω

(1− |w|2)2 div (S2ε X),

(8.14)where we recall that

∂ν := νx ∂x + νy ∂y and ∂τ := νy ∂x − νx ∂y,

and where ν := (νx, νy) is the outward unit normal to ∂ω.

In the subsequent sections of this Chapter, we will explain how this identitycan be used to prove that w is identically equal to 1 and hence that u = Sε eiθ.But, before we do so, we explain how to generalize this identity.

160

8.3 ρ−conformal vector fields

In order to be able to derive a Pohozaev formula in order to compare twosolutions of the Ginzburg-Landau equation in the general case and not only inthe special case where uε is the axially symmetric solution, we have to introducethe notion of ρ−conformal vector fields.

These ρ−conformal vector fields are a generalization of the well known con-formal vector fields. As will be explained later, this notion is only relevant intwo dimensions. We define :

Definition 8.9 Let ρ be a function defined in a 2 dimensional domain ω. Wewill say that a vector-field X is ρ-conformal in ω if there exists a vector field X(which may eventually be singular) solution of

div (ρ2 X) = 0 in D′(ω)

ρ2 curl X = 0 in D′(ω),(8.15)

such that X is given by

X =X⊥

|X|2 . (8.16)

We recall that, for any vector X = (Xx, Xy), we have defined

X⊥ := (Xy,−Xx).

Observe that, when the function ρ is taken to be a constant, say ρ = 1, thenρ-conformal vector fields are just conformal vector fields (i.e. the infinitesimalaction of a conformal transformation). Indeed, in this case, the vector field Xis 1-conformal if and only if the vector field X verifies both

div X = 0 and curl X = 0.

It is then an easy exercise (see Lemma 8.1 below) to show that these condi-tions are satisfied if and only if the complex valued function f := Xx + iXy isholomorphic, which in turn is equivalent to the fact that

|∂xf |2 = |∂yf |2 and ∂xf · ∂yf = 0,

which means that X is conformal.

As another example of a ρ-conformal vector field, let us simply consider

X = (x, y),

which is a conformal vector field. Let us show that this is also a ρ-conformalvector field for any function ρ only depending on r. More precisely, let us

161

assume that ρ is a smooth function in C \ 0, only depending on r, whichsatisfies ρ(0) = 0. Then, we define

X =1

(x2 + y2)1/2(y,−x).

Clearly, we have

div (ρ2 X) = 0 in D′(C)

ρ2 curl X = 0 in D′(C).(8.17)

In particular X = (x, y) is a ρ-conformal vector field.

Remark 8.1 Assume ρ does not vanish on a simply connected domain ω, thenon ω we can write X := ∇φ. Clearly, div (ρ2∇φ) = 0 and hence φ is a criticalpoint of the functional

φ −→∫

Ω

ρ2 |dφ|2 dx.

Observe that, if we were in dimension n ≥ 3, then a conformal change of metricwould make φ an harmonic function for that new metric. However, in dimension2, such change of metric does not exists, this is why ρ-conformal vector fieldsare not simply conformal fields in dimension 2.

8.4 Conservation laws

Let us start with the following basic computation :

Lemma 8.1 Let X be any smooth field which does not vanish and let X be thefield given by

X =X⊥

|X|2 .

Then, we have

(∂xXx − ∂yXy) = 2 Xx Xy div X + ((Xy)2 − (Xx)2) curl X,

and(∂yXx + ∂xXy) =

((Xy)2 − (Xx)2

)div X − 2 Xx Xy curl X.

Proof : Observe that the following identities hold

∂

∂z(Xx − iXy) =

12

(∂xXx − ∂yXy)− i

2(∂yXx + ∂xXy) , (8.18)

and

∂

∂z

(Xx − i Xy

)=

12

(∂xXx − ∂yXy

)− i

2

(∂yXx + ∂xXy

). (8.19)

162

Moreover, given the definition of the vector fields X and X we have

Xx − iXy =Xy + i Xx

(Xx)2 + (Xy)2=

i

Xx + i Xy.

Therefore,

i∂

∂z

(1

Xx + iXy

)= − i

(Xx + iXy)2∂

∂z

(Xx + iXy

).

Inserting (8.18) and (8.19) into this last identity leads to the desired result. 2

Identity (8.9) is a particular case of a more general identity we now state.

Lemma 8.2 Let ω ⊂ C and let w be a complex valued function defined in ω.Then, for all vector field X := (Xx, Xy) defined in ω, we have

∂Xw ·∆w = div(

∂Xw · ∇w − 12|∇w|2 X

)

− 12

(|∂xw|2 − |∂yw|2) (∂xXx − ∂yXy)

− ∂xw · ∂yw (∂xXy + ∂yXx).

(8.20)

Proof : The proof is straightforward and left to the reader. 2

As a first application, we obtain the following conservation law for conformalvector fields in dimension 2 :

Corollary 8.1 Let X be a conformal vector field, then

∂Xw ·∆w = div(

∂Xw · ∇w − 12|∇w|2 X

)(8.21)

Proof : Simply use Lemma 8.1 and Lemma 8.2. 2

The denomination conservation law is justified by the following identity :

d

dt

∫

ω

|∇w (x + tX(x))|2 = −2∫

ω

∂Xw ·∆w + 2∫

∂ω

∂Xw · ∂νw, (8.22)

which holds provided w is smooth enough.

Combining Lemma 8.1 and Lemma 8.2, we obtain :

Lemma 8.3 Let ω ⊂ C and let w be a complex valued function defined in ω.Let X := (Xx, Xy) be a vector field defined in ω, we set

X := − X⊥

|X|2 .

Then, provided curl X = 0, we have

∂Xw ·∆w = div(

∂Xw · ∇w − 12|∇w|2 X

)− ∂Xw · ∂X⊥w div X. (8.23)

163

Proof : Observe that

∂Xw · ∂X⊥w = (Xx∂xw + Xy∂yw) (Xy∂xw −Xx∂yw)

= XxXy(|∂xw|2 − |∂yw|2)

+((Xy)2 − (Xx)2)∂xw · ∂yw.

The result is then a simple Corollary of Lemma 8.2 and Lemma 8.1. 2

For further reference, we state :

Lemma 8.4 Assume that X is a vector field and that the function ρ does notvanish. Define

X := − X⊥

|X|2 ,

Thendiv X =

2|X|2

∂X⊥ρ

ρ+

1ρ2

div(ρ2 X).

Moreover,div X⊥ = 0,

provided curl X = 0.

Proof : For the first formula, we simply compute

divX = div(ρ−2 ρ2X) = −2∂Xρ

ρ+

1ρ2

div(ρ2 X).

The second formula follows from the identity

divX⊥ = curlX,

which holds for any vector field. 2

Thanks to all the above preliminary computations, we can derive the follow-ing conservation law :

Proposition 8.1 Let X be a vector field and let ρ be a function which does notvanish. We set

X := − X⊥

|X|2 .

Provided curl X = 0, we have

∂Xw ·(

∆w +2ρ∇ρ∇w

)= div (∂Xw · ∇w − 1

2|∇w|2 X)

+2

|X|2∂Xρ

ρ|∂Xw|2

− 1ρ2

∂Xw · ∂X⊥w div (ρ2 X).

(8.24)

164

Proof : Observe that the following identity holds

∂X⊥ρ ∂X⊥w + ∂Xρ ∂Xw = |X|2∇ρ∇w.

The result is then a simple consequence of Lemma 8.3 and Lemma 8.4. 2

Observe that, when X is a ρ-conformal vector field, identity (8.24) can besimplified and we obtain the :

Corollary 8.2 Let X be a ρ-conformal vector field, then

ρ−2∂Xw · div (ρ2∇w

)= div (∂Xw · ∇w − 1

2|∇w|2X)

+2

|X|2∂Xρ

ρ|∂Xw|2.

(8.25)

It is interesting to compare this formula with the corresponding formula (8.21)we have obtained in Corollary 8.1 when X is a conformal vector field. Clearly,the (8.25) involves the extra term

2|X|2

∂Xρ

ρ|∂Xw|2.

For later applications, the sign of this term will be of crucial importance. Thiswill be further commented in the last section of this Chapter in which we willaslo give a geometric meaning to this quantity.

The denomination conservation law is justified by the following identity :

d

dt

∫

ω

ρ2

∣∣∣∣∇w (x + tX

ρ2(x))

∣∣∣∣2

= 2∫

ω

∂Xw · div (ρ2∇w)ρ2

+2∫

∂ω

∂Xw · ∂w

∂ν,

(8.26)

which holds provided w is smooth enough.

8.4.1 Comparing solutions through a Pohozaev type for-mula : the general case

We now apply all the previous computation to the Ginzburg-Landau equation.

8.4.2 Conservation laws for Ginzburg-Landau equation

Assume that v is a solution of the Ginzburg-Landau equation

∆v +v

ε2(1− |v|2) = 0. (8.27)

We setX :=

1|v|2 v ∧∇v,

165

and

X := |v|2 v ∧∇⊥v

|v ∧∇⊥v|2. (8.28)

These two functions are well defined away from he points where v = 0 or v∧∇v =0. We have the following simple but fundamental result :

Proposition 8.2 Assume that v is a solution of (8.27). Then the vector fieldX defined above is a |v|-conformal vector field.

To check this, it is easier to write

v := |v| eiφ.

This is always possible, at least locally, away from the zero set of v. The factthat v is a solution of the Ginzburg-Landau equation implies that

div(|v|2 φ) = 0.

However, with this definitions, we have the alternate definition of X which isgiven by

X =∇⊥φ

|∇φ|2 , and X = ∇φ.

Hence, X is a |v|-conformal vector field.

Applying the result of Proposition 8.1, we obtain the following

Corollary 8.3 Let w be any regular map and v a solution of (8.27) in ω. Fur-ther assume that

v 6= 0 and v ∧∇v 6= 0,

in ω. Then the following identity holds

∂Xw ·(

∆w +2|v| ∇|v|∇w

)= div

(∂Xw · ∇w − 1

2|∇w|2 X

)

+2

|X|2∂X |v||v| |∂Xw|2,

(8.29)

where X is the vector fields defined in (8.28).

It is interesting to investigate the behavior of the above defined vector fieldfor Ginzburg-Landau equations, as the parameter ε tends to 0. To this aim, letus assume that vε is a sequence of solutions of (8.27) and further assume that,as ε tends to 0, the sequence vε converges to the harmonic S1 valued functionv?, given by

v? =n∏

j=1

(z − aj

|z − aj |)dj

eiφ,

where the function φ is harmonic. For example, this is the case if the vε areminimizers of the Ginzburg-Landau functional. Naturally, the convergence of

166

the sequence is understood away from the points aj . It is of interest to compute,using (8.28), the vector field X? corresponding to v?. Since v? is S1 valued, wefind

X? =v? ∧∇⊥v?

|v? ∧∇⊥v?|2.

Away from the points aj we can write, at least locally

v? := eiφ? .

Hence we find the alternate formula for X?

X? =∇⊥φ?

|∇φ?|2 .

Observe that, near aj (which is taken to be equal to 0 in order to simplify thenotations), we have

φ? = d θ + H

where d is the degree of v? around 0 and H is a harmonic function which has acritical point at the origin. It is then an easy exercise to see that

∇φ? = d(− y

r2,

x

r2

)+O(r).

And henceX? = ± (x, y) +O(r3).

In particular, near each aj , the vector field X? is close to the ”usual” conformalvector field which is used in Pohozaev formula.

8.4.3 The Pohozaev formula

In this section, we generalize (8.13) to the case where Ω is an arbitrary boundeddomain and where v is not necessary the axially symmetric solution of theGinzburg-Landau equation.

Let us give a definition which will be useful throughout this Chapter.

Definition 8.10 We will say that a C1 complex valued function, defined insome domain ω, is non degenerate if

v 6= 0 and v ∧∇v 6= 0,

in all ω.

Observe that, if v 6= 0, we can write, at least locally,

v := |v| eiφ,

and we findv ∧∇v = ∇φ.

167

Hence, the second condition in the Definition simply becomes

∇φ 6= 0.

Assume that u is a solution of the Ginzburg-Landau equation and assumethat v is an arbitrary complex valued function. If we define the function

w :=u

v,

we have

Lemma 8.5 The function w is a solution of

∆w +2v∇w∇v + w |v|2 (1− |w|2)

ε2= −w

v

(∆v + v

(1− |v|2)ε2

). (8.30)

Proof : The proof follows from the identity

∆(u

v

)+

2v∇v∇w =

1v2

(v ∆u− u ∆v) .

The result of the Lemma follows at once. 2

If in addition we assume that the function v is also a solution of the Ginzburg-Landau equation, the above formula simplifies and we obtain

∆w +2v∇w∇v + w |v|2 (1− |w|2)

ε2= 0. (8.31)

Using this, we get the

Proposition 8.3 Assume that u and v are two solutions of the Ginzburg-Landauequation, which are defined in ω. Further assume that v is nondegenerate in ω.Define

X = |v|2 v ∧∇⊥v

|v ∧∇⊥v|2and X :=

1|v|2 v ∧∇v.

Then the following identity holds :

div(

∂Xw · ∇w − 12|∇w|2 X − w ∧∇⊥w − 1

4ε2|v|2 (1− |w|2)2 X

)

= − 14ε2

(1− |w|2)2 div (|v|2 X)− 2|X|2

∂X |v||v| |∂Xw|2.

(8.32)

Proof : Observe that

2|v| ∇w∇|v| = 2

v∇w∇v − 2 i ∂Xw,

168

and also∂Xw · (2 i ∂Xw) =

2|X|2 ∂Xw ∧ ∂X⊥w

= −2 ∂xw ∧ ∂yw

= −div (w ∧∇⊥w).

Using these two identities together with Corollary 8.3, we obtain

∂Xw ·(

∆w +2v∇v∇w

)= div

(∂Xw · ∇w − 1

2|∇w|2X − w ∧∇⊥w

)

+2

|X|2∂X |v||v| |∂Xw|2.

Now, using the equation satisfied by w we find

|v|2ε2

∂Xw · w(1− |w|2) = − div(

∂Xw · ∇w − 12|∇w|2X − w ∧∇⊥w

)

− 2|X|2

∂X |v||v| |∂Xw|2.

Finally, we have

∂Xw · w |v|2(1− |w|2) = −14div

(|v|2 (1− |w|2)2 X)

+14

(1− |w|2)2 div (|v|2 X).

which together with the previous identity completes the proof. 2

The identity (8.32) is the exact generalization of the one established in (8.13)in the axially symmetric case. Again, this is a conservation law type identitywhich could have been obtained starting from the fact that w is a critical pointwith respect to the variations of the domain ω, of the functional

E(w) :=∫

ω

|v|2 |∇w|2 +2∫

ω

(v∧∇v) (w∧∇w)+1

2ε2

∫

ω

|v|4(1−|w|2)2. (8.33)

When we do not assume that v is a solution of the Ginzburg-Landau equationanymore, we have :

Proposition 8.4 Assume that u is a solution of the Ginzburg-Landau equation,which is defined in ω. Further assume that v is a complex valued function whichis nondegenerate in ω. Define

X = |v|2 v ∧∇⊥v

|v ∧∇⊥v|2and X :=

1|v|2 v ∧∇v.

169

If we assume that curl X = 0 then the following identity holds :

div(

∂Xw · ∇w − 12|∇w|2 X − w ∧∇⊥w − 1

4ε2|v|2 (1− |w|2)2 X

)

= − 14ε2

(1− |w|2)2 div (|v|2 X)− 2|X|2

∂X |v||v| |∂Xw|2

− 1|v|2 ∂Xw · ∂X⊥w div (v ∧∇v)

+1v

∂Xw · w(∆v +

v

ε2(1− |v|2)

).

(8.34)

Proof : The proof is similar to the proof of Proposition 8.3. However, since wedo not assume that v is a solution of the Ginzburg-Landau equation, the vectorfield X is not a |v|-conformal vector field anymore. Hence div (|v|2 X) 6= 0. Theis the reason why the quantity

− 1|v|2 ∂Xw · ∂X⊥w div(X |v|2),

which comes from Proposition 8.1, has to appear in the right hand side of theidentity. Finally, observe that

X |v|2 = v ∧∇v,

in order to get the desired formula. 2

8.4.4 Integration of the Pohozaev formula

In proposition 8.3, we have seen that if u and v are two solutions of the Ginzburg-Landau equation, which are defined in Ω. If we assume that v is nondegeneratein a subset ω ⊂ Ω and if we define

X = |v|2 v ∧∇⊥v

|v ∧∇⊥v|2and X :=

1|v|2 v ∧∇v.

Then the following identity holds :∫

∂ω

(∂Xw · ∂νw − 1

2|∇w|2X · ν + w ∧ ∂τw − 1

4ε2|v|2(1− |w|2)2X · ν

)

= − 14ε2

∫

ω

(1− |w|2)2 div (|v|2 X)−∫

ω

2|X|2

∂X |v||v| |∂Xw|2,

where ν denotes the outward unit normal to ∂ω and τ = ν⊥.

Now, we would like to understand what happens when ω tends to Ω. Inaddition to all the above assumptions, let us assume that the following propertiesare satisfies :

170

1. The functions u and v have the same boundary data on ∂Ω.

2. All the zeros of v are isolated and 0 is a regular value of v. We will denoteby Σ the zero set of v.

3. Away from the zero set of v

v ∧∇v 6= 0.

For all s > 0 small enough, we define

ωs = z ∈ Ω : dist(z, Σ) > s.

Granted all the above definitions, we prove

Lemma 8.6 Under the above assumptions, we have

lims→0

∫

∂ωs

(∂Xw · ∂νw −

(12|∇w|2 +

|v|24ε2

(1− |w|2)2)

X · ν + w ∧ ∂τw

)

=12

∫

∂Ω

|∇w|2 X · ν.

Proof : The proof of this Lemma follows at once since, on ∂Ω, we have |w| = 1,∂τw = 0 and

∂νw · ∂Xw = |∂νw|2 X · ν = |∇w|2 X · ν.

The details are left to the reader. 2

Let aj ∈ Σ be a zero of v. For the sake of simplicity, let us assume thataj = 0. Then we have

Lemma 8.7 Under the above assumptions, we have

lims→0

∫

∂Bs

(∂Xw · ∂νw −(

12|∇w|2 +

|v|24ε2

(1− |w|2)2)

X · ν+w ∧ ∂τw) = 0,

where ν := −1r(x, y) and τ = ν⊥.

Proof : In order to simplify the discussion, we will write

v = f(θ) r +O(r2) and w = h(θ) +O(r).

These expansions do hold since we have assumed that 0 is a regular value of v.

Near 0 we can also write

v = a x + b y +O(r2),

171

where a, b ∈ C. Going back to the definition of X and X, we find

X = |f |2 (a ∧ b)−1 (x, y) +O(r2),

andX = |f |−2 (a ∧ b) (−y, x) +O(1).

In particular, since the leading term in w only depends on θ, we see that

∂Xw = O(r) and ∂νw = O(1),

and finallyX · ν = O(1).

Thanks to these expansions, we already obtain

lims→0

∫

∂Bs

∂Xw · ∂νw − 14ε2

∫

∂Bs

|v|2 (1− |w|2)2 X · ν = 0.

With the above notations, we have

lims→0

s

(12(X · ν) |∇w|2 + w ∧ (∇⊥w · ν)

)

|∂Bs

= −12(a ∧ b)−1 |f |2 |∂θh|2 − (ih) · (∂θh).

The function w is a solution of (8.31), moreover, we have already seen theidentity

1v∇v∇w =

1|v|∇|v|∇w + 2 i ∂Xw.

Hence, we get

r2 ∆w +r2

|v|∇|v|∇w + 2 i r2 ∂Xw + r2 w |v|2 (1− |w|2)ε2

= 0.

Granted the above notations, we may let r tends to 0 in the above identity andwe find

∂θ

(|f |2 ∂θh)

+ +2 i (a ∧ b) ∂θh = 0.

Taking the scalar product with |g|2 h and integrating over ∂Bs we conclude that

lims→0

∫

∂Bs

(12|∇w|2X · ν − w ∧ ∂τw

)= 0,

as claimed. 2

Collecting the results of the last two Lemmas, we find :

172

Theorem 8.3 Assume that u and v are two solutions of the Ginzburg-Landauequation, which are defined in Ω and have the same boundary data on ∂Ω.Further assume that 0 is a regular value of v and, away from the zero set of v

v ∧∇v 6= 0.

Then the following identity holds :

2∫

∂Ω

∂Xw · ∂νw −∫

∂Ω

|∇w|2X · ν − 12ε2

∫

Ω

(1− |w|2)2 div (|v|2 X)

−∫

Ω

4|X|2

∂X |v||v| |∂Xw|2 = 0,

where as usual ν denotes the outward unit normal to ∂Ω and where

X = |v|2 v ∧∇⊥v

|v ∧∇⊥v|2.

8.5 Uniqueness Results

8.5.1 A few uniqueness results

We can already obtain some uniqueness results, thanks to the Pohozaev formulagiven in Theorem 8.3. These uniqueness results were originally established byP. Mironescu. The arguments used by P. Mironescu are exactly those which aredeveloped in the second section of this Chapter. Let us mention that the ideaof using Pohozaev formula to study uniqueness questions for Ginzburg-Landauequations goes back to the work of S. Chanillo and M. Kiessling in [20].

For example, as a beautiful application of Theorem 8.3, we have the followingresult due to P. Mironescu :

Theorem 8.4 [64] Assume that u is a solution of the Ginzburg-Landau equa-tion (8.6). Further assume that u(0) = 0 then u is radially symmetric, namely

u = Sε eiθ.

Proof : We set v := Sε eiθ and we find explicitely

X = (x, y).

We setw :=

u

v.

Observe that, since w ≡ 1 on ∂B1, we can write |∇w| = |∂νw| on ∂B1. Hence

2∫

∂B1

∂Xw · ∂νw −∫

∂B1

|∇w|2 =∫

∂B1

|∇w|2.

173

The assumptions of Theorem 8.3 hold and thus, we obtain∫

∂B1

|∇w|2 +1ε2

∫

B1

(1− |w|2)2 (S2ε + Sε ∂rSε) + 4

∫

B1

r∂rSε

Sε

|∂rw|2 = 0.

The function Sε being increasing, we conclude that ∂rw ≡ 0. Since w = 1 on∂B1, we get u = v. 2

We also obtain the :

Theorem 8.5 [64] Assume that u is a solution of

∆u + u (1− |u|2) = 0,

in C. Further assume that∫

C(1− |u|2)2 < +∞, and deg

(u

|u| ,∞)

= 1.

Then, there exists τ0 ∈ R and a ∈ C such that

e−iτ0 u(· − a) = S eiθ,

where S is defined in Theorem 3.1.

Proof : Since the degree of u at ∞ is nonzero, the function u has to vanish inC. Without loss of generality, we may assume that u(0) = 0.

Now, we use the radially symmetric solution defined in Theorem 3.1. We set

v := S eiθ.

In this case we haveX = (x, y).

Integrating (8.34) over BR\Bs and letting s tend to 0, we obtain using Lemma 8.6∫

∂BR

(r |∂rw|2 − r−1 |∂θw|2 + 2 w ∧ ∂θw − r

2ε2S2(1− |w|2)2

)

= − 1ε2

∫

BR

(1− |w|2)2 (S2 + S ∂rS)− 4∫

BR

r∂rS

S|∂rw|2.

(8.35)

Finally, we let R tend to +∞. Using the assumptions together with informationsabout the behavior at infinity of u which are provided by [15] and [90], one provesthat the left hand side of (8.35) tends to zero. This implies that |w| ≡ 1 and∂rw ≡ 0. Going back to the equation satisfied by w we now find

∂2θθw + 2 i ∂θw = 0.

Since |w| ≡ 1 we find that, either

w = eiτ0

174

for some τ0 ∈ R, orw = ei(τ0−2θ),

for some τ0 ∈ R. However, the later case is not possible otherwise the degree ofu at infinity would be −1, which contradicts our assumption. Hence w ≡ eiτ0

as claimed. 2

Remark 8.2 The uniqueness question for degrees different from ±1 is largelyopen and could happen not to be true as it is conjectured in [69].

8.5.2 Uniqueness results for semilinear elliptic problems

To our knowledge, the first time where a uniqueness result was established forsemilinear elliptic problems using the quotient of two solutions u and v arose ina work by H. Brezis and L. Oswald in [16], where they gave a different proof ofthe following result of A. M. Krasnosel’skii :

Theorem 8.6 Let Ω be a bounded regular domain of Rn and f be a smoothfunction defined in [0,+∞) such that f(t)/t is strictly decreasing. Assume u1

and u2 are positive solutions of −∆u = f(u) in Ω

u = 0 on ∂Ω.(8.36)

Then u1 = u2.

Proof : We define as usualw :=

u2

u1,

which is a solution of

div(u21∇w) = u2 f(u1)− u1 f(u2).

Observe that, thanks to Hopf Lemma, ∂νu1 6= 0 on ∂Ω and hence we see that

w =∂νu2

∂νu1,

on ∂Ω.

Now, we multiply the equation satisfied by w by w itself and integrate byparts over Ω. We obtain

∫

Ω

u21 |∇w|2 +

∫

Ω

(f(u1)

u1− f(u2)

u2

)(u2 − u1)u2 = 0.

Since we have assumed that f(t)/t is strictly decreasing we conclude that nec-essarily u2 = u1 in Ω. 2

For Ginzburg-Landau equations, a similar argument arose first in the work ofL. Lassoued and P. Mironescu [41]. The comparison argument through divisionis also very efficient, as it was observed in [21], to prove the following uniquenessresult of D. Ye and F. Zhou [100].

175

Proposition 8.5 Let Ω be a simply connected domain. Assume that u and vare two solutions of the Ginzburg-Landau equation

∆u +u

ε2(1− |u|2) = 0,

in Ω such that u = v on ∂Ω and which do not vanish in Ω. Further assume that

|u|2 + |v|2 − ε2 (|∇v|2 + |∇u|2) > 0, (8.37)

in Ω. Then u = v.

Proof : We setw :=

u

v,

andX :=

v ∧∇v

|v|2 .

We have already seen that w satisfies

∆w +2v∇v∇w + w |v|2 (1− |w|2)

ε2= 0.

Furthermore, we have also seen that

2v∇w∇v =

2|v| ∇w∇|v|+ 2 i ∂Xw.

Hence

div(|v|2∇w) + 2 i |v|2 ∂Xw + w |v|2 (1− |w|2)ε2

= 0.

Taking the scalar product with w and integrating by parts over Ω, we find∫

∂Ω

|v|2 w · ∂νw −∫

Ω

|v|2 |∇w|2 + 2∫

Ω

|v|2 w · (i∂Xw)

+1ε2

∫

Ω

|u|2 (1− |w|2) = 0.

Now, since u = v on ∂Ω we can write∫

∂Ω

|v|2 w · ∂νw =∫

∂Ω

(|u|2<(u−1 ∂νu)− |v|2 <(v−1 ∂νv)).

In addition, observe that

|v|2 w · (i∂Xw) = (v ∧∇v) (∇w ∧ w).

Hence we get∫

∂Ω

(|u|2 <(u−1 ∂νu)− |v|2 <(v−1 ∂νv))−

∫

Ω

|v|2 |∇w|2

+2∫

Ω

(v ∧∇v) (∇w ∧ w) +1ε2

∫

Ω

|u|2 (1− |w|2) = 0.

(8.38)

176

We claim that ∫

Ω

1|w|2 (v ∧∇v) (∇w ∧ w) = 0.

Indeed, Ω being simply connected, we can write

w := |w| eiψ and v := |v|eiφ.

Using theses notations, we obtain

1|w|2 (v ∧∇v) (∇w ∧ w) = |v|2∇φ∇ψ.

Therefore∫

Ω

1|w|2 (v ∧∇v) (∇w ∧ w) =

∫

∂Ω

|v|2ψ ∂νΦ−∫

Ω

div(|v|2∇φ)ψ.

Now, w ≡ 1 on ∂Ω and hence ψ ≡ 0 on ∂Ω. Furthermore, v is a solution of theGinzburg-Landau equation and thus

div(|v|2∇φ) = 0.

This ends the proof of the claim.

Going back to (8.38), we obtain∫

∂Ω

(|u|2 <(u−1 ∂νu)− |v|2 <(v−1 ∂νv))−

∫

Ω

|v|2 |∇w|2

+2∫

Ω

(|v|2 − |u|2)|u|2 (v ∧∇v) (∇w ∧ w) +

1ε2

∫

Ω

|u|2 (1− |w|2) = 0.

(8.39)

We make use of the inequality 2 a b ≤ a2 + b2, to get

2∫

Ω

(|v|2 − |u|2)|u|2 (v ∧∇v) (∇w ∧ w) ≤

∫

Ω

|v|2 |∇w|2

+∫

Ω

|∇v|2|u|2 |v|2 (|u|2 − |v|2)2,

which together with the previous identity yields∫

∂Ω

(|u|2<(u−1 ∂νu)− |v|2 <(v−1 ∂νv))

+1ε2

∫

Ω

|u|2|v|2 (|v|2 − |u|2)

≥∫

Ω

|∇v|2|u|2 |v|2 (|u|2 − |v|2)2.

Interchanging u and v, we also obtain∫

∂Ω

(|u|2<(u−1 ∂νu)− |v|2 <(v−1 ∂νv))

+1ε2

∫

Ω

|v|2|u|2 (|u|2 − |v|2)

≥∫

Ω

|∇u|2|u|2 |v|2 (|u|2 − |v|2)2.

177

Making the sum of the two inequalities we get

1ε2

∫

Ω

|u|2 + |v|2|u|2 |v|2 (|v|2 − |u|2)2 ≥

∫

Ω

|∇u|2 + |∇v|2|u|2 |v|2 (|u|2 − |v|2)2.

Using the assumption (8.37), we conclude that |u| = |v| in Ω.

We now obtain from (8.39)∫

∂Ω

(|u|2 <(u−1 ∂νu)− |v|2 <(v−1 ∂νv))−

∫

Ω

|v|2 |∇w|2 = 0.

Interchanging u and v and summing the two identities we conclude that∫

Ω

|v|2∣∣∣∇

(u

v

)∣∣∣2

+∫

Ω

|u|2∣∣∣∇

( v

u

)∣∣∣2

= 0.

Hence, w is constant equal to 1 in Ω. 2

As a first corollary of the previous Proposition, we have the :

Theorem 8.7 Assume that Ω is a simply connected bounded domain of C andassume that g : ∂Ω −→ S1 is a smooth map of degree 0. Then there exists ε0

(only depending on Ω and g) such that for all ε ∈ (0, ε0), the functional

Eε(u) :=12

∫

Ω

|∇u|2 +1

4ε2

∫

Ω

(1− |u|2)2,

admits a unique minimizer among maps in H1(Ω) which satisfy u = g on ∂Ω.

Proof : Since the boundary data g has degree 0, it is easy to see that thereexists a constant c > 0 only depending on Ω and g such that

12

∫

Ω

|∇u|2 +1

4ε2

∫

Ω

(1− |u|2)2 ≤ c,

for any minimizer. Arguing as in [10], one proves that all minimizers satisfythe lower bound

|u| ≥ 1/2,

for all ε small enough, say ε ∈ (0, ε0). Still using the analysis of [10], we canprove that there exists a constant c > 0 such that |∇u| ≤ c provided u is aminimizer of Eε and ε ∈ (0, ε0). Obviously, for ε small enough the assumptionsof Proposition 8.5 are fulfilled and the result follows. 2

Proposition 8.5 also yields the following result of M. Comte and P. Mironescu[21].

Theorem 8.8 Assume that Ω be a bounded simply connected domain of C andassume that g : ∂Ω −→ S1 is a smooth map of degree 0. For all α ∈ (0, 1), thereexists ε0 > 0 such that, for all ε ∈ (0, ε0) the solution of

∆u + u(1− |u|2)

ε2= 0 in Ω

u = g on ∂Ω,

178

satisfying|u| ≥ α > 0 in Ω,

is unique and minimizes Eε among maps in H1(Ω) whose boundary data is givenby g.

Proof : It is proven in [10] that there exists c > 0 such that for all u is a criticalpoint of Eε and if |u| ≥ 1/2, then |∇u| ≤ c. The result is again a consequenceof Proposition 8.5. 2

8.6 Dealing with general nonlinearities

We explain how the previous analysis can be carried out for a large class ofequations. In doing so, we show that our analysis is not completely specific tothe Ginzburg-Landau equation.

8.6.1 A pohozaev formula for general nonlinearities

In this section, we extend the Pohozaev identity we have obtained for solutionsof the Ginzburg-Landau equation to critical points of any functional of the form

E(u) :=∫

Ω

|∇u|2 −∫

Ω

F (|u|2), (8.40)

where the function F is assumed to be smooth in [0,∞) and u is a complexvalued function. Critical points of this functional are solution of the followingEuler-Lagrange equation

∆u + u f(|u|2) = 0, (8.41)

where f denotes the derivative of F . Observe that the Ginzburg-Landau equa-tion corresponds to the choice

F (t) = − 12ε2

(1− t)2.

Following closely the strategy of the proof of Theorem 8.3, we establish thefollowing result :

Theorem 8.9 Assume that u and v are two solutions of (8.41), which aredefined in Ω and have the same boundary data on ∂Ω. Further assume that 0 isa regular value of v and, away from the zero set of v

v ∧∇v 6= 0.

179

Then the following identity holds :∫

Ω

divX

|v|2(F (|v|2)− F (|u|2)− f(|v|2) (|v|2 − |u|2))

+∫

Ω

∂X |v|2|v|2

(f(|v|2)− f(|u|2)− f ′(|v|2) (|v|2 − |u|2))

+∫

Ω

∂X |v|2|v|4

(F (|u|2)− F (|v|2)− f(|u|2) (|u|2 − |v|2))

+ 4∫

ω

1|X|2

∂X |v||v| |∂Xw|2 = 2

∫

∂Ω

∂Xw · ∂νw −∫

∂Ω

|∇w|2 X · ν,

(8.42)

where, by definition

X := |v|2 v ∧∇⊥v

|v ∧∇⊥v|2.

Proof : To begin with, observe that the vector field X defined in the state-ment of the result is a |v|-conformal vector field. In particular the result ofProposition 8.1 still holds.

The functionw :=

u

v,

verifies the equation

∆w + 2∇w∇v

v+ w

(f(|u|2)− f(|v|2)) = 0.

Hence, Proposition 8.1 yields

∂Xw · w (f(|v|2)− f(|u|2)) = div (∂Xw · ∇w − 1

2|∇w|2 X)

+2

|X|2∂X |v||v| |∂Xw|2.

If we write∂Xw · w =

12∂X |w|2,

it remains to compute

12

∂X

( |u|2|v|2

) (f(|v|2)− f(|u|2)) .

For the sake of simplicity in the notations, we set

U := |u|2 and V := |v|2.

180

It is a simple exercise to check that the following identity holds

∂X

(U

V

)(f(U)− f(V )) = div(((F (U)− F (V ))− f(V )(U − V ))

X

V)

− 1V

(F (U)− F (V )− f(V )(U − V ))divX

− 1V

∂XV (f(U)− f(V )− f ′(V )(U − V ))

− 1V 2

∂XV (F (V )− F (U)− f(U) (V − U)).

Hence, we conclude that

div(2 ∂Xw · ∇w − |∇w|2 X

)+

4|X|2

∂X |v||v| |∂Xw|2

+div(((F (|u|2)− F (|v|2))− f(|v|2)(|u|2 − |v|2)) X

|v|2)

− 1|v|2 (div X)

(F (|u|2)− F (|v|2)− f(|v|2)(|u|2 − |v|2))

− 1|v|2 ∂X |v|2

(f(|u|2)− f(|v|2)− f ′(|v|2) (|u|2 − |v|2))

− 1|v|4 ∂X |v|2

(F (|v|2)− F (|u|2)− f(|u|2) (|v|2 − |u|2)) = 0.

We now integrate this identity over ω ⊂ Ω, where ω does not contain any zero ofv and let ω tend to Ω to find the desired formula. The details been identical towhat we have already done in the special case of the Ginzburg-Landau equation,we omit them. 2

Remark 8.3 Observe that if we assume that both F and its derivative f areconcave then

F (a)− F (b)− f(a) (a− b) ≥ 0,

andf(a)− f(b)− f ′(a) (a− b) ≥ 0.

8.6.2 Uniqueness results for general nonlinearities

The uniqueness results we have obtained in Theorem 8.4 and Theorem 8.5 forGinzburg-Landau equation hold for more general nonlinearities. Recall that theGinzburg-Landau equation corresponds to the choice

F (t) = − 12ε2

(1− t)2 and f(t) =1ε2

(1− t).

Using Theorem 8.9 and following the same strategy of proof, we obtain thecounterpart of Theorem 8.4 for a wide class of nonlinearities :

181

Theorem 8.10 Assume that both F and f are concave, f(0) > 0 and f(1) = 0.Let u be a solution of

∆u + u f(|u|) = 0 in B1

u = eiθ on ∂B1.

Further assume that u(0) = 0 then u is radially symmetric, namely that u canbe written as

u = ρ eiθ,

where ρ only depends on r.

Similarly we also obtain the counterpart of Theorem 8.5 :

Theorem 8.11 Assume that both F and f are concave, f(0) > 0 and f(1) = 0.Let u be a solution of

∆u + u f(|u|2) = 0,

in C. Further assume that∫

CF (|u|2) < +∞, and deg

(u

|u| ,∞)

= 1.

Then, there exists τ0 ∈ R and a ∈ C such that

e−i(θ+τ0) u(· − a)

is a radial function.

8.6.3 More about the quantities involved in the Pohozaevidentity

We conclude this Chapter by some remark concerning some quantities whichappear in the Pohozaev formula.

Proposition 8.6 Let v be a nondegenerate complex valued function defined insome open subset of C. Assume that the vector field X is defined by

X := |v|2 v ∧∇⊥v

|v ∧∇⊥v|2 .

Then∂X |v||v| =

|X|2|v|2 Area (v). (8.43)

Where the Area has been defined at the end of Chapter 7.

If we further assume that curl X = 0, then we have

divX = 2 |X| k, (8.44)

where k is the curvature of the flow-lines of X, namely the solutions of

γ(t) = X(γ(t)). (8.45)

182

Proof : Locally, we may assume that

v := |v| eiφ.

Then X = |∇φ|−2∇⊥φ and hence

∂X |v||v| =

1|∇φ|2 |v|2

(|v| ∇|v|∇⊥φ)

=|X|2|v|2 det (Dv),

and this yields (8.43).

Assume that γ is a solution of

γ = X(γ).

We identify γ with a point of R2 × 0. In which case it is well known that thecurvature k of γ is given by the third component of the vector

γ × γ

|γ|3 ,

where × denotes the cross product in R3. With little work, we obtain explicitely

|X|3 k = Xx Xy (∂yXy − ∂xXx) + (Xx)2 ∂xXy − (Xy)2 ∂yXx.

Since we have assumed that curlX = 0 we have ∂xXy = ∂yXx. Hence we alsohave

2 |X|3 k = 2 Xx Xy (∂yXy − ∂xXx) + ((Xx)2 − (Xy)2) (∂xXy + ∂yXx).

Finally, we use the fact that

X = − X⊥

|X|2 ,

which, together with Lemma 8.1, implies that(∂xXx − ∂yXy

)= −2 Xx Xy divX − ((Xy)2 − (Xx)2) curlX,

and(∂yXx + ∂xXy

)= −

((Xy)2 − (Xx)2

)div X + 2 Xx Xy curl X.

We conclude that2 |X|3 k = |X|4 divX,

which is the desired result. 2

It is of interest to put in perspective the result of this last Proposition andthe Pohozaev formula for general nonlinearities such which has been derived inTheorem 8.9, with the uniqueness results of Theorem 8.10 and Theorem 8.11.Indeed, it should be clear that global uniqueness results such as these deeply

183

rely on the fact that for the radially symmetric solution the Area density ispositive and also that the curvature of the flow lines associated to the vectorfield X (which is nothing but the gradient of the phase of the radially symmetricsolution) is also positive.

Another key feature of theses uniqueness theorems is that we do not assumethat the other solution which is compared to the radially symmetric solutionhas only one zero at the origin but simply that the set of zeros of u contains theorigin.

Hence, we can more generally state the :

Theorem 8.12 Assume that both F and f are concave. Let u and v be twosolutions of

∆u + u f(|u|2) = 0 in Ω

u = g on ∂Ω.

Assume that v ∧ ∇v 6= 0 away from the zero set of v, that the density area ofv is positive in Ω and that the curvature of the flow lines of the gradient of thephase of v is positive. If in addition

v−1(0) ⊂ u−1(0),

then u = v.

184

Chapter 9

The role of the zeros in theuniqueness question

Using the results of Chapter 8 and Chapter 9, we prove that, under some naturalupper bound on the energy, two solutions of the Ginzburg-Landau equationwhich have the same boundary data and the same zeros are necessarily identicalfor small values of the parameter ε.

9.1 The zero set of solutions of Ginzburg-Landauequations

In this part we study the zero set of a solution of

∆u +u

ε2(1− |u|2) = 0 in Ω

u = g on ∂Ω,(9.1)

where g takes its values into S1. More precisely, we prove that, provided theenergy corresponding to u is appropriately bounded, then u has a finite numberof isolated zeros.

Proposition 9.1 Let c0 > 0 be given. Assume that, for all ε ∈ (0, ε0), thefunction uε is a solution of (9.1) which satisfies

Eε(u) :=12

∫

Ω

|∇u|2 +1

4ε2

∫

Ω

(1− |u|2)2 ≤ c0 log1ε. (9.2)

Further assume that there exist Σ := a1, . . . , aN ⊂ Ω such that, as ε tends to0, the sequence uε converges in C2,α

loc (Ω \ Σ) to the S1 valued harmonic map

u? :=N∏

j=1

(z − aj

|z − aj |)dj

eiφ,

185

where the function φ is harmonic in Ω and where dj = ±1. Then, for ε smallenough, there exists N distinct points a1(ε), . . . , aN (ε) such that :

1. The zero set of uε is given by a1(ε), . . . , aN (ε).2. For all j = 1, . . . , N , limε→0 |aj(ε)− aj | = 0.

3. For all j = 1, . . . , N , the degree of uε at aj(ε) is equal to dj.

In addition, for all j = 1, . . . , N , if we define the sequence of rescaled func-tions

uε(z) := uε(εz + aj(ε)).

Then, there exists τj ∈ R such that, up to a subsequence, for all γ > 0 and forall k ∈ N, the sequence of rescaled functions uε converges as ε tends to 0 toS ei (dj θ+τj) in Ck

loc(Bγ). Where S eiθ is the radially symmetric solution of theGinzburg-Landau equation, which has been defined in Theorem 3.1.

The proof of this result is decomposed in several steps. To begin with let usrecall the following Pohozaev formula which was derived in [11].

Lemma 9.1 Assume that u is a solution of

∆u +u

ε2(1− |u|2) = 0,

and define X := (x, y). Then

div(

∂Xu · ∇u− 12|∇u|2 X − 1

4ε2(1− |u|2)2 X

)+

12ε2

(1− |u|2)2 = 0.

Proof : Simply take the scalar product of the equation satisfied by u with ∂Xuand proceed exactly as in the previous Chapter. 2

Now, we prove the following η-compactness Lemma :

Lemma 9.2 [η-compactness Lemma] Let c1 > 1 be given. There exists η > 0such that if u is a solution of

∆u +u

ε2(1− |u|2) = 0,

which satisfies‖∇u‖L∞ ≤ c1

4ε, (9.3)

and if there exists z ∈ C and R > 2ε for which∫

BR(z)

|∇u|2 +1

2ε2

∫

BR(z)

(1− |u|2)2 ≤ η logR

2ε, (9.4)

then|u| ≥ 1

2in BR/2(z).

186

Proof : We choose η =π

16 c21

and argue by contradiction. Assume that, for

some z′ ∈ BR/2(z), we have |u|(z′) < 1/2. For the sake of simplicity in thenotations, let us assume that z′ = 0. Thanks to (9.3) we see that

|u| < 3/4, in Bε/c1 .

Hence, we have1ε2

∫

Bε/c1

(1− |u|2)2 >π

16 c21

. (9.5)

Now, we integrate the Pohozaev formula which was derived in the previousLemma, over the ball of radius r, we find

1ε2

∫

Br

(1− |u|2)2 = r

∫

∂Br

(1r2|∂θu|2 − |∂ru|2 +

12ε2

(1− |u|2)2)

≤ r

∫

∂Br

|∇u|2 +r

2ε2

∫

∂Br

(1− |u|2)2.

Integrating this inequality over r ∈ (ε,R/2), we find, using (9.4),

∫ R/2

ε

1r

(∫

Br

(1− |u|2)2ε2

)dr ≤ η log

(R

2ε

).

Thanks to the mean value formula, we can conclude that there exists r0 ∈[ε,R/2] such that

1ε2

∫

Br0

(1− |u|2)2 ≤ η,

which is not possible thanks to our choice of η and (9.5). Hence |u|(z′) ≥ 1/2.2

Remark 9.1 There exists a corresponding version of the η-compactness Lemmafor the Ginzburg-Landau equation in higher dimensions. However the proof isnot as straightforward as the one in dimension 2. We refer to [82], [46] andand [47] for further details. Notice also that there is a corresponding versionof the η-compactness Lemma up to the boundary ∂Ω for any smooth S1 valuedDirichlet boundary condition independent of ε. We refer to [46] for a proof ofsuch a result for minimizers in dimension greater or equal to 3.

Using the previous Lemma, we obtain the following zero-set covering Lemma.Observe that, at this stage, the results do not depend on the fact that the degreesof the limiting vortices are ±1.

Lemma 9.3 Under the assumptions of Proposition 9.1, there exist ε0 > 0,n ∈ N and λ ≥ 1 only depending on Ω, g and c0 and, for all ε ∈ (0, ε0), thereexist nε points z1, . . . , znε ∈ Ω such that :

1. z ∈ Ω : |u| < 1/2 ∩Bλ ε(zj) 6= ∅.

187

2. z ∈ Ω : |u| < 1/2 ⊂ ∪nεj=1Bλ ε(zj).

3. For all ε ∈ (0, ε0), nε ≤ n.

4. For all k 6= l, B2λε(zk) ∩B2λε(zl) = ∅.Proof : The proof of this Lemma combines arguments from [10], [11], [12], [96]and [21]. In order to simplify the presentation we work away from the boundaryof Ω. However, similar arguments can be carried out up to ∂Ω.

Step 1. First of all, it follows from [10] that

‖∇u‖L∞ ≤ c1

4ε, (9.6)

where the constant c1 only depends on Ω, g and c0.

Step 2. We have already established the following Pohozaev identity inLemma 9.1

1ε2

∫

Br(z)

(1− |u|2)2 = r

∫

∂Br(z)

(1r2|∂θu|2 − |∂ru|2

)

+r

2ε2

∫

∂Br(z)

(1− |u|2)2.(9.7)

The key point, which was introduced in [12] and independently in [96], isto apply the above Pohozaev formula on ball of size εα, for some α ∈ (0, 1).Assume that α ∈ (0, 1) and β ∈ (0, α) are fixed. As we have already done in theproof of Lemma 9.2, we use the bound of energy (9.2) together with the abovePohozaev identity (9.7), to obtain

1ε2

∫ εβ

εα

1r

(∫

Br(z)

(1− |u|2)2)

dr ≤ c0 log1ε,

and using the mean value formula we deduce the existence of r ∈ (εα, εβ) suchthat

1ε2

∫

Br(z)

(1− |u|2)2 ≤ c0

α− β. (9.8)

Step 3. We claim that there exists a positive integer nα which does notdepend on ε such that the set

Z := z′ ∈ Bεα(z) : |u|(z′) < 1/2,is included in at most nα balls of radius ε. The proof of this claim is close towhat we have already done in the proof of Lemma 9.2. Let z′ ∈ Bεα(z) suchthat |u|(z′) < 1/2, then using (9.6), we see that |u| ≤ 3/4 in the ball of centerz and radius ε/c1 . Hence we have the lower bound

1ε2

∫

Bε/c1 (z′)(1− |u|2)2 ≥ π

16 c21

. (9.9)

188

However, we have proved that

1ε2

∫

Bεα

(1− |u|2)2 ≤ c0

α− β.

An easy covering argument yields that there are at most nα := 43c21c0

(α−β)π , balls ofradius ε which are needed to cover Z.

Granted the η-compactness Lemma and the global bound of the energy (9.2),we conclude that, for any α ∈ (0, 1), the set of points where |u| < 1/2 can becovered by at most Nα balls of radius εα, where Nα only depend on Ω, g andc0 but does not depend on ε.

Step 4. It is not difficult to see that we can choose λ ≥ 1 in such a way that,if we use balls of radius λ ε instead of balls of radius ε to cover Z, we can askthat

z′ ∈ Bεα(z) : |u|(z′) < 1/2 ⊂ ∪nα

l=1Bλε(zl), (9.10)

and∀k 6= l B2λε(zl) ∩B2λε(zk) = ∅. (9.11)

Indeed if (9.11) is not satisfied for λ = 1, there exists zl 6= zk such that

B2ε(zl) ∩B2ε(zk) 6= ∅.In which case we take λ = 4 and eliminate zk from the list of points. Thenwe argue by induction until (9.11) is satisfied. Since the number of points isbounded by nα, the process end up with some λ ≤ 4nα .

To conclude, we have obtained the existence of n ∈ N independent of ε anda set of points z1, . . . , znε in Ω, such that nε ≤ n and

z ∈ Ω : |u|(z) < 1/2 ⊂ ∪nε

l=1Bλε(zl).

Which together with (9.11) completes the proof of the Lemma. 2

Remark 9.2 For further use, it is useful to observe that, increasing λ if nec-essary, we may assume that in addition to the properties of Lemma 9.3, we canassume that

S(r) ≥ 3/4,

for all r ≥ λ, where S is the function which has been defined in Theorem 3.1.

Before we proceed to the proof of Proposition 9.1, we need a last Lemma

Lemma 9.4 Under the hypothesis of Proposition 9.1, if σ > 0 is fixed suchthat, for all j = 1, . . . , N , B2σ(aj) ⊂ Ω and

B2σ(aj) ∩B2σ(ak) = ∅,then

limε→0

1ε2

∫

Bσ(aj)

(1− |u|2)2 ≤ 2π +O(σ2). (9.12)

189

Proof : It follows from Theorem X.3 in [11] that, for all σ > 0,

||u| − 1| ≤ cσ ε2, (9.13)

on ∂Bσ(aj). Moreover, we have already seen that the fact that (aj)j=1,...,N isa critical point of the renormalized energy Wg can be characterized by the factthat the S1 valued harmonic map

u? :=N∏

j=1

(z − aj

|z − aj |)dj

eiφ,

can be written in the neighborhood of each aj in the following way

u? =(

z − aj

|z − aj |)dj

eiHj ,

with∇Hj(aj) = 0. (9.14)

Here Hj is a harmonic function in a neighborhood of aj (see [11] CorollaryVIII.1).

Now, Pohozaev formula (9.7), yields

1ε2

∫

Bσ(zl)

(1− |u|2)2 = r

∫

∂Bσ(zl)

(|∂ru|2 − 1

r2|∂θu|2

)

+r

2ε2

∫

∂Bσ(zl)

(1− |u|2)2.

Using (9.13) together with (9.14), we may pass to the limit as ε tends to 0 inthis identity and obtain

limε→0

1ε2

∫

Bη(zl)

(1− |u|2)2 ≤ 2 π d2j +O(σ2). (9.15)

Finally, we use the fact that |dj | = 1 to conclude. 2

We are now in a position to prove Proposition 9.1.

Proof of proposition 9.1 : We keep the notations of the proof of Lemma 9.3.A priori many zl may converge, as ε tends to 0 to the same vortex aj . Let usdenote by Zj the set of points of z1, . . . , znε which converge to aj . We shallprove that, provided ε is chosen small enough, the set Zj contains exactly onepoint, hence we will have nε = N . The other statements of the Proposition willthen follow immediately.

Step 1. We claim that

supj=1,...,nε

(sup

zj,k,zj,l∈Zj

|zj,k − zj,l|ε

), (9.16)

190

stays bounded independently of ε. To prove this, we argue by contradictionand assume that the claim is not true. Then, we may assume that (up to asubsequence) there exists j, k and l such that

limε→0

|zj,k − zj,l|ε

= +∞.

We set

uε := uε(ε z + zj,k), and rε :=|zj,k − zj,l|

2ε.

Thanks to (9.8), we see that

∆uε + uε (1− |uε|2) = 0,

in Brε and ∫

Brε

(1− |uε|2)2 ≤ c0

α− β.

Extracting some subsequence, if this is necessary, we may assume that the se-quence uε converges in Ck topology, for any k ∈ N and on any compact subsetof C, to u solution of

∆u + u (1− |u|2) = 0 in C,∫

C(1− |u|2)2 < +∞.

(9.17)

Solutions of (9.17) are well understood thanks to the work of H. Brezis, F. Merleand T. Riviere [15], and also to the work of I. Shafrir [90], where it is proventhat |u| tends to 1 at ∞ and also that

∫

C(1− |u|2)2 = 2 π (deg(u,∞))2 .

Observe that this identity implies that u is constant if its degree at infinity is 0.However, since infBλε

uε < 1/2, this implies that infBλ|u| ≤ 1/2. This implies

that u is not constant and hence, we have∫

C(1− |u|2)2 ≥ 2π.

Therefore, we have

1ε2

∫

Brε (zj,k)

(1− |uε|2)2 ≥ 2 π + oε(1).

Similarly, we also have

1ε2

∫

Brε (zj,l)

(1− |uε|2)2 ≥ 2 π + oε(1).

191

Combining this two inequalities, we get that

limε→0

1ε2

∫

Bσ(aj)

(1− |uε|2)2 ≥ 4 π,

which clearly contradicts the result of Lemma 9.4. This ends the proof of theclaim.

Step 2. Thanks to the claim proved in Step 1, we can say that there existsa constant c > 0 such that for any two points zj,k, zj,l ∈ Zj , we have

|zj,k − zj,l| ≤ c ε.

As already mentioned in Remark 9.2, the constant λ which appears in thestatement of Lemma 9.3 can be chosen in such a way that

S(r) ≥ 3/4,

for all r ≥ λ, where the function S has been defined in Theorem 3.1.

We claim that|uε| ≥ 1/2,

in Bcε(zj,k) \Bλε(zj,k).

We argue by contradiction and assume that there exists a sequence uε sat-isfying all the above hypothesis for which

infBcε(zj,k)\Bλε(zj,k)

|uε| < 1/2,

We setuε := uε(ε z + zj,k).

Arguing as in Step 1, we see that, up to a subsequence, the sequence uε convergesto u solution of

∆u + u (1− |u|2) = 0,

in C, and thanks to (9.8), we also have∫

C(1− |u|2)2 ≤ c0

α− β.

Moreover, by construction u(0) = 0 and

deg(u,∞) = limε→0

deg(

uε

|uε| , ∂Bσ(aj))

= dj .

We can then use the uniqueness result we have already proved in Theorem 8.5and deduce that u is given by S ei(θ+τj) for some τj ∈ R, where S eiθ is theradially symmetric solution defined in Theorem 3.1. In particular, we have

|u| ≥ 3/4,

192

in C \Bλ. However, by assumption

infBcε(zj,k)\Bλε(zj,k)

|uε| < 1/2,

which implies thatinf

Bc\Bλ

|u| ≤ 1/2,

which is the desired contradiction.

In particular, thanks to the above claim, we can state that the set Zj reducesto a point.

Step 3. It follows from Step 3 that nε = N as claimed. Moreover, we havealso proved that near aj the solution uε has a unique zero aj(ε) and necessarilythe degree of uε/|uε| at aj(ε) is given by the degree of u? at aj . Hence 1, 2and 3 in Proposition 9.1 are proved. The last statement of the Proposition alsofollows from Step 2. 2

9.2 A uniqueness result

In all this section, we will assume that the following assumptions hold :

1. (a1, . . . , aN ) is a critical point of the renormalized energy Wg, for dj = ±1.

2. For all ε ∈ (0, ε0), uε and vε are solutions of the Ginzburg-Landau equation

∆u +u

ε2(1− |u|2) = 0 in Ω

u = g on ∂Ω.

3. There exists c0 > 0 such that, for all ε ∈ (0, ε0)

12

∫

Ω

|∇u|2 +1

4ε2

∫

Ω

(1− |u|2)2 < c0 log1ε,

holds for u = uε and u = vε.

4. As ε tends to 0, both sequences uε and vε converge to u?, the S1 valuedharmonic map associated to a1, . . . , aN , which is explicitely given by

u? :=N∏

j=1

(z − aj

|z − aj |)dj

eiφ.

Under these assumptions, we prove the following Theorem which is one ofthe main achievement of the two previous Chapters.

193

Theorem 9.1 Assume that the above assumptions (1)-(4) hold. Further as-sume that

u−1ε (0) = v−1

ε (0). (9.18)

Then, there exists ε0 > 0, such that, for any ε ∈ (0, ε0), we have

uε = vε.

It will be convenient to denote by a1(ε), . . . , aN (ε) the common zero setof uε and vε. We will assume that the constant σ > 0 is chosen in such a waythat, for all ε ∈ (0, ε0)

B2σ(aj(ε)) ⊂ Ω,

and, for all j 6= kB2σ(aj(ε)) ∩ B2σ(ak(ε)) = ∅.

Finally, for all r ∈ (0, σ), we set

Ωr := Ω \ ∪Nj=1Br(aj(ε)).

Before we proceed to the proof of Theorem 9.1, we will need pointwise esti-mates on quantities depending on uε and vε. This is the content of the followingsection.

9.2.1 Preliminary results

This Lemma is concerned with universal estimates for uε and vε.

Lemma 9.5 Assume that the above assumptions (1)-(4) hold. Then, for allk ∈ N, there exists ck > 0 such that for all k ≥ 0 and for all ε ∈ (0, ε0)

|∇k(1− |uε|)|+ |∇k(1− |vε|)| ≤ ck ε2 r−2−k

|∇kuε|+ |∇kvε| ≤ ck r−k,(9.19)

where r denotes the distance function to the zero set of vε.

Proof : The proof of (9.19) was first established for minimizers of the Ginzburg-Landau functional in [21] and was then extended to critical points in [22] (seeTheorem 1). As explained in [22], the main step consists in proving that, for allj = 1, . . . , N and for all ε ∈ (0, ε0)

∫

Bσ(aj(ε))\Bε(aj(ε))

|∇ψj |2 ≤ c,

for some σ independent of ε, where ψj is the “excess of phase” defined by theidentity

eiψj :=(

z − aj(ε)|z − aj(ε)|

)−dj v

|v| ,

194

in Bσ(aj(ε)) and where a1(ε), . . . , aN (ε) are the zeros of v.

The proof of this later fact makes use of ideas from [15] where a similarestimate is proved for the “excess of phase” but on all of C. The result forarbitrary k follows by induction. We refer to the step B.6 of the proof ofTheorem 1 in [10] for the details. 2

In the next Lemma, we collect some information about the |vε|-conformalvector field related to vε.

Lemma 9.6 Assume that the above assumptions (1)-(4) hold. Denote by Xthe vector field

X := |vε|2 vε ∧∇⊥vε

|vε ∧∇⊥vε|2 .

Then for all α > 0, there exist ε0 > 0, c > 0 and σ0 > 0 such that, for allε ∈ (0, ε0), we have

|X| ≤ c r and dj divX ≥ 2− α, (9.20)

in each Bσ0(aj(ε)), where r denotes the distance to a1(ε), . . . , aN (ε), the zeroset of vε and

dj∂X |vε||vε| ≥ 1

c, (9.21)

in each B2ε(aj(ε)).

Moreover, for all j = 1, . . . , N and for all γ > 0, there exists c > 0, inde-pendent of ε ∈ (0, ε0), such that

dj∂X |vε||vε| ≥ c, (9.22)

in each Bγε(aj(ε)).

Proof : Take a point aj(ε) and assume dj = +1 (the case where dj = −1 canbe treated similarly). To keep the notations short, we will omit the ε indices.

Step 1. From the arguments in the proof of Lemma 9.3, we know thatvε := v(ε z + aj(ε)) converges in Ck norm, on any bounded set of C, to theradially symmetric solution S ei(θ+τj) defined in Theorem 3.1.

Thus it is not difficult to see that, if we denote

v

|v| ∧ ∇⊥ v

|v| = ∇⊥θ +∇⊥ω, (9.23)

we have, for any fixed c > 0

limε→0

(‖r2∇2ω‖L∞(Bc) + ‖r∇ω‖L∞(Bc)

)= 0. (9.24)

195

We set

X := |v|2 v ∧∇⊥v∣∣v ∧∇⊥v∣∣2 =

1εX(εz + aj(ε)). (9.25)

We have, on one hand,∣∣∣∣X − ∇⊥θ

|∇θ|2∣∣∣∣ ≤ c |r2∇ω|, (9.26)

and on the other hand∣∣∣∣div

(X − ∇⊥θ

|∇θ|2)∣∣∣∣ ≤ c

(|r2∇2ω|+ |r∇ω|) , (9.27)

where we have used the fact that |r2∇2ω|+ |r∇ω| is uniformly bounded, inde-pendently of ε. Observe that

∂x

(∂yθ

|∇θ|2)− ∂y

(∂xθ

|∇θ|2)

= 2. (9.28)

Combining (9.23)-(9.28), we establish (9.20) and (9.22) in Bcε(aj(ε)) for anyfixed c.

Step 2. We claim that the first inequality which appears in (9.20) still holdsin Bσ(aj(ε))\Bcε(aj(ε)). We argue by contradiction and assume that the resultis not true. There would exists a sequence zε such that |zε|−1 |X(zε)| tends to+∞. Since we have already proved that (9.20) holds on any ball of size c ε, wecan assume that

limε→0

rε

ε= +∞.

Furthermore, since vε converges to u? away from the aj , this implies thatlimε→0 |zε| = 0. We set rε := |zε| and

vε := v(rε z + aj(ε)).

Obviously vε verifies

∆vε +(rε

ε

)2

vε (1− |vε|2) = 0, (9.29)

in Bσ/rε. In particular ∆vε and vε are collinear. Moreover, we know from

Lemma 9.5 that|∇kvε| ≤ ck |z|−k.

In particular

|vε| |1− |vε|2| ≤ c

(ε

rε

)2

|z|−2. (9.30)

Up to a subsequence, we may then pass to the limit as ε tends to 0 and obtain amap v, defined in C \ 0 such that ∆v and v are collinear. Moreover, it followsfrom (9.30) that v is either identically 0 or S1 valued. However, observe that

196

|vε| is bounded from below by 1/2 outside Bλ. Hence v cannot be identicallyequal to 0 and we can conclude that v is a S1 valued harmonic map defined inC \ 0.

We also have an additional information : since the degree of vε at aj(ε) is+1, we can state that the degree of v is also 1. Hence

v := ei(θ+τ).

However,

Xε :=1rε

X(rε z + aj(ε)),

also converges to

X :=∇⊥θ

|∇θ|2 ,

which contradicts the fact that |r−1ε X(zε)| → +∞. This ends the proof of the

first inequality in (9.20).

Step 3. We prove now that the second inequality in (9.20) also holds in Bσ0

provided σ0 is fixed small enough. To begin with, let us denote

X0 :=u? ∧∇⊥u?

|u? ∧∇⊥u?|2 .

Then, we may choose σ0 > 0 in such way that

divX0 > 2− α, (9.31)

in each Bσ0(aj). Indeed, if (r, θ) are radial coordinates about aj , we may writeu? := ei(θ+Hj) near aj where Hj is harmonic and ∇Hj(aj) = 0. Hence

X0 =∇⊥θ

|∇⊥θ|2 +O(r2),

near each aj . The existence of σ0 is then immediate.

Again, we argue by contradiction and assume that for some α > 0, thereexists a sequence zε ∈ Bσ0(aj(ε)) such that div X(zε) < 2−α. We set rε := |zε|.Since we have already proved that (9.20) holds on any ball of size c ε, we canassume that

limε→0

rε

ε= +∞.

Furthermore, since vε converges to u? away from the aj , (9.31) implies thatlimε→0 |zε| = 0. We set

vε := v(rε z + aj(ε)).

The analysis which is detailed in Step 2 shows that

Xε :=1rε

X(rε z + aj(ε)),

197

also converges to

X :=∇⊥θ

|∇θ|2 .

Observe that div X = 2.

By assumption div Xε(zε/rε) < 2 − α and letting ε tends to 0 we concludethat there exists z ∈ ∂B1 such that div X(z) ≤ 2 − α. This is the desiredcontradiction, hence the second inequality in (9.20) is proved. 2

Let us assume that u is a solution of the Ginzburg-Landau in Ω. Then wehave

div (u ∧∇u) = 0.

Hence, there exist a function ψu defined in Ω such that

∇⊥ψu = u ∧∇u.

Let us denote by b1, . . . , bN the zero set of u, and let dj ∈ Z be the degree ofu

|u| at bj . Away from the zeros of u we may write

u := |u| eiφu .

In which case we find that

|u|−2∇ψu = −∇⊥φu.

It is now a simple exercise to see that

div(

1|u|2∇ψu

)= 2 π

N∑

j=1

dj δbj .

Moreover, if u := g on ∂Ω, we have

∂νψu = − (u ∧∇⊥u) · ν= u ∧ ∂τu

= g ∧ ∂τg.

To summarize, we have proved that ψu is a solution of

div(

1|u|2∇ψu

)= 2 π

N∑

j=1

dj δbj in Ω

∂νψu = g ∧ ∂τg on ∂Ω.

(9.32)

It will be important to observe that also we have, for all σ > 0 small enough∫

∂Bσ(bj)

1|u|2 ∂rψu = 2 π σ dj . (9.33)

198

where r = |z − bj |. This identity simply follows from the fact that

1|u|2 ∂rψu = ∂τφu,

together with the fact that dj is the degree ofu

|u| on ∂Bσ(bj).

In the following Lemma we obtain an estimate for the weighted L2 norm of∇(ψu − ψv) in all Ω in terms of the L2 norm |uε| − |vε| in Ωε and the weightedL2 norm of ∂X(u/v) in Ω \ Ω2ε. Observe that, the use of weighted L2 spaces iscrucial since similar estimates would not hold in L2. More precisely we have :

Lemma 9.7 Assume that ν ∈ (1 − 2−1/2, 1) and further assume that the hy-pothesis of Theorem 9.1 hold. Then, there exists c > 0 and ε0 > 0 such that,for all ε ∈ (0, ε0), we have

∫

Ω

r2−2ν

|uε|2 |∇(ψuε− ψvε

)|2 ≤ c

∫

Ωε

r−2ν (|uε| − |vε|)2

+c

∫

Ω\Ω2ε

r2−2ν

|X|2∣∣∣∣∂X

(uε

vε

)∣∣∣∣2

.

(9.34)

Where r denotes the distance from z to the zero set of vε and where X is thevector field defined by

X := |vε|2 vε ∧∇⊥vε

|vε ∧∇⊥vε|2 .

Proof : For the sake of simplicity in the notations, let us drop the ε indices.

Step 1. Since u and v have the same zero set and both converge to u?, wefind, using (9.32) that

div(

1|u|2∇ψu

)= div

(1|v|2∇ψv

).

Hence, the function ξ := ψv − ψu satisfies−∆ξ +

2|u|∇|u|∇ξ = |u|2 div

((1|v|2 −

1|u|2

)∇ψv

)in Ω

∂νξ = 0 on ∂Ω.(9.35)

Moreover, using (9.33) we see that∫

∂Bσ(aj(ε))

1|u|2 ∂rψu =

∫

∂Bσ(aj(ε))

1|v|2 ∂rψv, (9.36)

where r := |z − aj(ε)|.Step 2. Let us denote by

Sj := S

(z − aj(ε)

ε

),

199

where S has been defined in Theorem 3.1. We claim that

limε→0

(sup

Bσ(aj(ε))

∣∣∣∣|u|Sj

− 1∣∣∣∣)

= 0. (9.37)

We first prove that, for all fixed γ > 1, (9.37) holds when Bσ(aj(ε)) isreplaced by Bγε(aj(ε)). We argue by contradiction and assume that there existsa sequence uε for which

supBγε(aj(ε))

∣∣∣∣|uε|Sj

− 1∣∣∣∣ ≥ α > 0.

As usual, we setuε := uε(ε z + aj(ε)).

We have already seen that, up to a subsequence, uε converges in Ck topologyto S ei(θ+τj) on any compact of C. Using the properties of S together with thefact that both uε and S vanish at the origin, it is easy to see that

limε→0

supBγ

r−1 (|uε| − S) = 0.

Hence we find

limε→0

supBγ

∣∣∣∣|uε|S

− 1∣∣∣∣ = 0,

which clearly contradicts our assumption. This proves (9.37) on any ball ofradius γ ε.

Now that we have obtained (9.37) on any ball of radius γ ε, we can argueas in Step 2 of the proof of Lemma 9.6 to obtain the corresponding result onBσ(aj(ε)).

Step 3. Let η be the cutoff function equal to 1 in B1 and equal to 0 outsideB2. We set

ηj := η

(z − aj(ε)

ε

).

Using these notations we decompose the right hand side of (9.35) in the followingway

div((

1|v|2 −

1|u|2

)∇ψv

)= div

((1− ηj)

( |u|2 − |v|2|u|2 |v|2

)∇ψv

)

+1|v|2∇ψv ∇

(ηj|u|2 − |v|2|u|2

).

(9.38)

If we set

f =|u|2|v|2∇ψv ∇

(ηj|u|2 − |v|2|u|2

),

200

and

g = (1− ηj)(

1|v|2 −

1|u|2

)∇ψv.

We can write−div(|u|−2∇ξ) = |u|−2 f + divg.

We would like to apply the result of Proposition 7.5. However the function |u|does not satisfy all the required hypothesis since this function does not dependonly on |z − aj(ε)| near each aj(ε). This is the reason why we introduce thefunction

ρ :=N∑

j=1

ηj Sj +

1−

N∑

j=1

ηj

|u|,

where we have set

ηj := η

(z − aj(ε)

σ

).

Obviously, this time, the function ρ fulfills all the required assumptions. Grantedthis, we will write

−div(ρ−2∇ξ) = |u|−2 f + divg + div((|u|−2 − ρ−2)∇ξ).

Hence, for any 1− 2−1/2 < ν < 1, we obtain from Proposition 7.5, the estimate∫

Ω

r2−2ν ρ−2 |∇ξ|2 ≤ c

(∫

Ω

r4−2ν ρ2 |u|−4 |f |2 +∫

Ω

r2−2ν ρ2 |g|2)

+ c

∫

Ω

r2−2ν ρ2 (|u|−2 − ρ−2)2 |∇ξ|2

+ c

N∑

j=1

(∫

∂Bσ(aj)

ρ−2 ∂rj ξ

)2

+ c

N∑

j=1

(∫

∂Bσ(aj)

g · (z − aj)

)2

,

(9.39)

where r is the distance from z to a1(ε), . . . , aN (ε), rj := |z−aj(ε)| and whereσ is fixed in [σ/2, σ]. Observe that the constant c > 0 only depends on ν anddoes not depend on σ provided σ stays bounded from below and from above,say σ ∈ [σ/2, σ].

Step 4. It follows from (9.37) that

limε→0

supBσ(aj(ε))

∣∣∣∣|u|ρ− 1

∣∣∣∣ = 0.

Therefore, for ε small enough

c

∫

Ω

r2−2ν ρ2 (|u|−2 − ρ−2)2|∇ξ|2 ≤ 14

∫

Ω

r2−2ν ρ−2 |∇ξ|2. (9.40)

201

We now make use of (9.36) which implies that∫

∂Bσ(aj(ε))

ρ−2 ∂rj ξ =∫

∂Bσ(aj(ε))

(ρ−2 − |v|−2) ∂rj ψv,

−∫

∂Bσ(aj(ε))

(ρ−2 − |u|−2) ∂rjψu.

Using (9.19) in Lemma 9.5, we see that ∂rjψu and ∂rj

ψv are bounded fromabove and |u| and |v| are bounded from below, independently of ε. Thus, wehave (∫

∂Bσ(aj(ε))

ρ−2∂rjξ

)2

≤ c

∫

∂Bσ(aj(ε))

(|u| − |v|)2. (9.41)

Still using (9.19) in Lemma 9.5, we obtain

(∫

∂Bσ(aj)

g · (z − aj)

)2

≤ c

∫

∂Bσ(aj(ε))

(|u| − |v|)2. (9.42)

Collecting (9.40)-(9.42) into (9.39) we obtain∫

Ω

r2−2ν ρ−2 |∇ξ|2 ≤ c

(∫

Ω

r4−2ν ρ2 |u|−4 |f |2 +∫

Ω

r2−2ν ρ2 |g|2)

+ c

N∑

j=1

∫

∂Bσ(aj(ε))

(|u| − |v|)2.(9.43)

Using (9.37) together with the fact that u and v converge to u?, we get

|v|/2 ≤ |u| ≤ 2 |v|, (9.44)

for all ε small enough. Moreover, since we have |∇ψv| ≤ c |v| |∇v|, (9.19) inLemma 9.5 implies that

|∇ψv| ≤ c r−1 |v|.Hence (9.43) becomes

∫

Ω

r2−2ν |u|−2 |∇ξ|2 ≤ c

N∑

j=1

∫

B2ε(aj(ε))

r4−2ν 1|v|2

∣∣∣∣∇ψv ∇( |v||u|

)∣∣∣∣2

+ c

∫

Ωε

r−2ν (|u| − |v|)2

+ c

N∑

j=1

∫

∂Bσ(aj(ε))

(|u| − |v|)2.

(9.45)In order to obtain this estimate, we have extensively used the fact that both |u|and |v| can be assumed to be bounded from below by 1/2 in Ω \ ∪jBε(aj(ε))provided ε is chosen small enough.

202

Step 5. For all j = 1, . . . , N , we can choose σ ∈ [σ/2, σ] in such a way that∫

∂Bσ(aj(ε))

(|u| − |v|)2 ≤ 2σ

∫

Bσ(aj(ε))\Bσ/2(aj(ε))

(|u| − |v|)2.

The choice of σ may depend on u and v but this is irrelevant since in (9.39),the constant c does not depend on σ ∈ [σ/2, σ]. Thus, (9.45) yields

∫

Ω

r2−2ν |u|−2 |∇ξ|2 ≤ c

N∑

j=1

∫

B2ε(aj(ε))

r4−2ν 1|v|2

∣∣∣∣∇ψv ∇( |v||u|

)∣∣∣∣2

+ c

∫

Ωε

r−2ν (|u| − |v|)2.(9.46)

This is not exactly the estimate which appears in the statement of the Lemma.To obtain the relevant estimate, we begin by the following simple observation

∇( |v||u|

)= −|v|

2

|u|2 ∇( |u||v|

),

which together with (9.44) yields

∣∣∣∣∇ψv ∇( |v||u|

)∣∣∣∣2

≤ c

∣∣∣∣∇ψv ∇( |u||v|

)∣∣∣∣2

.

Now, we compute

∇(u

v

)=

(∇

( |u||v|

)+ i

|u||v| ∇(φu − φv)

)ei(φu−φv).

Hence ∣∣∣∣∇ψv ∇( |v||u|

)∣∣∣∣2

≤∣∣∣∇ψv ∇

(u

v

)∣∣∣2

.

Therefore, we obtain

∫

Ω

r2−2ν |u|−2 |∇ξ|2 ≤ c

N∑

j=1

∫

B2ε(aj(ε))

r4−2ν 1|v|2

∣∣∣∇ψv ∇(u

v

)∣∣∣2

+ c

∫

Ωε

r−2ν (|u| − |v|)2.

We end up using the fact that

X = |v|2 ∇ψv

|∇ψv|2 ,

and hence ∇ψv

|∇ψv| =X

|X| .

203

Since |∇ψv| ≤ c r−1 |v|, we get

1|v|

∣∣∣∇ψv ∇(u

v

)∣∣∣ =1|v|

|∇ψv||X|

∣∣∣∂X

(u

v

)∣∣∣ ≤ cc

r |X|∣∣∣∂X

(u

v

)∣∣∣ .

Using this, the estimate follows at once. 2

In the next Lemma, we obtain an L2 bound for ε−1 (|uε| − |vε|) and also∇(|uε| − |vε|) in ΩR in terms of the L2 norm of |uε| − |vε| and ∇(ψu − ψv) in aslightly larger set. Let us recall that

Ω := Ω \ ∪jBR(aj(ε)).

Lemma 9.8 Assume that the assumptions of Theorem 9.1 hold. Then, thereexists c > 0, γ0 > 0 and ε0 > 0 such that, for all ε ∈ (0, ε0) and for allR ∈ [γ0 ε, σ], we have

1ε2

∫

ΩR

(|uε| − |vε|)2 +∫

ΩR

|∇(|uε| − |vε|)|2

≤ cε2

R2

(1ε2

∫

ΩR/2

(|uε| − |vε|)2 +∫

ΩR/2

|∇(ψuε − ψvε)|2)

.

(9.47)

Proof : As usual, we drop the ε indices in order to simplify the notations. InΩ \ a1(ε), . . . , aN (ε), the following equations hold

∆|u|+ |u|(

1− |u|2ε2

)− |u| |∇φu|2 = 0,

and also

∆|v|+ |v|(

1− |v|2ε2

)− |v| |∇φv|2 = 0.

Since |∇ψu| = |u|2 |∇φu| and |∇ψv| = |v|2 |∇φv|, we see that the function|u| − |v| satisfies

∆(|u| − |v|)− V (|u| − |v|) = F, (9.48)

where we have set

V :=1ε2

(|u|2 + |u| |v|+ |v|2 − 1) +|∇ψu|2|u|4 − |∇ψv|2 |u|

2 + |v|2|u|4|v|3 (|u|+ |v|),

and

F :=|v||u|4 (|∇ψu|2 − |∇ψv|2).

Obviously, if ε is small enough and if γ0 is sufficiently large, the potential Vsatisfies

V ≥ 1ε2

, (9.49)

204

in Ωγ0ε/2. If this is the case, we claim that

1ε2

∫

ΩR

(|u| − |v|)2 ≤ c

(∫

ΩR/2

|∇(ψu − ψv)|2 +1

R2

∫

ΩR/2

(|u| − |v|)2)

. (9.50)

for some constant c > 0 independent of R ∈ [γ0 ε, σ].We set

χ := 1−N∑

j=1

η

(2z − aj(ε)

R

),

where η is some cutoff function identically equal to 1 in B1 and equal to 0outside B2.

We set D := |u| − |v| and we compute

∆(χ2D) = V χ2 D + χ2 F + 2∇(χ2)∇D + ∆(χ2)D.

Now, we multiply by D and integrate by parts over Ω to obtain∫

Ω

V χ2 |D|2 +∫

Ω

χ2 |∇D|2 + 2∫

Ω

χD∇χ∇D +∫

Ω

χ2 F D = 0.

Using the inequality −A2

2− 2 B2 ≤ 2 AB, we conclude that

∫

Ω

V χ2 |D|2 +12

∫

Ω

χ2 |∇D|2 − 2∫

Ω

|D|2 |∇χ|2 +∫

Ω

χ2 F D ≤ 0.

Now

a b ≤ 12ε2

a2 +ε2

2b2

and hence, we get∣∣∣∣∫

Ω

χ2 F D

∣∣∣∣ ≤1

2ε2

∫

Ω

χ2 |D|2 +ε2

2

∫

Ω

χ2 |F |2.

Thanks to (9.49), we obtain

1ε2

∫

ΩR

|D|2 +∫

ΩR

|∇D|2 ≤ c

(1

R2

∫

ΩR\ΩR/2

|D|2 + ε2

∫

ΩR/2

|F |2)

,

as desired. The estimate is then an easy consequence of (9.19) which impliesthat |∇ψv| is bounded by a constant times r−1 and |u| and |v| are bounded frombelow by 1/2 in Ωγ0 ε provided ε is small enough and γ0 is chosen large enough.2

The main results of this section are stated in the next two Corollaries inwhich we control the L2 norms of ε−1 (|uε| − |vε|) and ∇(|uε| − |vε|) in thecomplement of small ball centered at the aj(ε) in terms of the L2 norm of|uε| − |vε| and ∇(ψu − ψv) in small balls centered at the aj(ε).

205

Corollary 9.1 Assume that the hypothesis of Theorem 9.1 hold. Then, thereexists c > 0, γ0 > 0 and ε0 > 0 such that, for all ε ∈ (0, ε0) and for allR ∈ [γ0 ε, σ], we have

1ε2

∫

ΩR

(|uε| − |vε|)2 +∫

ΩR

|∇(|uε| − |vε|)|2

≤ c( ε

R

)2(

1ε2

∫

Ωε\ΩR

(|uε| − |vε|)2 +∫

Ω\Ω2ε

1|X|2

∣∣∣∣∂X

(uε

vε

)∣∣∣∣2)

.

(9.51)where X is the vector field defined by

X := |vε|2 vε ∧∇⊥vε

|vε ∧∇⊥vε|2 .

Proof : It follows from Lemma 9.8 that

1ε2

∫

ΩR

(|uε| − |vε|)2 +∫

ΩR

|∇(|uε| − |vε|)|2 ≤ c

R2

∫

ΩR/2

(|uε| − |vε|)2

+ cε2

R2

∫

ΩR/2

|∇(ψu − ψv)|2 .

(9.52)We make use of Lemma 9.7 to get

∫

ΩR/2

|∇(ψu − ψv)|2

≤ c( ε

R

)2−2ν(

1ε2

∫

ΩR/2

(|uε| − |vε|)2 +∫

Ω\Ω2ε

r2−2ν

|X|2∣∣∣∣∂X

(uε

vε

)∣∣∣∣2)

.

which, together with (9.52) yields

1ε2

∫

ΩR

(|uε| − |vε|)2 +∫

ΩR

|∇(|uε| − |vε|)|2 ≤ c

R2

∫

ΩR/2

(|uε| − |vε|)2

+ cε2−2ν

R4−2ν

∫

Ωε

(|uε| − |vε|)2

+ c( ε

R

)4−2ν∫

Ω\Ω2ε

1|X|2

∣∣∣∣∂X

(uε

vε

)∣∣∣∣2

.

Now, if γ0 is chosen large enough, we conclude that

c

R2

∫

ΩR

(|uε| − |vε|)2 + cε2−2ν

R4−2ν

∫

ΩR

(|uε| − |vε|)2 ≤ 12ε2

∫

ΩR

(|uε| − |vε|)2,

206

for all R ∈ [γ0 ε, σ]. Hence we have

1ε2

∫

ΩR

(|uε| − |vε|)2 +∫

ΩR

|∇(|uε| − |vε|)|2

≤ c

R2

∫

ΩR\ΩR/2

(|uε| − |vε|)2

+ cε2−2ν

R4−2ν

∫

ΩR\Ωε

(|uε| − |vε|)2

+ c( ε

R

)4−2ν∫

Ω\Ω2ε

1|X|2

∣∣∣∣∂X

(uε

vε

)∣∣∣∣2

.

The proof of the result then follows at once since ν < 1. 2

Using similar arguments, we also have

Corollary 9.2 Assume that ν ∈ (1 − 2−1/2, 1). Then, under the assumptionsof the previous Corollary, we have

∫

ΩR

|∇(ψuε − ψvε)|2 ≤ c( ε

R

)2−2ν(

1ε2

∫

ΩR\Ωε

(|uε| − |vε|)2

+∫Ω\Ω2ε

1|X|2

∣∣∣∂X

(uε

vε

)∣∣∣2)

.

Proof : Indeed, we already have from Lemma 9.7∫

ΩR

|∇(ψu − ψv)|2 ≤ +c

R2

∫

ΩR

(|uε| − |vε|)2

+c( ε

R

)2−2ν(

1ε2

∫

Ωε\ΩR

(|uε| − |vε|)2 +∫

Ω\Ω2ε

r2−2ν

|X|2∣∣∣∣∂X

(uε

vε

)∣∣∣∣2)

,

the result is then a straightforward consequence of the previous Corollary whichallows us to bound the first term on the right hand side of the inequality. 2

9.2.2 The proof of Theorem 9.1

The proof of Theorem 9.1 relies on a combination of the Pohozaev identity wehave obtained in Chapter 9 with the estimates in weighted Sobolev spaces whichhas been established in Lemma 9.7.

To begin with, for any σ ∈ [σ, 2σ], we apply Proposition 8.1 to w :=u

v,

on any Bσ(aj(ε)). To keep the notations short we will write Br instead ofBr(aj(ε)). We get

14 ε2

∫

Bσ

div(|v|2 X) (1− |w|2)2 + 2∫

Bσ

1|X|2

∂X |v||v| |∂Xw|2

= −∫

∂Bσ

∂Xw · ∂νw +12

∫

∂Bσ

|∇w|2 X · ν

+1

4 ε2

∫

∂Bσ

(1− |w|2)2 |v|2 X · ν −∫

∂Bσ

w ∧ ∂τw,

(9.53)

207

where

X := |v|2 v ∧∇⊥v

|v ∧∇⊥v|2 .

Step 1. Observe that ∫

∂Bσ

1 ∧ ∂τw = 0,

since on ∂Bσ we have

∂τ = − 1σ

∂θ.

Hence we can write∫

∂Bσ

w ∧ ∂τw =∫

∂Bσ

(w − 1) ∧ ∂τw.

We may now multiply (9.53) by dj (which is the degree of u? at aj) and obtain,using (9.19), (9.20) together with the above identity

dj

4ε2

∫

Bσ

div(|v|2 X) (1− |w|2)2 + 2 dj

∫

Bσ

1|X|2

∂X |v||v| |∂Xw|2

≤ c

∫

∂Bσ

(|∇(u− v)|2 + |u− v|2)

+c

ε2

∫

∂Bσ

(|u| − |v|)2,

for some constant c > 0 which does not depend on σ ∈ [σ, 2σ]. Notice that, wecan always choose σ in such a way that

∫

∂Bσ

(|∇(u− v)|2 + |u− v|2) ≤ 4σ

∫

B2σ\Bσ

(|∇(u− v)|2 + |u− v|2),

and also ∫

∂Bσ

(|u| − |v|)2 ≤ 4σ

∫

B2σ\Bσ

(|u| − |v|)2.

Hence, we conclude that

dj

4ε2

∫

Bσ

div(|v|2 X) (1− |w|2)2 + 2 dj

∫

Bσ

1|X|2

∂X |v||v| |∂Xw|2

≤ c

∫

B2σ\Bσ

(|∇(u− v)|2 + |u− v|2)

+c

ε2

∫

B2σ\Bσ

(|u| − |v|)2.(9.54)

Step 2. Away from the zero set, we may write

u := |u| eiφu and v := |v| eiφv .

208

Hence

|u− v|2 + |∇(u− v)|2 ≤ c ((|u| − |v|)2 + |φu − φv|2

+ |∇(|u| − |v|)|2 + |∇(φu − φv)|2),

in Ωσ.

Since φu = φv on ∂Ω, we can estimate the L2 norm of φu − φv in Ωσ, interms of the L2 norm of ∇(φu − φv) in the same set. Using this together with

∇φu =∇⊥ψu

|u|2 and ∇φv =∇⊥ψv

|v|2 ,

we easily obtain∫

B2σ\Bσ

(|∇(u− v)|2 + |u− v|2) ≤∫

Ωσ

((|u| − |v|)2 + |∇(|u| − |v|)|2)

+∫

Ωσ

|∇(ψu − ψv)|2.

Where we have used the fact that |v| and |u| are uniformly bounded from belowin Ωσ. Therefore, (9.54) yields

dj

4 ε2

∫

Bσ

div(|v|2 X) (1− |w|2)2 + 2 dj

∫

Bσ

1|X|2

∂X |v||v| |∂Xw|2

≤ c

ε2

∫

Ωσ

(|u| − |v|)2

+ c

∫

Ωσ

|∇(|u| − |v|)|2

+ c

∫

Ωσ

|∇(ψu − ψv)|2.

(9.55)

Step 3. Let γ be a positive number chosen larger than γ0 which appearsin Corollary 9.1 and Corollary 9.2. Using Lemma 9.5 and Lemma 9.6, we canbound

|∂X |v|| ≤ cε2

r2.

Moreover, we have1

|X|2 |∂Xw| ≤ |∇w|.

Hence we can estimate∣∣∣∣∣∫

Bσ\Bγ ε

1|X|2

∂X |v||v| |∂Xw|2

∣∣∣∣∣ ≤ c

∫

Bσ\Bγ ε

ε2

r2|∇w|2 .

We have already seen that

∇(u

v

)=

(∇

( |u||v|

)+ i

|u||v| ∇(φu − φv)

)ei(φu−φv).

209

Therefore, we can estimate

|∇w|2 ≤ c

(1r2||u| − |v||2 + |∇(|u| − |v|)|2 + |∇(ψv − ψu)|2

),

on Ωε, since we have from Lemma 9.5

|∇|u||+ |∇|v|| ≤ cε2 r−3 ≤ c r−1,

in Ωε. We conclude that∣∣∣∣∣∫

Bσ\Bγ ε

1|X|2

∂X |v||v| |∂Xw|2

∣∣∣∣∣ ≤ c

γ2

∫

Bσ\Bγ ε

|∇(|u| − |v|)|2

+c

γ4 ε2

∫

Bσ\Bγ ε

(|u| − |v|)2

+c

γ2

∫

Bσ\Bγ ε

|∇(ψu − ψv)|2.

(9.56)

This inequality, together with (9.55) yields

dj

4 ε2

∫

Bσ

div(|v|2 X) (1− |w|2)2 + 2 dj

∫

Bγε

1|X|2

∂X |v||v| |∂Xw|2

≤ c

ε2

∫

Ωσ

(|u| − |v|)2 + c

∫

Ωσ

|∇(|u| − |v|)|2

+c

∫

Ωσ

|∇(ψu − ψv)|2 +c

γ2

∫

Bσ\Bγ ε

|∇(|u| − |v|)|2

+c

γ2

∫

Bσ\Bγ ε

(|u| − |v|)2 +c

γ2 ε2

∫

Bσ\Bγ ε

|∇(ψu − ψv)|2.

In order to conclude, we sum the previous inequality over j and use Corol-lary 9.1 and Corollary 9.2 to get

N∑

j=1

dj

4ε2

∫

Bσ

div(|v|2 X) (1− |w|2)2 + 2N∑

j=1

dj

∫

Bγε

1|X|2

∂X |v||v| |∂Xw|2

≤ c(ε2−2ν + γ2ν−4)

(1ε2

∫

Ωσ\Ωε

(|u| − |v|)2 +∫

Ω\Ω2ε

1|X|2 |∂Xw|2

).

(9.57)We have

dj div(|v|2 X) = 2 dj |v| ∂X |v|+ dj |v|2 divX.

Granted Lemma 9.6 we already have

dj div(|v|2 X) ≥ 0, (9.58)

in Bγε, provided ε is small enough. Moreover, we also have

|2 dj |v| ∂X |v|| ≤ cε2

r2,

210

anddj |v|2 divX ≥ 1

2,

in Bσ. Hence we find that

dj div(|v|2 X) ≥ 14, (9.59)

in Bσ \Bγ ε provided γ is chosen large enough.

Finally, we also have∂X |v||v| ≥ 0, (9.60)

in Bγ ε and∂X |v||v| ≥ 1

c, (9.61)

in B2ε provided ε is small enough.

Using (9.58)-(9.61) we see that, for all ε small enough, the right hand sideof (9.57) is less than the left hand side of (9.57) provided the constant γ hasbeen fixed large enough. In particular, w ≡ 0 and therefore u = v for all ε smallenough. 2

211

Chapter 10

Solving uniquenessquestions

In this Chapter, we prove the main uniqueness result which is stated in Chapter1. The proof of this result combines the construction of the solutions of theGinzburg-Landau equation we have performed in Chapter 3 through Chapter 7with the Pohozaev formula for the Ginzburg-Landau equation as established inChapter 9. Indeed, using the machinery developed in Chapter 10, we are goingto compare any sequence of solutions of the Ginzburg-Landau equation to thesolutions we have constructed in Chapter 7. One of the major new difficulty weare confronted with is that we do not assume that the two solutions have thesame zero set.

10.1 Statement of the uniqueness result

Let (aj)j=1,...,N be a non degenerate critical point of the renormalized energyWg, where we assume that all dj = ±1. For all ε ∈ (0, ε0), we denote by vε thesolutions of the Ginzburg-Landau equation

∆u +u

ε2(1− |u|2) = 0 in Ω

u = g on ∂Ω,(10.1)

which we have constructed in Chapter 7. The result we want to prove reads :

Theorem 10.1 Let εn be a sequence which tends to 0 and uεn a sequence ofsolutions of the Ginzburg-Landau equation (10.1) with ε = εn. Assume that,as n tends to +∞, the sequence uεn converges to the S1 valued harmonic mapu? which is associated to (a1, . . . , aN ). Further assume that there exists c0 > 0such that

12

∫

Ω

|∇uεn |2 +1

4 εn2

∫

Ω

(1− |uεn |2)2 ≤ c0 log 1/εn,

212

holds. Then, for n large enough, we have

uεn = vεn .

In the sequel, it will be convenient to skip the n indices and simply denoteuε instead of uεn

and vε instead of vεn.

10.2 Proof of the uniqueness result

10.2.1 Geometric modification of the family vε

Thanks to the result of Proposition 9.1, we know that the zero set of uε is givenby a1(ε), . . . , aN (ε) where

limε→0

aj(ε) = aj .

Moreover, by construction, the zero set of vε is given by b1(ε), . . . , bN (ε) where

limε→0

bj(ε) = aj .

There is a priori no reason why aj(ε) should be equal to bj(ε) moreover, weknow that both solutions uε and vε look like a rescaled version of S eiθ, thesolution on all C, up to some shift in the phase. Our aim is here to modify thesolution vε in such a way that the zeros of the two functions do match and thephases at each aj(ε) also match. We will denote by d the distance between thezeros of uε and the zeros of vε. Namely

d := maxj|aj(ε)− bj(ε)|.

It will be convenient to denote by

r := dist(z; a1(ε), . . . , aN (ε)).and, as in the previous Chapter, the parameter σ > 0 is assumed to be chosenin such a way that, for all j = 1, . . . , N , B4σ(aj(ε)) ⊂ Ω and, if j 6= k

B2σ(aj(ε)) ∩B2σ(ak(ε)) = ∅.We also set, for all R ∈ (0, σ)

ΩR := Ω \ ∪Nj=1BR(aj(ε)).

Step 1. Translation of the zeros of vε. As in Chapter 6, we introduce forall c = (c1, . . . , cN ) ∈ CN small enough, the diffeomorphism

ϕc := z +N∑

j=1

η

(2z − aj

σ

)cj ,

213

where η is a cutoff function equal to 1 in B1 and equal to 0 outside B2.

We definevε := vε ϕb(ε)−a(ε),

where we have set

a(ε) := (a1(ε), . . . , aN (ε)) and b(ε) := (b1(ε), . . . , bN (ε)).

Step 2. Phase shift near each aj(ε). Observe that, since vε is a solution ofthe Ginzburg-Landau equation, we know that

div(vε ∧∇vε) = 0.

Thereforediv(vε ∧∇vε),

has compact support in ∪Nj=1Bσ(aj(ε))\Bσ/2(aj(ε)) and we easily find that, for

all k ≥ 0, there exists ck > 0 such that

supΩ

∣∣∇kdiv(vε ∧∇vε)∣∣ ≤ ck d. (10.2)

Let us denote by K the solution of

∆K =1

|vε|2 div (vε ∧∇vε) in Ω

K = 0 on ∂Ω.

(10.3)

It follows at once from (10.2) that, for all k ≥ 0 there exists ck > 0 such that

supΩ|∇kK| ≤ ck d. (10.4)

We define

K :=

1−

N∑

j=1

η

(z − aj(ε)

ε

) K +

N∑

j=1

η

(z − aj(ε)

ε

)K(aj(ε)). (10.5)

With this definition, K is constant in each Bε(aj(ε)) \Bε/2(aj(ε)) and, thanksto (10.4) we have, for all k ≥ 1

|∇kK| ≤ c d εk−1, (10.6)

in each Bε(aj(ε)) \ Bε/2(aj(ε)). Furthermore, (10.4) still holds in Ωε. Thesecond geometric modification of vε is defined by

vε := vε e−iK .

Let us state some simple properties of vε. First of all we have

∆vε +vε

ε2(1− |vε|2) = 0, (10.7)

214

in each Bε/2(aj(ε)). Away from the zero set of vε we may write (at least locally)

vε := |vε| eiφε .

Hence

∆vε +vε

ε2(1− |vε|2) =

(∆|vε|+ |vε|

ε2(1− |vε|2)− |∇φε|2 |vε|

)eiφε

+ i1|vε| div(|vε|2∇φε)eiφε .

We would like to estimate this expression in Ωε/2.

A simple computation yields

div (vε ∧∇vε) = |vε|2 ∆(K − K)−∇|vε|2∇K. (10.8)

Observe that it follows from the construction of vε in Chapter 7 that, for allk ≥ 1, there exists ck > 0 such that

|∇k|vε|| ≤ ck ε2 r−2−k,

in Ωε/2 \ Ω2σ. We may now use the previous estimates together with the es-timates for K, to get, for all k ≥ 0, the existence of a constant ck > 0 suchthat ∣∣∇k (div (v ∧∇v))

∣∣ ≤ ck ε2 d r−3−k, (10.9)

in Ωε/2

We setvε = |vε| eiφε .

Since |v| = |v|, we have the identity

∆|vε|+ |vε|ε2

(1− |vε|2)− |∇φε|2 |vε| = ∆|vε|+ |vε|ε2

(1− |vε|2)− |∇φε|2 |vε|.

Since v is a solution of the Ginzburg-Landau equation in Ωσ \ Ωσ/2, we get

∆|vε|+ |vε|ε2

(1− |vε|2)− |∇φε|2 |vε| = (|∇φε|2 − |∇φε|2) |vε|,

in Ωσ \ Ωσ/2. It is then not very difficult to see that, for all k ≥ 0, there existsck > 0 such that

∣∣∣∣∇k

(∆|vε|+ |vε|

ε2(1− |vε|2)− |∇φε|2 |vε|

)∣∣∣∣ ≤ ck d r−1−k. (10.10)

in Ωε/2.

We are now going to obtain some estimates measuring how far uε is from vε.

215

10.2.2 Estimating the L2 norm of |uε| − |vε|In the next Proposition, whose proof follows closely the strategy of the proof ofTheorem 9.1 in the last Chapter, we estimate the L2 norm of |uε|−|vε|. Since vε

is not assumed to be a solution of the Ginzburg-Landau equation the strategydeveloped in the last Chapter does not lead directly to the conclusion uε = vε

but provides us with an estimate.

Proposition 10.1 Assume that the assumptions of Theorem 10.1 hold. Then,there exists c > 0 such that

∫

Ω

(|u| − |v|)2 ≤ c ε4 d2 log 1/ε, (10.11)

for all ε small enough.

As already mentioned, we will follow step by step the proof of Theorem 9.1.The new difficulty here comes from the fact that vε is not assumed to be asolution of the Ginzburg-Landau equation. Hence, we will have to computecarefully the error term we have introduced when we have translated the zerosof vε and changed the phase of vε near each zero.

As in Chapter 10, we define the function ψuε by

∇⊥ψuε := uε ∧∇uε.

This function is well defined since u is a solution of the Ginzburg-Landau equa-tion and hence div(u ∧∇u) = 0.

Let Hε be the solution of

∆Hε = div (v ∧∇v) in Ω

Hε = 0 on ∂Ω.(10.12)

We also obtain the existence of ψvε such that

∇⊥ψvε := vε ∧∇vε −∇Hε.

Lemma 10.1 Assume that ν ∈ (0, 1) and further assume that the hypothesisof Theorem 10.1 hold. Then, there exists c > 0 and ε0 > 0 such that, for allε ∈ (0, ε0), we have ∫

Ω

r2−2ν |∇Hε|2 ≤ c ε4−2ν d2. (10.13)

In addition, we have‖r∇Hε‖L∞ ≤ c ε d. (10.14)

216

Proof : Combining (10.9) and Lemma 7.1 we get for all ν ∈ (0, 1)

∫

Ω

r2−2ν |∇H|2 ≤ c ε4−2ν d2 + c

N∑

j=1

(∫

Bσ/4(aj(ε))

∆H

)2

. (10.15)

In addition we have∫

Bσ/4(aj(ε))

∆H =∫

Bσ/4(aj(ε))

div (v ∧∇v).

Since v = v e−i K and since we have K = K on ∂Bσ/4(aj(ε)), we obtain

v ∧ ∂ν v = −|v|2 ∂νK + v ∧ ∂ν v.

Since div(v ∧∇v) = 0 in Bσ/2(aj(ε)), we can write the previous equality as∫

Bσ/4(aj(ε))

∆H = −∫

∂Bσ/4(aj(ε))

|v|2 ∂νK.

Since ∆K = 0 in Bσ/2(aj(ε)), we finally get that∫

Bσ/2(aj(ε))

∆H =∫

∂Bσ/4(aj(ε))

(|v|2 − 1) ∂νK.

Finally, observe that, on Bσ/2(aj(ε)), the function v is just a translation of v andusing the estimates obtained for |v| and the estimate (10.6) for K, we concludethat ∫

Ω

r2−2ν |∇H|2 ≤ c ε4−2ν d2 + c ε4 d2 ≤ c ε4−2ν d2.

In order to derive the last estimate, we make use of the fact that for all p > 1and all function U ∈ W 2,p(B2 \B1/2), we have

sup∂B1

|∇U | ≤ c

(∫

B2\B1/2

|∆U |p)1/p

+ c

(∫

B2\B1/2

|U |2)1/2

.

We refer to Theorem 9.11 in [26] for a proof of this result. Using a simple scalingargument we get

r sup∂Br

|∇U | ≤ c

(r2p−2

∫

B2r\Br/2

|∆U |p)1/p

+ c

(∫

B2r\Br/2

|U |2)1/2

,

for all U ∈ W 2,p(B2r \ Br/2). We apply this inequality to H and use the factthat |∆H| ≤ c ε2 r−3 together with (10.13) to get the pointwise bound

r |∇H| ≤ c (ε2 r−1 + ε2−ν rν−1) ≤ c d ε,

217

in each Bσ(aj(ε)).

Finally, we also have

supΩσ

|∇H| ≤ c

(∫

Ωσ/2

|∆H|p)1/p

+

(∫

Ωσ/2

|H|2)1/2

.

Which easily leads tosupΩσ

|∇H| ≤ c d ε.

The proof of the Lemma is therefore complete. 2

Paralleling Lemma 9.7, we have

Lemma 10.2 Assume that ν ∈ (1 − 2−1/2, 1) and further assume that the hy-pothesis of Theorem 10.1 hold. Then, there exists c > 0 and ε0 > 0 such that,for all ε ∈ (0, ε0), we have

∫

Ω

r2−2ν

|uε|2 |∇(ψuε − ψvε)|2 ≤ c

∫

Ωε

r−2ν (|uε| − |vε|)2

+c

∫

Ω\Ω2ε

r2−2ν

|X|2∣∣∣∣∂X

(uε

vε

)∣∣∣∣2

+c ε4−2ν d2.

(10.16)

Where r denotes the distance from z to the zero set of uε and where X is thevector field defined by

X := |vε|2 vε ∧∇⊥vε

|vε ∧∇⊥vε|2 .

Proof : For the sake of simplicity in the notations, we drop the ε indices.

Since u and v have the same zeros a1(ε), . . . , aN (ε) and the same multi-plicities d1, . . . , dN, the following holds

div(

u ∧∇⊥u

|u|2)

= div(

v ∧∇⊥v

|v|2)

= 2 π

N∑

j=1

dj δaj(ε), (10.17)

hence ψu and ψv satisfy

div(∇ψu

|u|2)

= div

(∇ψv

|v|2 − ∇⊥H

|v|2)

.

Moreover, we have on ∂Ω

∂νψu = u ∧ ∂τu := g ∧ ∂τg,

and∂νψv + ∂τH = v ∧ ∂ν v.

218

Since H = 0 on ∂Ω, we obtain

∂νψv = v ∧ ∂ν v := g ∧ ∂τg.

As in the proof of Lemma 9.7, we set ξ = ψv − ψu. This time ξ satisfies

−∆ξ + 2∇|u||u| ∇ξ = |u|2 div

((1|v|2 −

1|u|2

)∇ψv

)− |u|2 div

(∇⊥H

|v|2)

,

in Ω together with∂νξ = 0,

on ∂Ω. Observe that

|u|2 div(∇⊥H

|v|2)

= −2|u|2|v|3 ∇

⊥H∇|v|.

Since there exists c > 0 such that

sup(

r∇|v||v|

)≤ c,

we get ∫

Ω

r4−2ν

∣∣∣∣|u|2 div(∇⊥H

|v|2)∣∣∣∣

2

≤ c

∫

Ω

r2−2ν |∇H|2. (10.18)

The result follows at once from (10.14), (10.13) and (10.18), using the argu-ments developed in the proof of Lemma 9.7, and taking into account the newterm involving H. 2

As in the proof of Theorem 9.1, we will need results corresponding to Corol-laries 9.1 and 9.2.

Corollary 10.1 Assume that ν ∈ (1 − 2−1/2, 1) and further assume that theassumptions of Theorem 9.1 hold. Then, there exists c > 0, γ0 > 0 and ε0 > 0such that, for all ε ∈ (0, ε0) and for all R ∈ [γ0 ε, σ], we have

1ε2

∫

ΩR

(|uε| − |vε|)2 +∫

ΩR

|∇(|uε| − |vε|)|2 ≤ c d2 ε2 (1 + | log R|)

+ c( ε

R

)2(

1ε2

∫

Ωε\ΩR

(|uε| − |vε|)2 +∫

Ω\Ω2ε

1|X|2

∣∣∣∣∂X

(uε

vε

)∣∣∣∣2)

,

(10.19)and also

∫

ΩR

|∇(ψuε − ψvε)|2 ≤ c( ε

R

)2−2ν

d2ε2 (1 + | log R|)

+ c( ε

R

)2(

1ε2

∫

Ωε\ΩR

(|uε| − |vε|)2 +∫

Ω\Ω2ε

1|X|2

∣∣∣∣∂X

(uε

vε

)∣∣∣∣2)

.

219

Where X is the vector field defined by

X := |vε|2 vε ∧∇⊥vε

|vε ∧∇⊥vε|2 .

Proof : The proof follows at once from a straightforward modification ofLemma 9.8. Indeed, we still have

∆|u|+ |u|(

1− |u|2ε2

− |∇ψu|2|u|4

)= 0. (10.20)

However, this time

R := ∆|v|+ |v|(

1− |v|2ε2

− |∇ψv|2|v|4

), (10.21)

is not equal to 0 anymore, since v is not a solution of the Ginzburg-Landauequation and also since

∇⊥ψv 6= v ∧∇v.

This induces an extra term on the right hand side of the inequalities of Lemma 9.8.

This extra term can be estimated using (10.10) we know that

|R| ≤ c

rd +

|∇H|r

+ |∇H|2. (10.22)

Hence∫

ΩR/2

|R|2 ≤ c d2 (1 + | log R|) +∫

ΩR/2

|∇H|2r2

+∫

ΩR/2

|∇H|4.

Using (10.13), we estimate∫

ΩR/2

|∇H|2r2

≤ cR2ν−4

∫

Ω

r2−2ν |∇H|2 ≤ c d2( ε

R

)4−2ν

.

Finally, (10.2) implies that∫

Ω

|∆H|2 ≤ c d2.

Therefore, we obtain ∫

Ω

|∇H|4 ≤ c d4.

Collecting these estimates we can conclude that∫

ΩR/2

|R|2 ≤ c d2 (1 + | log R|).

220

Observe that we have implicitly assumed that d is bounded ! The remaining ofthe proof is identical to the proof of Lemma 9.8. We set

D := |u| − |v|,

and find that∆D − V D = F ,

where

V :=1ε2

(|u|2 + |u| |v|+ |v|2 − 1) +|∇ψu|2|u|4 − |∇ψv|2 |u|

2 + |v|2|u|4|v|3 (|u|+ |v|),

and

F :=|v||u|4 (|∇ψu|2 − |∇ψv|2)−R.

The remaining of the proof is now straightforward and is left to the reader. 2

Proof of Proposition 10.1 A priori the function v is not a solution ofthe Ginzburg-Landau equation. Hence, instead of the Pohozaev formula ofProposition 8.3, we need to apply to

w :=u

v,

the Pohozaev formula we have obtained in Proposition 8.4. This provides uswith the following identity

14ε2

∫

ω

div(|v|2 X) (1− |w|2)2 + 2∫

ω

1|X|2

∂X |v||v| |∂Xw|2

= −∫

∂ω

∂Xw · ∂νw +12

∫

∂ω

|∇w|2 X · ν

+1

4ε2

∫

∂ω

(1− |w|2)2|v|2X · ν −∫

∂ω

w ∧ ∂τw

+∫

ω

1|v|2 div (v ∧∇v) ∂Xw · ∂X⊥w

−∫

ω

∂Xw · w

v

(∆v + v

(1− |v|2)ε2

).

(10.23)

where ω is any of the Bσ(aj(ε)) and where σ ∈ [σ, 2σ]. As we have alreadydone in the proof of Theorem 9.1, we are going to estimate all the terms on theright hand side of this identity. To simplify the formula, we avoid the mentionof aj(ε) in Bσ(aj(ε)).

As usual, it will be convenient to write

u := |u| eiφu , and v := |v| eiφv .

221

Step 1. Arguing as in Step 1 and Step 2 of the proof of Theorem 9.1, it isan easy exercise to see that (10.23) leads to

dj

4ε2

∫

Bσ

div(|v|2 X) (1− |w|2)2 + 2 dj

∫

Bσ

1|X|2

∂X |v||v| |∂Xw|2

≤ c1ε2

∫

Ωσ

(|u| − |v|)2 + c

∫

Ωσ

|∇(|u| − |v|)|2

+c

∫

Ωσ

|∇(ψu − ψv)|2 + c

∫

Ωσ

|∇H|2

+∣∣∣∣∫

B2σ

1|v|2 div (v ∧∇v) ∂Xw · ∂X⊥w

∣∣∣∣

+∣∣∣∣∫

Bσ

∂Xw · w

v

(∆v + v

(1− |v|2)ε2

)∣∣∣∣ .

(10.24)

Notice that this time we have estimated the L2 norm of ∇(φu − φv) in Ωσ bythe L2 norm of ∇(ψu − ψv), |u| − |v| and ∇(|u| − |v|) as before, but also the L2

norm of ∇H on the same set.

In addition, let us recall that σ is chosen in [σ, 2σ] in such a way that∫

∂Bσ

(|u| − |v|)2 ≤ 4σ

∫

B2σ\Bσ

(|u| − |v|)2. (10.25)

Step 2. We claim that∫

B2σ\Bε/2

|∇(|u| − |v|)|2 ≤ c

ε2

∫

B4σ\Bε/4

(|u| − |v|)2

+∫

B4σ\Bε/4

|∇(ψu − ψv)|2 + c d2 ε2 log 1/ε.

This inequality is obtained following closely the strategy of the proof of Lemma 9.8.As in the proof of Corollary 10.1, we set D := |u| − |v|, we have already seenthat D solves

∆D − V D = F .

By opposition with what we have done in the proof of Lemma 9.8, we considerthe function χ defined by

χ := η

(z − aj(ε)

4σ

)− η

(z − aj(ε)

ε/2

),

where η is a cutoff function equal to 1 in B1/2 and equal to 0 outside B1.Following closely the arguments of Lemma 9.8 and using the estimates for Rwhich are derived in the proof of Corollary 10.1, we obtain

∫

Ω

χ2 |∇(|u| − |v|)|2 ≤ c

ε2

∫

B4σ\Bε/4

(|u| − |v|)2

+ c

∫

B4σ\Bε/4

|∇(ψu − ψv)|2 + c ε2 d2 log 1/ε.

222

The proof of the claim is therefore complete.

Step 3. Using (10.9), we can write∣∣∣∣∫

B2σ

1|v|2 div (v ∧∇v) ∂Xw · ∂X⊥w

∣∣∣∣ ≤ c d

∫

B2σ\Bε/2

ε2

r|∇w|2. (10.26)

Notice that div (v ∧∇v) = 0 in Bε/2. Thanks to (9.19), we have

|∇|u|| ≤ cε2

r3.

Moreover, both u and v are uniformly bounded from below in B4σ\Bε/4. Hence,we have the bound

|∇w|2 ≤ c

(ε4

r6(|u| − |v|)2 + |∇(|u| − |v|)|2 + |∇(φu − φv)|2

),

in B4σ \Bε/4. Using the claim proved in Step 2, we conclude that∣∣∣∣∫

B2σ

1|v|2 div(v ∧∇v)∂Xw · ∂X⊥w

∣∣∣∣ ≤ cd

ε

∫

B4σ\Bε/4

(|u| − |v|)2

+ c dε

∫

B4σ\Bε/4

|∇(ψu − ψv)|2

+ c d3 ε3 log 1/ε.(10.27)

Step 4. Observe that, for all fixed γ, we may decompose∫

Bσ

1|X|2

∂X |v||v| |∂Xw|2 =

∫

Bγε

1|X|2

∂X |v||v| |∂Xw|2

+∫

Bσ\Bγε

1|X|2

∂X |v||v| |∂Xw|2.

Now, the last term can be estimated as follows∣∣∣∣∣∫

Bσ\Bγε

1|X|2

∂X |v||v| |∂Xw|2

∣∣∣∣∣ ≤ c

∫

Bσ\Bγε

ε2

r2|∇w|2

≤ c

γ2

∫

Bσ\Bγε

|∇(|u| − |v|)|2

+c

γ4 ε2

∫

Bσ\Bγε

(|u| − |v|)2

+c

γ2

∫

Bσ\Bγε

|∇(ψu − ψv)|2

+c

γ2

∫

Bσ\Bγε

|∇H|2.

223

We conclude using the result of Step 2, together with (10.13) that∣∣∣∣∣∫

Bσ\Bγε

1|X|2

∂X |v||v| |∂Xw|2

∣∣∣∣∣ ≤ c

γ2 ε2

∫

B4σ\Bε/4

(|u| − |v|)2

+c

γ2

∫

B4σ\Bε/4

|∇(ψu − ψv)|2

+c

γ4−2νd2 ε2.

(10.28)

Step 5. We bound ∫

Ωσ

|∇H|2 ≤ c d2 ε4−2ν ,

thanks to (10.13). Further observe that we have from Lemma 10.2∫

Bσ\Bε/4

|∇(ψu − ψv)|2 ≤ c

ε2

∫

Ωε

(|u| − |v|)2

+ c

∫

Ω\Ω2ε

1|X|2 |∂Xw|2 + c d2 ε2.

Now, we return to (10.24) and use all the estimates we have obtained in Step2,3 and 4, together with the results of Corollary 10.1 and Lemma 10.2. Arguingas in the proof of Theorem 9.1, we conclude that

dj

4ε2

∫

Bσ

div(|v|2 X) (1− |w|2)2 + 2 dj

∫

Bγε

1|X|2

∂X |v||v| |∂Xw|2

≤ c (ε2−2ν + γ2ν−4 + d ε)1ε2

∫

Ωε/2\Ωσ

(|u| − |v|)2

+c (ε2−2ν + γ2ν−4 + d ε)∫

Ω\Ω2ε

1|X|2

∂X |v||v| |∂Xw|2

+c d2 ε2 +∣∣∣∣∫

Bσ

∂Xw · w

v

(∆v + v

(1− |v|2)ε2

)∣∣∣∣ .

(10.29)

Step 6. It remains to estimate the last term of the right hand side of (10.29).To this aim, we write

1v

(∆v + v

(1− |v|2)ε2

)=

1|v|

(∆|v|+ |v|

(1− |v|2

ε2− |∇φv|2

))

+i

|v|2 div(v ∧∇v).

For notational convenience, the first term on the right hand side of this identitywill be denoted by R1 and the second one by R2.

224

Observe that R1 ≡ 0 in all Bε/2 and also that |R1| + r |∇R1| ≤ c d r−1 inBσ \Bε/2. Hence, we can integrate by parts and obtain

∫

Bσ

∂Xw · wR1 =∫

Bσ

(|w|2 − 1) div(XR1) +∫

∂Bσ

(|w|2 − 1)R1X · ν

≤ c d

∫

Bσ\Bε/2

1r

(|w|2 − 1) + c d

∫

∂Bσ

(|w|2 − 1)

≤ c d

∫

B2σ\Bε/2

1r

(|w|2 − 1).

Here, we have used (10.25) to obtain the last estimate. Now, for all κ1 > 0, weuse the inequality

a b ≤ κ1 a2 +1

4κ1b2, (10.30)

to obtain∫

Bσ

∂Xw · w R1 ≤ c

κ1ε2 d2 log 1/ε + c

κ1

ε2

∫

B2σ\Bε/2

(|u| − |v|)2. (10.31)

Recall that, by definition of ∧, we have

∂Xw · (iw) = w ∧ ∂Xw.

Furthermore, we have

w ∧∇w =|u|2|v|2 ∇(φu − φv).

Using this together with (10.9), we already obtain the estimate∫

Bσ\Bε/2

w ∧∇wR2 ≤ c d

∫

Bσ\Bε/2

ε2

r2|∇(φu − φv)|

≤ cd

∫

Bσ\Bε/2

ε2

r2

(|∇(ψu − ψv)|+ 1

r||u| − |v||

)

+ c d

∫

Bσ\Bε/2

ε2

r2|∇H|.

Making use of (10.30) and Cauchy-Schwarz inequality, we obtain, for all κ1 > 0∫

Bσ\Bε/2

w ∧∇wR2 ≤ c κ1

∫

Bσ\Bε/2

|∇(ψu − ψv)|2

+cκ1

ε2

∫

Bσ\Bε/2

(|u| − |v|)2 + cκ1 d2 ε2

+c d εν

(∫

Bσ\Bε/2

r2−2ν |∇H|2)1/2

≤ c κ1

∫

Bσ\Bε/2

|∇(ψu − ψv)|2

+cκ1

ε2

∫

Bσ\Bε/2

(|u| − |v|)2 + cκ1 d2ε2.

225

Here, we have used (10.13) to obtain the last estimate.

We conclude, using (10.29) that

dj

4ε2

∫

Bσ

div(|v|2 X) (1− |w|2)2 + 2 dj

∫

Bγε

1|X|2

∂X |v||v| |∂Xw|2

≤ c (ε2−2ν + γ2ν−4 + d ε + κ1)1ε2

∫

Ωε/2\Ωσ

(|u| − |v|)2

+ c (ε2−2ν + γ2ν−4 + d ε + κ1)∫

Ω\Ω2ε

1|X|2

∂X |v||v| |∂Xw|2

+ c d2 ε2 log 1/ε.

(10.32)

To complete the proof it suffices to chose κ1 small enough but fixed, γ largeenough but fixed and then ε small enough. The end of the proof being identicalto Step 3 in the proof of Theorem 9.1, we omit it. 2

Observe that, as a byproduct of the previous proof we have also obtainedthe following inequality

∫

B2ε(aj(ε))

1|X|2

∣∣∣∣∂X

(uε

vε

)∣∣∣∣2

≤ c d2 ε2 log 1/ε. (10.33)

This implies, using Lemma 10.2, that∫

Ω

r2−2ν

|uε|2 |∇(ψuε − ψvε)|2 ≤ c d2 ε4−2ν log 1/ε. (10.34)

Combining this together with the result of Corollary 10.1 yields∫

Ωσ

|∇(|u| − |v|)|2 ≤ c d2 ε2 log 1/ε.

Moreover, arguing as in Step 2 of the proof of Proposition 10.1, we also get, forall λ > 0, the existence of a constant cλ > 0 such that

∫

Ωλε\Ωσ

|∇(|u| − |v|)|2 ≤ cλ d2 ε2 log 1/ε.

Hence, we conclude that∫

Ωλε

|∇(|u| − |v|)|2 ≤ cλ d2 ε2 log 1/ε. (10.35)

10.2.3 Pointwise estimates for uε − vε and |uε| − |vε|.Based on the results of the previous section, we derive pointwise estimates foruε − vε and |uε| − |vε|. This is the content of the following :

226

Proposition 10.2 Assume that the assumptions of Theorem 10.1 hold. Then,for all k ∈ N, for all ν ∈ (0, 1) and for all p ∈ (2, 4/(ν + 1)), there exists c > 0such that

supΩ

(rk |∇k(uε − vε)|

)+ sup

Ω

(rk |∇k(φuε

− φvε)|) ≤ c d2/p εν , (10.36)

andsupΩ

(r2+k−ν |∇k(|uε| − |vε|)|

) ≤ c d2/p ε2, (10.37)

Moreover we havesupΩ\Ωε

|∇k(uε − vε)| ≤ c d2/p εν−k. (10.38)

Proof : We choose µ ∈ (ν, 1). Let λ ∈ (0, 1/2) be fixed small enough, namelywe want λ2 to be smaller than 1/8 times the first eigenvalue of ∆ on B1, for theDirichlet boundary condition.

Step 1. We first establish the estimates in Ωλε. As usual, we set

u := |u| eiφu and v := |v| eiφv .

With these notations, we may write

u− v =(|u|

(ei(φu−φv) − 1

)+ (|u| − |v|)

)eiφv . (10.39)

Recall that, by definition, we have

∇(φu − φv) =1|u|2 ∇

⊥ψu − 1|v|2 ∇

⊥ψv +1|v|2 ∇H. (10.40)

We already know from (10.14) that

supΩ

(r |∇H|) ≤ c d ε.

In order to prove the result, it remains to obtain a pointwise estimate for∇(ψu−ψv) and also for |u| − |v|.

Step 2. We claim that the following pointwise estimate holds

supΩλε

(r |∇(ψu − ψv)|

)≤ cp d2/p εµ, (10.41)

provided p > 2 satisfies

µ <2p.

We set ξ := ψv − ψu. We have already seen that

−∆ξ + 2∇|u||u| ∇ξ = |u|2 div

((1|v|2 −

1|u|2

)∇ψv

)− |u|2 div

(∇⊥H

|v|2)

.

227

Using the fact that |∇|u||+ |∇|v|| ≤ c ε2 r−3, |∇ψv| ≤ c r−1, together with thepointwise estimate for ∇H, we get

|∆ξ| ≤ cε2

r3|∇ξ|+ c

r|∇(|u| − |v|)|+ c

ε2

r4||u| − |v||+ c d

ε3

r4.

We make use of the trick we have already used in the proof of Lemma 10.1.Namely, for all p > 2 and all r ∈ [ε, σ], we have

sup∂Br

r |∇ξ| ≤ c

(r2p−2

∫

B2r\Br/2

|∆ξ|p)1/p

+ c

(∫

B2r\Br/2

|∇ξ|2)1/2

.

Using the fact that|∇ξ| ≤ c r−1,

we easily get(

r2p−2

∫

B2r\Br/2

(ε2

r3|∇ξ|

)p)1/p

≤ cε

r

(∫

B2r\Br/2

|∇ξ|2)1/p

.

Now we make use of (10.34) to conclude that

(r2p−2

∫

B2r\Br/2

(ε2

r3|∇ξ|

)p)1/p

≤ c(d2 ε2 log 1/ε

)1/p.

Using similar arguments, (10.35), Proposition 10.1 and also

||u| − |v||+ r |∇(|u| − |v|)| ≤ c ε2 r−2,

we conclude easily that, for all r ≥ ε

(r2p−2

∫

B2r\Br/2

|∆ξ|p)1/p

≤ c(d2 ε2 log 1/ε

)1/p.

Finally, we also have(∫

B2r\Br/2

|∇ξ|2)1/2

≤ c d2 ε2 log 1/ε.

The claim follows at once since

(ε2 log 1/ε)1/p ≤ c d2/p εµ,

if µ < 2/p.

Step 3. We claim now that we have

supΩλε

(r2−µ ||u| − |v||) ≤ ε2 d2/p, (10.42)

228

provided p > 2 satisfies

µ <4− p

p.

In particular, this will show that (10.37) holds in Ωλε, for k = 0.

Indeed, let us denote by

χ := 1−N∑

j=1

η

(2z − aj(ε)

λε

)

where η is a cutoff function equal to 1 in B1 and equal to 0 outside B2. Thanksto (10.35), Proposition 10.1 and also to the fact that

||u| − |v||+ r |∇(|u| − |v|)| ≤ c ε2 r−2,

we get‖χ(|u| − |v|)‖W 1,p(Ω) ≤ c

(d2 ε4−p log 1/ε

)1/p. (10.43)

Invoquing Sobolev embedding, we conclude that

‖χ(|u| − |v|)‖L∞(Ω) ≤ c(d2 ε4−p log 1/ε

)1/p. (10.44)

this already proves the claim in BMε \Bλε for any fixed M > 0.

Recall that we have already seen in the proof of Corollary 10.1 that D :=|u| − |v| satisfies an equation of the form

∆D − V D = F .

Moreover, provided λ0 > 0 is fixed large enough we may assume that

V ≥ 1ε2

,

in Ωλ0ε.

Thanks to Step 1 and also to (10.14), we know that

|F | ≤ c((d2 ε2 log 1/ε)1/p r−2 + d r−1

).

In view of (10.44) we can conclude that the function

z ∈ Ω −→ c ε2((d2 ε2 log 1/ε)1/p r−2 + d r−1

)

+(d2 ε4−p log 1/ε

)1/p (e(λ0ε−r)/ε + e(r−2σ)/ε

),

can be used as a barrier function in B2σ \Bλ0ε to show that

|D| ≤ c ε2 r2−µ,

229

in Bσ \Bλ0ε, provided

µ <4− p

p.

In particular |D| ≤ cε2 on ∂Ωσ.

Finally, the functionz ∈ Ω −→ c ε2,

can be used as a barrier function in Ωσ to show that

|D| ≤ c ε2,

in Ωσ. This ends the proof of the claim.

Step 4. It follows from (10.41) and (10.42) that

supΩλε

(r |∇(φu − φv)|) ≤ c d2/p εµ.

Since φu − φv = 0 on ∂Ω, this estimates yields

supΩλε

|φu − φv| ≤ c d2/p εµ log 1/ε.

In particular we get

supΩλε

|u− v| ≤ c d2/p εµ log 1/ε ≤ c d2/p εν .

To conclude, we have already proved (10.36) in Ωλε, for k = 0.

Step 5. Observe that

‖u− v‖L∞(∂Ωλε) ≤ c d2/p εµ. (10.45)

Further observe that v is a solution of the Ginzburg-Landau equation in eachBε/2. We set

U := u(λεz) and V := v(λεz).

These functions are solutions of

∆U + λ2U(1− |U |2) = 0, (10.46)

and∆V + λ2V (1− |V |2) = 0,

in B1.

Let W denote the harmonic extension of (V − U)∂B1 in B1. We denote

T := ∆(V + W ) + λ2(V + W )(1− |V + W |2). (10.47)

230

Thanks to (10.45), we know that

‖W‖L∞(B1) ≤ c d2/p εµ.

Hence,|T | ≤ c d2/p εµ.

in B1. Taking the scalar product of V + W − U with the difference between(10.47) and (10.46), we obtain after an integration by part

∫

B1

|∇(V + W − U)|2 − λ2

∫

B1

|V + W − U |2

+λ2

∫

B1

(V + W − U) · (U |U |2 − (V + W ) |V + W |2)

≤ c (d2/p εµ)1/2

(∫

B1

|V + W − U |2)1/2

.

Since λ2 has been chosen less than 1/8 times the first eigenvalue of ∆ in B1,it is an easy exercise to conclude that

∫

B1

|V + W − U |2 ≤ c d2/p εµ.

A classical bootstrap argument then shows that

‖V + W − U‖L∞(B1) ≤ c d2/p εµ.

Performing the scaling backward, we obtain

‖u− v‖L∞(Bλε) ≤ d2/p εµ.

which proves (10.38) and completes the proof of (10.36) and (10.37) when k = 0.

The proof of the estimates for arbitrary k relies on a induction argument.2

10.2.4 The final arguments to prove that uε = vε.

We keep the notations of Chapter 7 and use all the function spaces defined inChapter 4, Chapter 6 and Chapter 7. As in Chapter 7, the parameter µ ischosen in (1, 2).

To begin with, let us recall that the solution vε we have constructed, can bewritten in the following way

vε = eiψd0

(eiφ

((|u|+ wr) ei(1−χε)

wi|u| + i χε wi

)) ϕd±1 , (10.48)

where d0 ∈ RN , d±1 ∈ (R2)N and w ∈ E . In addition, we have the bound

|d0|RN + |d±1|RN×RN + ‖w‖E ≤ c ε1−α (10.49)

231

where α is any fixed number α ∈ (0, 1) (see Theorem 6.1).

Observe that we can also write

uε = eiψd0

(eiφ

((|u|+ wr) ei(1−χε)

wi|u| + i χε wi

)) ϕd±1

(10.50)

for some d0 ∈ RN , d±1 ∈ (R2)N and w ∈ E . However, this time we do not haveany a priori control on the parameters. Notice that the same would be true forany complex valued map u defined in Ω whose zero set has exactly N pointsclose to the aj , with corresponding degree dj .

Though we have not made this explicit in the notation, the parametersd0, d±1, w and d0, d±1, w all depend on ε since uε and vε do.

Since uε and vε are solutions of the Ginzburg-Landau problem, we obviouslyhave

(d0, d±1, w) = − (DM(0,0,0)

)−1 (M(0, 0, 0) + Q1(d0, d±1)

+Q2(d0, d±1, w) + Q3(d0, d±1, w)),

and

(d0, d±1, w) = − (DM(0,0,0)

)−1 (M(0, 0, 0) + Q1(d0, d±1)

+Q2(d0, d±1, w) + Q3(d0, d±1, w)).

Hence

(d0, d±1, w)− (d0, d±1, w) = − (DM(0,0,0)

)−1 (

Q1(d0, d±1)−Q1(d0, d±1)

+Q2(d0, d±1, w)−Q2(d0, d±1, w)

+Q3(d0, d±1, w)−Q3(d0, d±1, w)).(10.51)

The strategy is now to prove that the parameters l1, l2 and l3 which are definedin Lemma 6.1, Lemma 6.2 and Lemma 6.3 can be assumed to be arbitrarily smallprovided ε is small enough. As a consequence, we will obtain using (10.51)

‖(d0, d±1, w)− (d0, d±1, w)‖RN×(RN×RN )×E

≤ 12‖(d0, d±1, w)− (d0, d±1, w)‖RN×(RN×RN )×E ,

provided ε is small enough. In particular, this will imply that

(d0, d±1, w) = (d0, d±1, w).

Henceuε = vε,

232

as wished.

Estimate for d±1. By definition, the zero set of the function uε is given bya1(ε), . . . , aN (ε) but also by

aj − (d+1,j + i d−1,j) : j = 1, . . . , N.

Similarly, the zero set of vε is given by b1(ε), . . . , bN (ε) but also by

aj − (d+1,j + id−1,j) : j = 1, . . . , N.

Hence we have

aj(ε)− bj(ε) = (d+1,j + id−1,j)− (d+1,j + i d−1,j).

Since by assumptionlimε→0

aj(ε) = aj ,

we conclude thatlimε→0

d±1 = 0. (10.52)

Estimate for d0. By definition, we have

vε := e−iK vε ϕb(ε)−a(ε).

Moreover, since χ ≡ 1 in Bε(aj(ε)), we can write

uε − vε = eiφ(eid0(|u|+ w)

)− ei(φ−K)

(eid0(|u|+ w)

). (10.53)

For the sake of convenience, we have dropped all the j indices and we willimplicitly assume that aj(ε) = 0.

Thanks to Proposition 9.1, we already know that the sequence of functionsuε(·+aj(ε)) converges uniformly to S ei(θ+τj) in B2ε. Similarly, the sequence offunctions vε(·+ aj(ε)) converges uniformly to S ei(θ+τj) in B2ε. Hence, we have

limε→0

|u|+ w

|u| = limε→0

|u|+ w

|u| = 1,

uniformly in B2ε.

Integrating (10.53) divided by u over ∂Bε, we find

limε→0

1ε

∫

∂Bε

uε − vε

u= lim

ε→0

1ε

∫

∂Bε

(eid0 − ei(d0−K)).

It follows from Proposition 10.2 that the left hand side of this expression isbounded by ∣∣∣∣

1ε

∫

∂Bε

uε − vε

u

∣∣∣∣ ≤ c d2/p εν .

233

So this expression tends to 0 as ε does. Moreover, we have from (10.5)

sup∂Bε

|K| ≤ c d.

Since d tends to 0 as ε does, this implies that

limε→0

|d0 − d0| = 0.

However, we already know that d0 tends to 0 hence

limε→0

d0 = 0. (10.54)

Estimate of w in Ω2ε. In this set, χ ≡ 0, hence we have

u = eiψd0

(eiφ (|u|+ wr) ei

wi|u|

) ϕa−a(ε),

andv = e−iK eiψd0

(eiφ (|u|+ wr) ei

wi|u|

) ϕa−a(ε).

In particular, we get

wr − wr = (|u| − |v|) ϕa(ε)−a.

Using the pointwise bound of Proposition 10.2, we conclude easily that

‖wr − wr‖C1,α0 (Ω2ε) + ‖wr − wr‖C2,α

µ−2(Ω2σ\Ωσ) ≤ c d2/p ε2.

Therefore

limε→0

‖wr − wr‖C1,α0 (Ω2ε) = lim

ε→0‖wr − wr‖C2,α

µ−2(Ω2σ\Ωσ) = 0. (10.55)

In addition, we also have

φu − φv = ψd0− ψd0 + i

(wi − wi

|u|) φa−a(ε) + iK.

Applying once more the result of Proposition 10.2, we get

‖wr − wr‖C1,α0 (Ω2ε) + ‖wr − wr‖C2,α

µ−2(Ω2σ\Ωσ) ≤ c(d + |d0 − d0|+ d2/pε2).

Therefore

limε→0

‖wr − wr‖C1,α0 (Ω2ε) = lim

ε→0‖wr − wr‖C2,α

µ−2(Ω2σ\Ωσ) = 0. (10.56)

Estimates for w in B2ε. In Bε, χ ≡ 1, hence we have

w − w = (e−iK − 1)w + e−φ (u− v)− (eid0 − ei(d0−K)) |u|.

234

Applying one last time Proposition 10.2, it is easy to see that

‖w − w‖C2,α0 (Ω\Ωε) ≤ c(d ‖w‖C2,α

0 (Ω\Ωε) + d2/p εν + d + |d0 − d0|).

Hence, we conclude that

limε→0

‖w − w‖C2,α0 (Ω\Ωε) = 0.

Though things are slightly more involved, we can prove using the same strategythat

limε→0

‖w − w‖C2,α0 (Ω\Ωε) = 0. (10.57)

Clearly, (10.54) and (10.52) imply that l1 (which is defined in Lemma 6.1)tends to 0 as ε does while (10.55), (10.56) and (10.57) imply that l2 (which isdefined in Lemma 6.2) and l3 (which is defined in Lemma 6.3) also tend to 0 asε does. This proves the desired result.

10.3 A conjecture of F. Bethuel, H. Brezis andF. Helein

In the special case where Ω = B1 and g = eiθ the previous Theorem allows usto give a positive (partial) answer to Brezis’s conjecture.

Theorem 10.2 Any critical point of the Ginzburg-Landau functional on theunit disk which is equal to eiθ on the boundary is, for ε sufficiently small, theaxially symmetric solution defined in Theorem 8.2.

Otherwise stated, any solution of

∆u +u

ε2(1− |u|2) = 0 in B1

u = eiθ on ∂B1

(10.58)

is, for ε small enough, the axially symmetric solution defined in Theorem 8.2.This result was conjectured by F. Bethuel, H. Brezis and F. Helein in [11].

The main difference between Theorem 10.1 and Theorem 10.2 is that, inthe latter, we do not assume any a priori bound on the energy of the solution.Recall that this bound is needed in theorem 10.1 as it was already needed inTheorem 9.1, to control the number of vortices of the solutions. However, sincethe unit ball is a starshaped domain, we can integrate over B1 the Pohozaevformula which was derived in Chapter 10, Lemma 9.1. As a result, we get

∫

∂B1

(∂Xu · ∂νu− 1

2|∇u|2 X · ν

)+

12ε2

∫

B1

(1− |u|2)2 = 0,

235

where the vector field X is given by X := (x, y). Hence

12

∫

∂B1

|∂νu|2 − 12

∫

∂B1

|∂τu|2 +1

2ε2

∫

B1

(1− |u|2)2 = 0.

Since ∂τu = ∂τeiθ on ∂B1 we conclude that∫

∂B1

|∂νu|2 +1ε2

∫

B1

(1− |u|2)2 = 2 π.

It is also proved in Chapter X of [11] that given any sequence of solutionsof (10.58), one can always extract a subsequence converging to a S1 valuedharmonic map of the form

u? =N∏

j=1

(z − aj

|z − aj |)dj

eiφ,

where φ is an harmonic function in B1. We then invoque Theorem X.5 of [11]which implies that

limε→0

∫

B1

|1− |u|2|2ε2

= 2 π

N∑

j

d2j . (10.59)

However, using (10.59), we deduce then that necessarily N = 1 and d1 = ±1.Using a degree argument, we conclude that dj = 1.

In addition, it is proved in [11] (see pages 96-97) that necessarily the uniquevortex a1 is located at the origin and also that it is a nondegenerate criticalpoint of the corresponding renormalized energy Wg (see also [42] where a sim-ple expression of the renormalized energy in the axially symmetric setting isderived). Hence,

u? =z

|z| .

Using all these informations, we see that Theorem 10.2 is now a Corollary ofTheorem 10.1.

236

Chapter 11

Towards Jaffe and Taubesconjectures

We prove that, under some natural energy bound, all solutions of the gaugeinvariant Ginzburg-Landau equation are gauge equivalent to the radially sym-metric solution.

11.1 Statement of the result

In this last Chapter, we explain how the results we have established in the pre-vious Chapters may be applied to study the gauge invariant Ginzburg-Landaufunctional

Gε(u,A) =∫

Ω

|dAu|2 +1

2ε2

∫

Ω

(1− |u|2)2 +∫

Ω

|dA|2,

for large values of the coupling parameter 1/ε. Here u is a complex valuedfunction and A can be understood either as a vector field

A := (Ax, Ay),

or as a 1-formA := Ax dx + Ay dy.

We recall that, by definition

dAu := da− i Au.

We will always restrict our attention to the case where

Ω = C.

237

11.1.1 Preliminary remarks

It is an easy exercise to show that, if (u,A) is a critical point of Gε, then (u,A)is a solution of the following Euler-Lagrange equation

∆u− 2 i ∂Au− |A|2 u− i divA u = − u

ε2(1− |u|2) in C

− ? d ? d A = u ∧ du− |u|2 A in C.

(11.1)Where ? is the Hodge operator defined on 2-forms by

? dx dy := 1,

on 1 forms by? dx := dy and ? dy := −dx,

and on 0-forms by? 1 := dx dy.

Given (u,A) solution of (11.1), we define the induced magnetic field h by

h := ? dA.

Observe that h is a 0-form, namely a function. It will be useful to notice that,if (u, A) is a solution of (11.1), then h satisfies

−div(∇h

|u|2)

+ h = 0, (11.2)

away from the zero set of u. Indeed, we have

− ? dh = u ∧ dAu = u ∧ (du− iAu).

Hence

? d ?

(1|u|2 dh

)= − ? d

(u

|u|2 ∧ (du− iAu))

= ? d

(u

|u|2 ∧ (iAu))

= ? dA = h.

Here we have used the fact that, away from the zeros of u, we have the identity

d

(u

|u|2 ∧ du

)= 0,

which may be proved by setting, at least locally away from the zero set of u, wecan write

u := |u| eiφ, (11.3)

238

and thus, we compute

u

|u|2 ∧ du =eiφ

|u| ∧(eiφ d|u|+ i |u| eiφ dφ

)

= −dφ.

Moreover, if (u,A) is a solution of (11.1), we can also prove that |u| satisfies

∆|u|+ 1ε2|u| (1− |u|2) =

|∇h|2|u|3 , (11.4)

away from the zero set of u. Indeed, using (11.3), we can multiply the firstequation of (11.1) by e−iφ and take the real part. We find

∆|u|+ 1ε2|u| (1− |u|2) = |A−∇φ|2 |u|.

Moreover, we can use the second equation of (11.1) which yields

|∇h|2 = |u ∧ (∇u− iAu)|2 = |∇φ−A|2,

and the result follows at once.

One of the main property of the functional Gε is the fact that it is invariantunder gauge transformation. More precisely, for any sufficiently smooth functionφ, if we define

uφ := eiφ u,

andAφ := A + dφ.

ThenGε(u,A) = Gε(uφ, Aφ).

In particular, if (u,A) is a solution of (11.1) so is (uφ, Aφ).

Some quantities do not depend on the particular choice of gauge. For exam-ple, the induced magnetic field h := ?dA, the current u ∧ dAu and |u| which donot depend on the choice of gauge.

Very often, it will be convenient to choose a particular gauge to work with :the Coulomb gauge. To define it on some open simply connected subset ω ⊂ C,we solve the following elliptic problem

∆φ = − ? d ? A in ω

∂νφ = −A · ν on ∂ω.(11.5)

And then, we set(u, A) := (eiφ u,A + dφ).

239

Observe that with this particular choice, we have

? d ? A = 0 in ω

A · ν = 0 on ∂ω.

It is interesting to notice that, since ω is simply connected, one may equivalentlydefine ξ as the solution of

∆ξ = ? dA in ω

ξ = 0 on ∂ω,(11.6)

and then define A = ? dξ. Obviously, we have the relation

? dξ = A + dφ.

And ∆ξ = 0 follows easily. Furthermore, since ∂ν + A · ν = 0 on ∂ω we see that∂τ ξ = 0 on ∂ω. Hence, ξ is constant on ∂ω. Since ?dξ does not depend on theparticular choice of this constant, we may take ξ = 0 on ∂ω.

11.1.2 The uniqueness result

The existence of a radially symmetric solution (up to gauge transformation) hasbeen proved by M.S. Berger and Y.Y. Chen.

Theorem 11.1 [9] There exists (v,B) a non constant solution of (11.1) whichis of the form

v := Sε eiθ and B := Tε dθ,

where Sε and Tε only depend on r. The functions Sε and Tε are strictly in-creasing and satisfy

Sε(0) = Tε(0) = 0, limr→+∞

Sε = limr→+∞

Tε = 1.

Finally 0 < Sε < 1 and 0 < Tε < 1 for all r > 0.

Observe that the degree of v/|v| at infinity is equal to +1. To obtain the radiallysymmetric solution whose degree at infinity is −1 is suffices to choose

v := S e−iθ and B := −T dθ.

The main observation is that, if we look for solutions of the form

v := Sε eiθ and B := Tε dθ,

then Sε and Tε have to solve the following coupled system of ordinary differentialequations

d2S

dr2+

1r

dS

dr+

S

ε2(1− S2)−

(T − 1

r

)2

S = 0

d2T

dr2− S2 (T − 1) = 0.

240

The existence of a solution can be obtained as follows. First, for any R > 1,we obtain a solution of this system defined in [1/R, R], with the boundaryconditions S(1/R) = T (1/R) = 0 and S(R) = T (R) = 1. For example, thissolution can be obtained by minimizing the corresponding energy. Then, weobtain a priori estimates which allow one to pass to the limit as R tend to +∞.

One of the main qualitative difference between the function Sε := S(·/ε),where S has been defined in Theorem 3.1,and the function Sε defined in thelast Theorem is that while Sε converges quadratically to 1 at ∞, the functionSε converges exponentially to 1 at ∞. This is an important observation whichhas many physical consequences [36]. Furthermore, observe that the identity

Sε = S1(·/ε),

which is obviously true, is no longer true for Sε.

We will need the following :

Lemma 11.1 There exists a constant c > 0 such that, for all ε ∈ (0, 1) we have

r ∂r|v||v| ≤ c,

in C where v is the solution defined in the previous Theorem.

Our main result asserts that any solution of (11.1) which satisfies someenergy bound, is gauge equivalent to the radially symmetric solution defined inthe previous Theorem. More precisely, we prove the

Theorem 11.2 Let c0 > 0 be given. Then, there exists ε0 > 0 such that, any(uε, Aε) solution of (11.1) satisfying

Gε(uε, Aε) ≤ 2π log 1/ε + c0, (11.7)

then (uε, Aε) is gauge equivalent to one of the two axially symmetric solutionsof (11.1) having respectively degree +1 or −1 at infinity.

11.2 Gauge-invariant Ginzburg-Landau criticalpoints with one zero.

Before we proceed to the proof of this result, we will give a thorough descriptionof the general shape of any solution of (11.1). What we want to obtain is aresult close to the result we have obtained in Proposition 9.1 for the non-gaugeinvariant Ginzburg-Landau equation.

Proposition 11.1 Let c0 > 0 be fixed. Assume that (uε, Aε) is a solution of(11.1) which satisfies (11.7). Further assume that uε(0) = 0. Then, for ε small

241

enough, u−1ε (0) = 0, the degree of uε at 0 is equal to ±1 and there exists

λ > 0 such thatinfC\Bλε

|uε| ≥ 12.

Moreover, up to a subsequence, there exists a gauge φε such that the sequenceof rescaled functions

uε := eφε uε(ε ·),converges in Ck,α

loc topology to S ei(θ+τ) for some τ ∈ R, for all k ∈ N.

The proof of this result is very close to the proof of the corresponding resultin Chapter 10, the main difference being that, this time, we have to prove alsothat the upper-bound (11.7) implies that uε has a unique zero. Again, one ofthe main ingredients is the Pohozaev formula, which, in the case of the gaugeinvariant Ginzburg-Landau equation, reads :

Lemma 11.2 [36], [83]. Assume that (u,A) is a solution of (11.1) and definethe vector field X by X := (x, y). Then for all R > 0 and all z0 ∈ C, we have

R

2

∫

∂BR(z0)

(|(∇u− iAu) · τ |2 − |(∇u− iAu) · ν|2)

+R

4ε2

∫

∂BR(z0)

(1− |u|2)2 X · ν − R

2

∫

∂BR(z0)

|dA|2

− 12ε2

∫

BR(z0)

(1− |u|2)2 +∫

BR(z0)

|dA|2 = 0.

Proof : Notice that, in order to simplify the notations, we can assume thatz0 = 0. In order to prove this formula, the simplest technique is to consider(u, A) to be a critical point of Gε with respect to the variations of the domainas we have done in the first section of Chapter 9.

Assume that η is a cutoff function. We consider, for all t small enough, theone parameter familly of functions

ut(z) := u(z + t η z),

and of one forms

At(z) = (1 + t η)A(z + t η z) + t (z ·A(z + t η z)) dη.

and write thatd

dtGε(ut, At)|t=0 = 0.

The reason why we have done such choices can be motivated as follows. Withthese formulas we get explicitely

∇ut − i At ut = (1 + t η) (∇u− i A u)(z + t η z)

+ t∇η z · ((∇u− i Au)(z + t η z)) .(11.8)

242

And observe that, under a gauge transformation

u′ := eiφ u and A′ := A + dφ,

the indentity (11.8) is transformed into

∇u′t − i A′t u′t = eiφ(z+tηz) (∇ut − i At ut) .

Hence the gauge invariance is preserved.

Now we setv := η ∂Xu,

andB := η A + η ∂XA + (X ·A)∇η,

where, as above, δ > 0 is some small parameter and η is a cutoff function whichis radial, identically equal to 1 in BR and equal to 0 outside BR+δ. Notice that,in order to simplify the notations, we have assumed that z0 = 0.

Since (u,A) is assumed to be a critical point of Gε, we may write∫

C(∇u− iAu) · (∇v − iAv − iBu)− 1

ε2

∫

C(1− |u|2)u · v +

∫

CdAdB = 0.

We now evaluate each term involved in this identity and then let δ tend to 0.

We find, with little work,∫

C(∇u− iAu) · (∇v − iAv − iBu) =

∫

C(∇u− iAu)

·(∂Xu− i(X ·A) u)∇η

− 12

∫

C∂Xη|∇u− iAu|2

Letting δ tend to 0 in this expression , we obtain

limδ−→0

∫

C(∇u− iAu) · (∇v −−iAv − iBu) =

R

2

∫

∂BR

|∇u− iAu|2

−R

∫

∂BR

(∇u− iAu) · (∂Xu− i(X ·A)u).

Next, we find

1ε2

∫

C(1− |u|2)u · v +

∫

CdAdB = − 1

2ε2

∫

Cη (1− |u|2)2

+1

4ε2

∫

C∂Xη (1− |u|2)2.

243

And, letting δ tend to 0, we get

limδ→0

1ε2

∫

C(1− |u|2) u · v +

∫

CdA dB = − 1

2ε2

∫

BR

(1− |u|2)2

+R

4ε2

∫

∂BR

(1− |u|2)2.

Finally, we obtain, after a few efforts∫

CdAdB =

∫

Cη |dA|2 +

12

∫

C∂Xη |dA|2.

Which leads to

limδ→0

∫

CdAdB =

∫

BR

|dA|2 − R

2

∫

∂BR

|dA|2.

The result then follows at once. 2

Let us assume that (u, A) is a solution of (11.1) which satisfies (11.7). Firstof all, combining the equation (11.1) and the energy upper-bound (11.7), we getas in the proof of Proposition II.6 of [12]

‖du− iAu‖L∞ ≤ c ε−1. (11.9)

Furthermore, observe that, we can write, at least locally away from the zeros ofu,

u := |u| eiφ,

in which case, we find

du− iAu = (d|u|+ i (dφ−A)u)) eiφ.

Therefore|du− iAu|2 = |d|u||2 + |dφ−A|2.

Hence we also have, as a consequence of the previous estimate

‖∇|u|‖L∞ ≤ ‖du− iAu‖L∞ ≤ c ε−1.

Using the second equation of (11.1) we deduce the pointwise estimate

|∇h| ≤ |du− iAu|. (11.10)

In particular, this implies that

‖∇h‖L∞ ≤ c ε−1, (11.11)

and, thanks to (11.7), we also get∫

C|∇h|2 +

∫

C|h|2 ≤

∫

C|du− iAu|2 +

∫

C|dA|2 ≤ 4π log 1/ε + 2 c0. (11.12)

244

We will also need a local estimate which plays a crucial role in this Chapter.For all p > 2, all z0 ∈ C and all r ∈ (0, 1) we have by Holder estimate

∫

Br(z0)

|h|2 ≤ |Br(z0)|(p−2)/p

(∫

Br(z0)

|h|p)2/p

.

Using Sobolev embedding together with (11.12), we find∫

Br(z0)

|h|2 ≤ cp r(2p−4)/p log 1/ε, (11.13)

for all ε < 1. Observe that 4π log 1/ε + 2 c0 ≤ c log 1/ε for all ε ∈ (0, 1) !

We will also need the η−compactness Lemma for the gauge-invariant Ginzburg-Landau equation. This is a modified version of Lemma 9.2.

Lemma 11.3 [η-compactness Lemma] Let c1 > 0 be given. There exists η > 0,R0 ∈ (0, 1) and ε0 > 0 such that, if (u,A) is a solution of (11.1), with ε ∈ (0, ε0),which satisfies

|∇|u|| ≤ c1

4ε, (11.14)

and if there exists z ∈ C and R ∈ (4ε,R0) such that∫

BR(z)

(|∇u− iAu|2 +

12ε2

(1− |u|2)2 + |dA|2)≤ η log

R

2ε, (11.15)

then|u| ≥ 1

2in BR/2(z). (11.16)

Proof : We choose η =π

32 c21

and argue by contradiction. Assume that, for

some z′ ∈ BR/2(z), we have |u|(z′) < 1/2. For the sake of simplicity in thenotations, let us assume that z′ = 0. Thanks to (11.14) we see that

|u| < 3/4, in Bε/c1 .

Hence, we have1ε2

∫

Bε/c1

(1− |u|2)2 >π

16 c21

= 2 η. (11.17)

Now, we integrate the Pohozaev formula which was derived in the previousLemma, over the ball of radius r, we find using (11.13)

1ε2

∫

Br

(1− |u|2)2 ≤ r

∫

∂Br

g(u,A) + cp r(p−4)/p log 1/ε.

Where we have set

g(u,A) := |∇u− iAu|2 +1

2ε2(1− |u|2)2 + |dA|2.

245

Integrating this inequality over r ∈ (ε,R/2), we find using (11.15)

∫ R/2

ε

1r

(∫

Br

(1− |u|2)2ε2

)dr ≤ η log

R

2ε+ cp R(2p−4)/p log 1/ε.

Thanks to the mean value formula, we can conclude that there exists r0 ∈[ε,R/2] such that

1ε2

∫

Br0

(1− |u|2)2 ≤ η + cp R(2p−4)/p (log 1/ε) (logR

2ε)−1. (11.18)

If we assume that R0 and ε0 are chosen so that

cp R(2p−4)/p log 1/ε(logR

2ε)−1 ≤ η,

for all R ∈ (4ε,R0) and all ε ∈ (0, ε0), we see that (11.18) clearly contradicts(11.17). Hence |u|(z′) ≥ 1/2. 2

Proof of Proposition 11.1 : To begin with, let us observe that the result ofLemma 9.3 remains true. Namely, there exist ε0 > 0, n ∈ N and λ ≥ 1 onlydepending on c0 and, for all ε ∈ (0, ε0), there exist nε points z1, . . . , znε ∈ Csuch that :

1. z ∈ C : |u| < 1/2 ∩Bλ ε(zj) 6= ∅.2. z ∈ C : |u| < 1/2 ⊂ ∪nε

j=1Bλ ε(zj).

3. For all ε ∈ (0, ε0), nε ≤ n.

4. For all k 6= l, B2λε(zk) ∩B2λε(zl) = ∅.As in the proof of Lemma 9.3, the proof of this result relies on the extensive useof the Pohozaev formula together with the η-compactness Lemma.

Recall that we may also assume that λ is chosen large enough in order toensure that S > 3/4 for all r > λ.

We prove now that nε = 1 provided ε is chosen small enough. We argueby contradiction and assume that this is not the case. Hence, we can find asequence εi → 0 such that nεi ≥ 2. To keep the notations short, we will dropthe i indices and write nε instead of nεi , uε instead of uεi , . . .

Up to a subsequence, we may always assume that either

supk 6=l

|zk − zl|ε

,

tends to +∞ as ε tends to 0 or

supk 6=l

|zk − zl|ε

,

246

stays bounded as ε tends to 0.

Step 1. We first consider the case where

supk 6=l

|zk − zl|ε

,

tends to +∞ as ε tends to 0. Observe that, increasing the value of λ if this isnecessary, we may assume that for all k 6= l

|zk − zl|ε

,

tends to +∞ as ε tends to 0. Obviously, this value of λ may depend on thesequence we are dealing with but this is not relevant for the remaining of theargument. We keep on assuming that the number of balls Bλε(zk) needed tocover the set where |uε| ≤ 1/2 is larger than or equal to two.

Given any Br(z), we will use the Coulomb gauge on it. Namely, we willreplace (uε, Aε) by the gauge equivalent solution (uε, Aε) where

Aε := ?dξ

and where ξ is the solution of

∆ξ = ?dAε in Br(z)

ξ = 0 on ∂Br(z).(11.19)

For all p > 2, classical elliptic estimates lead to

‖Aε‖pL∞(Br) ≤ c rp−2

∫

Br

|h|p.

However, Sobolev embedding implies that∫

Br

|h|p ≤ c ‖h‖pH1(C).

We now make use of (11.12) to conclude that

‖Aε‖L∞(Br) ≤ c r(p−2)/p (log 1/ε)1/2. (11.20)

To keep notations short, we will write (uε, Aε) instead of (uε, Aε), referring tothe latter one as the Coulomb gauge in Br.

We claim that, for all l = 1, . . . , nε, we have

deg(

uε

|uε| ; ∂Bλε(zl))6= 0, (11.21)

247

provided ε is small enough. Indeed, fix some index l ∈ 1, . . . , nε. By assump-tion, we can find a sequence Rε ∈ (0, 1/ε) which tends to +∞ such that

BRεε(zl) ∩Bλε(zk) = ∅,

for all k 6= l. We make use of the Coulomb gauge in BRεε(zl) and define thesequences of rescaled functions

uε := uε(ε ·+zl) and Aε := Aε(ε ·+zl),

in BRε(0). Obviously, (uε, Aε) satisfy the following equations

∆uε + uε (1− |uε|2)− ε2|Aε|2 uε − 2 i ε ∂Aεuε = 0.

Observe that we do not have the term −i ε divAε uε since we have chosen theCoulomb gauge.

Using (11.20), (11.7) together with the fact that |u| ≤ 1, we deduce that,given any R > 0, the sequence ∆uε is uniformly bounded in L2(B2R). Hence,∇uε is bounded in any Lp(B3R/2), for p > 1. Using this information back intothe equation, we see that, for all p > 1, the sequence ∇2uε is also boundedin any Lp(BR). In particular, we can conclude that, up to a subsequence, uε

converges at least in C0loc(C) topology to u which is a solution of

∆u + u(1− |u|2) = 0 in C∫

C(1− |u|2)2 < +∞.

(11.22)

Moreover, |u| > 12

in C \Bλ.

It is proved in [15] that either

deg(

u

|u| , ∂Bλ(0))6= 0,

or u is constant with |u| = 1 in C. However, in the latter case, because of theC0 convergence, this would imply that |uε‖ ≥ 1/2 in Bλ for ε small enough,which contradicts the definition of zl. Hence

deg(

u

|u| ; ∂Bλ(0))6= 0,

and, since degrees do converge, we have proved (11.21).

Let us recall that, by definition h = ?dAε. Arguing as in Theorem V.1 of[12] or as in part 5 of [83], we deduce that, for all p ∈ (1, 2)

‖ ? dAε‖W 1,p(C) ≤ c, (11.23)

248

for some constant c > 0 which does not depend on ε. Moreover, since (11.7)holds, we may find a sequence of radii ri → +∞ such that

limi−→+∞

ri

∫

∂Bri

(|∂Au|2 +

(1− |u|2)22ε2

+ |dA|2)

= 0.

This, together with Pohozaev identity proved in Lemma 11.2, implies that

12ε2

∫

C(1− |uε|2)2 =

∫

C|dAε|2. (11.24)

Hence we also conclude that

12ε2

∫

C(1− |uε|2)2 ≤ c, (11.25)

for some constant c > 0 which does not depend on ε.

It is easy to see that, up to a subsequence, there exists R > 0, which doesnot depend on ε, such that either BR(zl) ∩ BR(zk) = ∅ or |zk − zl| ≤ R/2.Assuming that, for each l, we are working with the Coulomb gauge in BR(zl),we obtain from (11.23) that

∫

BR(zl)

|Aε|2 ≤ c, (11.26)

where, once more, c > 0 is independent of ε. Combining this upper-boundtogether with (11.7) we deduce that

∫

BR(zl)

|∇uε|2 ≤ 2π log 1/ε + c. (11.27)

Let us denote by z1, . . . , zq the collection of points zk which are included inBR(zl). Because of the choice of R, of (11.25) and because |uε| ≥ 1/2 inC \ ∪nε

k=1Bλε(zk), we can apply Theorem 4 and Theorem 5 of [15] and deducethat

∫

BR(zl)

|∇uε|2 ≥ 2π

q∑

j=1

d2j log 1/ε−

∑

i 6=j

di dj log |zk − zl| − c, (11.28)

where di is the degree of uε at the point zi.

Since, by assumption

limε→0

|xi − xj |ε

= +∞,

we have ∑

i 6=j

di dj log |xk − xl| = o(log 1/ε).

249

We can now combine (11.27) together with (11.28), to conclude that (for ε smallenough) q = 1 and also that the degree of uε at zl is ±1. In particular we have,thanks to (11.28)

∫

BR(zl)

|∇uε − iAεuε|2 ≥ 2π log 1/ε− c. (11.29)

Since all the BR(zl) have been assumed to be disjoint, this last inequality to-gether with (11.7) gives the desired contradiction for ε small enough.

Step 2. We now assume that

supk 6=l

|zk − zl|ε

,

remains bounded. We fix R > 0 (independently of ε) in such a way that BRε(z1)contains all the zk. Using a blow up argument as in Step 1, it is easy to see that

deg(

uε

|uε| , ∂BRε(z1))6= 0.

Moreover, we have already seen in (11.25) that

1ε2

∫

C(1− |uε|2)2 < c,

for some constant c > 0 which is independent of ε. Using lower bounds for theenergy which was obtained in [15], we get that

∫

B1(z1)\BRε(z1)

|∇uε|2 ≥ 2π deg(

uε

|uε| , ∂BRε(z1))2

log 1/ε− c, (11.30)

which is valid independently of the choice of the gauge. In particular, if we takethe Coulomb gauge in B1(z1), we also have

∫

B1(z1)

|Aε|2 ≤ c,

as we have already seen in (11.26). This last inequality, together with (11.25)implies that

∫

B1(z1)\BRε(z1)

|∇uε − iAεuε|2 ≥ 2π deg(

uε

|uε| , ∂BRε(z1))2

log 1/ε− c.

Comparing this inequality with (11.7), we conclude that

deg(

uε

|uε| , ∂BRε(z1))

= ±1,

250

since we have already seen that the degree is non zero. In particular, thereexists a point where uε vanishes. Relabeling the points zk if this is necessary,we can assume that uε(z1) = 0.

In BRε(z1), we take once more the Coulomb gauge. Let us define the rescaledfunctions

uε := u(ε ·+z1) and Aε := Aε(ε ·+z1).

As we have already seen in Step 1, uε is a solution of

∆uε + uε (1− |uε|2)− ε2 |Aε|2 uε − 2 i ε Aε · ∇uε = 0,

in C, since we are working with the Coulomb gauge. Moreover uε(0) = 0,

deg(

uε

|uε| , ∂BR

)= ±1,

and finally ∫

C(1− |uε|2)2 ≤ c,

where c > 0 does not depend on ε. We can argue as in Step 1 to show that, upto a subsequence, the sequence uε converges at least in C0

loc norm, to u solutionof

∆u + u (1− |u|2) = 0 in C∫

C(1− |u|2)2 < +∞

u(0) = 0

deg(

u

|u| , ∂BR

)= ±1.

Using Corollary 8.3, we conclude that u is equal to S ei(θ+τ), for some constantτ ∈ R. In particular, this implies that |u| ≥ 3/4 in C \ Bλ. Therefore, for εsmall enough, |uε| ≥ 1/2 in BRε(z1) \Bλε(z1). This is again a contradiction.

Since we have obtained a contradiction in both cases, we have proved thatnε = 1.

The remaining of the proof is now similar to what we have already done inChapter 10, so we omit it. 2

One of the main observation established in [36] is the exponential decay atinfinity of various intrinsic (gauge invariant) quantities defined with any (u,A)solution of (11.1). More precisely, we have

Proposition 11.2 For all α ∈ (0, 1) and all k ∈ N, there exists c > 0 andε0 > 0, such that, for all ε ∈ (0, ε0) and for all (u,A) solution of (11.1) whichsatisfies (11.7) and u(0) = 0

supr≥1

(eαr |∇kh|) ≤ c

supr≥1

(e2αr |∇k(1− |u|)|) ≤ c ε2.

(11.31)

251

where as usual h = ?dA.

Proof : To begin with, we know from the result of Proposition 11.1, that|u| ≥ 1/2 in C \ Bλε. Where the constant λ is independent of ε. Moreover, aswe have seen in Step 1 of the proof of Proposition 11.1, for all p ∈ (1, 2), thereexists c > 0 such that

‖h‖W 1,p(C) ≤ c. (11.32)

Recall that h = ?dA satisfies

−div(∇h

|u|2)

+ h = 0, (11.33)

in C \ 0 and, thanks to (11.4), we see that 1− |u| satisfies

−∆(1− |u|) +(|u|+ |u|2)

ε2(1− |u|) =

|∇h|2|u|3 , (11.34)

in C \ 0.Using (11.33) and arguing as in Step 6 of the proof of Theorem 1 in [10], we

obtain for any k ∈ N, the existence of a constant ck > 0 such that

‖∇kh‖L∞(C\Bλ) ≤ ck

‖∇k(|u| − 1)‖L∞(C\Bλ) ≤ ck.(11.35)

Now, we can improve the second estimates. Indeed, it is not difficult to seethat the potential in the equation satisfied by 1− |u| satisfies

(|u|+ |u|2)ε2

≥ 14 ε2

,

in C \Bλε. And then that the function

r −→ c (e−r/(2ε) + ε2),

can be used as a barrier function to show that

|1− |u|| ≤ c (e−r/(2ε) + ε2),

in C \Bλε. In particular, this implies that

|1− |u|| ≤ c ε2, (11.36)

in C \B1/4, for all ε small enough. Using rescaled Schauder’s estimates we alsoconclude that

|∇(1− |u|)| ≤ c ε, (11.37)

in C \B1/2.

252

For all β ∈ (0, 1), we define Gβ to be the solution of

−∆Gβ + β2 Gβ = δ0 in D′(C). (11.38)

It is well known thatlim

r→+∞log Gβ = −β. (11.39)

Let us now fix β ∈ (0, 1) in such a way that

α < β2 < 1.

Next, we choose M > 0 large enough so that |h| ≤ M Gβ on ∂B1/2 and we set

K± := ±h−M Gβ .

Obviously, we have

−div(∇K±|u|2

)+ K± = M

(β2

|u|2 Gβ − 2∇|u||u|3 ∇Gβ −Gβ

).

Using (11.36) and (11.37), it is an easy exercise to show that, provided ε ischosen small enough, the right hand side of this identity is negative in C \B1/2.Hence, we conclude that both K± verify

−div

(∇K±|u|2

)+ K± ≤ 0 on C \B1/2

K± ≤ 0 on ∂B1/2.

(11.40)

Using the maximum principle, we deduce that |h| ≤ M Gβ in C \ B1/2. Thisalready proves the first estimate in (11.31) when k = 0. The general case followsat once by induction.

In order to prove the second estimate, we apply similar arguments. Firstobserve, using (11.39), that for all β′ ∈ (α, β), we can find a constant c > 0 suchthat

G2β′ ≥ cG2β ,

in C \B1/2. Since we have already proved |h| ≤ M Gβ , we see that it is possibleto choose M > 0 in such a way that

|∇h|2|u|3 ≤ M G2β′

in C \B1/2 and, thanks to (11.36), in such a way that we also have

M G2β′ ≥ 1− |u|2ε2

,

on ∂B1/2. We setH := 1− |u| − ε2MG2β′ .

253

A simple computation shows that

−∆H +(|u|+ |u|2)

ε2H =

|∇h|2|u|3 −M

((|u|+ |u|2)− 4ε2 (β′)2

)G2β′ .

For ε small enough, we can ensure that−∆H +

(|u|+ |u|2)ε2

H ≤ 0 in C \B1/2

H ≤ 0 on ∂B1/2.

(11.41)

Applying the maximum principle once more, we deduce that 1−|u| ≤ M ε2 G2β′

in C \B1/2 which already proves the second estimate of (11.31) for k = 0. Theestimate for general k is obtained by induction. 2

11.3 Proof of Theorem 11.1

Let (u,A) be a solution of (11.1) which satisfies (11.7). Thanks to Proposi-tion 11.1 we know that, if ε is small enough, then the degree of u/|u| at ∞ isequal to ±1. In the remaining of this section, we will assume that the degree isalways 1.

11.3.1 The Coulomb gauge

To begin with, let us make the following observation.

Lemma 11.4 Under the assumptions of Theorem 11.1, if the degree of u/|u|at ∞ is equal to 1, then ∫

Ch = 2π. (11.42)

Proof : Indeed, we know from Proposition 11.2 that dh tends exponentiallyfast to 0 at ∞. This implies that u ∧ (du − iAu) also tends to 0 exponentiallyfast at ∞. Now write

u := |u| eiφ,

to conclude that ∇φ−A tends to 0 exponentially fast at ∞.

Since we have assumed that the degree of u/|u| at ∞ is 1, we have∫

∂BR

∂θφ = 2 π R.

Observe that ∂θφ = A⊥ · ν. Hence, we conclude that

limR→+∞

1R

∫

∂BR

A⊥ · ν = 2 π.

254

Finally, integration by parts yields∫

∂BR

A⊥ · ν =∫

BR

divA⊥ =∫

BR

h.

And the result follows. 2.

Let (u,A) be a solution of (11.1) which satisfies (11.7). We will use thefollowing gauge in C

A(z) :=12π

−∂y

∫

Clog |z − z′|h(z′) + cx

∂x

∫

Clog |z − z′|h(z′) + cy,

, (11.43)

where cx and cy are chosen in such a way that A(0) = 0. Otherwise stated,

A +∇⊥(

12π

log r ∗ h

),

is a constant vector field.

Standard results on singular integrals [95], yields for any p ∈ (1, 2)

(∫

C|∇A|p∗

) 1p∗≤ c

(∫

C|∇h|p

) 1p

,

Where 1p∗

:= 1p − 1

2 . In addition, Morrey estimates yields [26], [95]

|A(z)−A(z′)| ≤ c |z − z′|α(∫

C|∇A|p∗

)1/p∗

,

where α := 1− 2p∗

.

Recall that we have proven in Step 1 of the proof of Proposition 11.1, that

‖ ? dAε‖W 1,p(C) ≤ c, (11.44)

provided p ∈ (1, 2). Combining this information with the above classical esti-mates, we conclude that, for all p ∈ (1, 2), there exists c > 0, independent of εsuch that

|A(z)−A(0)| ≤ c |z|α, (11.45)

whereα = 2− 2

p.

In the following Lemma, we give a precise description of the behavior of Aat ∞.

255

Lemma 11.5 Under the assumptions of Theorem 11.1, there exists A∞ ∈ R2

such that the following estimate holds in C \B1

∣∣∣∣A(z)−A∞ −∇θ − 12π∇⊥

(z

|z|2 ·∫

Cz′ h(z′)

)∣∣∣∣ ≤ c r−3, (11.46)

where the constant c > 0 does not depend on ε.

Proof : We define ζ = 12π log r ∗ h. Thanks to (11.42), we may write

ζ(z)− log r =12π

∫

Clog

|z − z′||z| h(z′).

With this definition, we see that

A +∇⊥ζ,

is a constant vector field.

Using the result of Proposition 11.2, we can estimate, for any α ∈ (0, 1)∣∣∣∣∣∫

|z−z′|≤|z|/2

∂z log|z − z′||z| h(z′)

∣∣∣∣∣ ≤ c

∣∣∣∣∣∫

|z−z′|≤|z|/2

∂z log|z − z′||z|

∣∣∣∣∣ e−α|z|

≤ c |z|2 e−α|z|,(11.47)

where c > 0 depends on α.

Now, for all |z′| ≤ |z|/2, we use the expansion∣∣∣∣∂z log

|z − z′|2|z|2 + 2∂z

z · z′|z|2

∣∣∣∣ ≤ c|z′|2|z|3 .

Therefore, we can estimate∣∣∣∣∣∫

|z′|≤|z|/2

∂z log|z − z′||z| h(z′) + 2∂z

z

|z|2 ·∫

Cz′ h(z′)

∣∣∣∣∣ ≤ c |z|−3. (11.48)

Finally, in ω where |z′| ≥ |z|/2 and |z′ − z| ≥ |z|/2, we use the fact that∣∣∣∣∂z log

|z′ − z||z|

∣∣∣∣ ≤ c1|z| ,

to get ∣∣∣∣∫

ω

∂z log|z − z′||z| h(z′)

∣∣∣∣ ≤ c e−α|z|. (11.49)

The result then follows from (11.47)-(11.49). 2

As a simple Corollary of the previous Lemma, we have

256

Corollary 11.1 Under the assumptions of Theorem 11.1, we have

limR→+∞

R

∫

∂BR

(|A · τ |2 − |A · ν|2) = 2π + π a∞ ·A⊥∞.

Wherea∞ :=

∫

Cz′ h(z′),

is identified with the corresponding vector of R2.

Proof : This simply follows from the fact that, thanks to the previous result,we can estimate

A · ν = A∞ · τ − 1r

+1

2πr2a∞ · ν +O

(1r3

),

and

A · ν = A∞ · ν − 12πr2

a∞ · τ +O(

1r3

).

The proof is then straightforward. 2

11.3.2 Preliminary results

We already know from (11.9) that |∇u− iAu| ≤ c ε−1. It follows from Proposi-tion 11.1 that

u−1 (0) = 0.We have also seen in Proposition 11.1, that, up to a gauge transformation,the rescaled function u := u(ε ·) converge in any Ck(BR) topology to S ei(θ+τ ).Arguing as in the proof of Lemma 9.6, we obtain that, for any fixed γ > 0, wehave

|u| ≥ c|z|ε

, (11.50)

in Bγε.

Notice that, by construction, divA = 0, hence, u satisfies the equation

∆u + u(1− |u|2)

ε2− |A|2u− 2i∂Au = 0. (11.51)

in C and we also have, thanks to Lemma 11.4

−div(∇h

|u|2)

+ h = 2π δ0, (11.52)

in C.

We now use the radially symmetric solutions defined in Theorem 11.1. Inorder to simplify the equations it will be convenient to drop the ε indices in thedefinition of Sε and Tε and simply write S and T . Also, we define

v = S eiθ B := T dθ,

257

andk := ? dB,

the magnetic field of the axially symmetric solution.To begin with, we have the

Lemma 11.6 Under the assumptions of Theorem 11.1, there exists ε0 > 0 andfor all l ≥ 0, there exists cl > 0 such that

|∇lk|+ |∇lh| ≤ cl sup(r, ε)−l,

and|∇l|u||+ |∇l|v|| ≤ cl ε

2 sup(r, ε)−l−2,

in C \ 0.Proof : We give the proof of these estimates for u. It follows from Propo-sition 11.1 that u has a unique 0, which may be assumed to be the origin.Moreover |u| ≥ 1/2 in C \Bλε, for some λ > 0 independent of ε.

It follows from [11], [12] and [82] that the result holds in any BRε∪ (C\BR),for any fixed R > 0 independent of ε.

We claim that there exists c > 0, independent of ε, such that, for all r ∈(ε, 1), we have ∫

B2r\Br

gε(u,A) ≤ c. (11.53)

wheregε(u,A) := |dAu|2 +

12ε2

(1− |u|2)2 +∫

Ω

|dA|2.

As usual, we consider the Coulomb gauge

A = −∇⊥(12π

? h) + A0

where A0 is chosen so that A(0) = 0. We know that ‖h‖W 1,p(C) ≤ c, for allp ∈ (1, 2), and also that

1ε2

∫

C(1− |u|2)2 ≤ c.

Therefore, in order to prove (11.53), it suffices to prove that, for all r ∈ (ε, 1),we have ∫

B2r\Br

|∇u|2 ≤ c.

We write, away from 0u := |u| eiθ+ζ .

Hence, we obtain∫

B2r\Br

|∇u|2 =∫

B2r\Br

(|∇|u||2 + |u|2 |∇(θ + ζ)|2).

258

So, ∫

B2r\Br

|∇u|2 ≥∫

B2r\Br

(|∇θ|2 + |u|2 |∇ζ|2 + 2|u|2∇θ∇ζ)

. +∫

B2r\Br

1r2

(|u|2 − 1).

Moreover, since we have∫

∂Br

∇θ∇ζ =1r2

∫

∂Br

∂θζ = 0,

hence, we can write∫

B2r\Br

|u|2∇θ∇ζ =∫

B2r\Br

(|u|2 − 1)∇θ∇ζ.

Hence, we conclude that∫

B2r\Br

|∇u|2 ≥∫

B2r\Br

|∇θ|2 +∫

B2r\Br

(|u|∇ζ +

(|u|2 − 1)|u| ∇θ

)2

. +∫

B2r\Br

(1− |u|2]2|u|2 |∇θ|2 +

∫

B2r\Br

1r2

(|u|2 − 1).

(11.54)Using the inequality 2 a b ≤ a2 + b2, we find that

∫

B2r\Br

(1− |u|2) 1r2≤ ε2

2

∫

B2r\Br

1r4

+1

2ε2

∫

B2r\Br

(1− |u|2)2.

Therefore, we finally get from (11.54)∫

B2r\Br

|∇u|2 ≥ 2 π log 1/ε− c, (11.55)

for some constant c > 0 only depending on c0.

Observe that∫

B2r\Br

|∇u|2 ≤∫

B1\B2r

|∇u|2 +∫

B2r\Br

|∇u|2 +∫

Br\Bε

|∇u|2,

and, thanks to (11.7) and (11.55), we can say that∫

B2r\Br

|∇u|2 ≤ c,

for some constant c > 0 independent of ε. This ends the proof of the claim.

Now that (11.53) is proven, we observe that, in B2r \Br the equations read

∆|u|+ |u|(

1− |u|2ε2

− |∇h|2|u|4

)= 0

259

and−div (|u|−2∇h) + h = 0,

and we know from (11.53) that∫

B2r\Br

(|∇h|2 + |∇|u||2 +

1ε2

(1− |u|2)2)≤ c.

We fix R ∈ (λε, 1) and define ε := ε/R,

h(z) := h(Rz) and s(z) = |u|(Rz).

Obviously, we have

∆s + s

(1− s2

ε2− |∇h|2

s4

)= 0,

and−div (s−2∇h) + R2 h = 0,

in B2 \B1. Furthermore,∫

B2\B1

(|∇h|2 + |∇s|2 +

1ε2

(1− s2)2)≤ c.

It is proven in [10] that, under such assumptions, we have

|∇lh| ≤ cl,

and|∇ls| ≤ cl ε

2.

The result then follows by simply performing the scaling backward. 2

As we have done in Lemma 9.7, we would like to compare h and k. This isthe content of the following

Lemma 11.7 Assume that ν ∈ (1 − 2−1/2, 1) and further assume that the hy-pothesis of Theorem 11.1 hold. Then, there exists c > 0 and ε0 > 0 such that,for all ε ∈ (0, ε0), we have

∫

Cr2−2ν |∇(h− k)|2

|v|2 ≤∫

C\Bε/2

r−2ν(|u| − |v|)2 +∫

Bε

r2−2ν

∣∣∣∣∂r

(u

|v|)∣∣∣∣

2

.

(11.56)

Proof : The strategy of the proof is very similar to what we have already donein the proof of Lemma 9.7. We define ξ = h− k. A simple computation showsthat ξ satisfies

−∆ξ + 2∇|v||v| ∇ξ + |v|2 ξ = |v|2

(div

((1|u|2 −

1|v|2

)∇h

)). (11.57)

260

The proof now follows exactly as in the proof of Lemma 9.7. However, thistime, instead of Proposition 7.5, we apply Proposition 7.6 and we find

∫

Cr2−2ν 1

|v|2 |∇ξ|2 ≤ c

∫

Cr4−2ν 1

|v|2 |f |2 + c

∫

Cr2−2ν |v|2 |g|2

+ c

(∫

∂Bς

1|v|2 ∂rξ −

∫

∂Bς

ξ

)2

+ c

(∫

∂Bς

g · x)

.

where we have set

g := div(

(1− ηε)|v|2 − |u|2|u|2|v|2 ∇h

),

and

f := ∇h∇(

ηε|v|2 − |u|2|u|2

).

Here, as in the proof of Lemma 9.7, η is a cutoff function equal to 1 in B1 andequal to 0 outside of B2 and, for all ε > 0 we have defined

ηε := η(2 · /ε).

Finally, ς ∈ [σ0/2, σ0].

Step 1. To begin with, we bound∫

Cr2−2ν |v|2 |g|2 ≤ c

∫

C\Bε/2

r2−2ν |∇h|2 (|u| − |v|)2. (11.58)

Step 2. Integrating the equation satisfied by ξ over Bς , we find

c

(∫

∂Bς

1|v|2 ∂rξ −

∫

∂Bς

ξ

)2

≤(∫

∂Bς

∣∣∣∣|v|2 − |u|2|u|2|v|2

∣∣∣∣ |∇h|)2

.

Using Cauchy-Schwarz inequality, we get

c

(∫

∂Bς

1|v|2 ∂rξ −

∫

∂Bς

ξ

)2

≤∫

∂Bς

(|v| − |u|)2 |∇h|2.

Now we may use the mean value formula to conclude that

c

(∫

∂Bς

1|v|2 ∂rξ −

∫

∂Bς

ξ

)2

+ c

(∫

∂Bς

g · x)

≤ c

∫

Bσ0\Bσ0/2

|∇h|2 (|u| − |v|)2

≤ c

∫

C\Bε/2

r2−2ν |∇h|2 (|u| − |v|)2.

(11.59)

261

Step 3. Finally,

∫

Cr4−2ν 1

|v|2 |f |2 ≤ c

∫

Bε

r4−2ν 1|v|2

∣∣∣∣∇h∇( |u||v|

)∣∣∣∣2

+ c

∫

Bε\Bε/2

r4−2ν 1ε2

1|v|2 |∇h|2 (|u| − |v|)2.

(11.60)

Obviously,∫

Bε\Bε/2

r4−2ν 1ε2

1|v|2 |∇h|2(|u| − |v|)2 ≤ c

∫

C\Bε/2

r2−2ν |∇h|2(|u| − |v|)2.

While, for the first term on the right hand side of (11.60), we have

∫

Bε

r4−2ν 1|v|2

∣∣∣∣∇h∇( |u||v|

)∣∣∣∣2

≤ c

∫

Bε

r4−2ν 1|v|2

∣∣∣∣∇k∇( |u||v|

)∣∣∣∣2

+c

∫

Bε

r4−2ν 1|v|2 |∇(h− k)|2

∣∣∣∣∇( |u||v|

)∣∣∣∣2

.

Since B = T dθ where T only depends on r, we can state that∣∣∣∣∇k∇

( |u||v|

)∣∣∣∣2

= |∇k|2(

∂r

( |u||v|

))2

.

Moreover, using a blow up argument, it is easy to show that

limε→0

(ε ‖∇|u| − |v|‖L∞(Bε) +

∥∥∥∥|u| − |v||v|

∥∥∥∥L∞(Bε)

)= 0,

which in particular implies that

limε→0

supBε

(r

∣∣∣∣∇( |u||v|

)∣∣∣∣)

= 0.

Hence, for ε small enough we have

∫

Bε

r4−2ν 1|v|2 |∇(h− k)|2

∣∣∣∣∇( |u||v|

)∣∣∣∣2

≤ 12

∫

Bε

r2−2ν 1|v|2 |∇ξ|2.

We conclude that∫

Bε

r4−2ν 1|v|2

∣∣∣∣∇h∇( |u||v|

)∣∣∣∣2

≤ c

∫

Bε

r4−2ν 1|v|2 |∇k|2

∣∣∣∣∂r

( |u||v|

)∣∣∣∣2

+12

∫

Bε

r4−2ν 1|v|2 |∇ξ|2.

(11.61)

262

Step 4. Collecting all these estimates, we obtain∫

Cr2−2ν 1

|v|2 |∇ξ|2 ≤ c

∫

C\Bε/2

r2−2ν |∇h|2 (|u| − |v|)2

+ c

∫

Bε

r4−2ν 1|v|2 |∇k|2

∣∣∣∣∂r

( |u||v|

)∣∣∣∣2

.

The result then follows from the fact that we know from Lemma 11.6 that

|∇h| ≤ c |u|/r and |∇k| ≤ c |v|/r,

in C \ 0.Without saying so, we have extensively used the fact that, for ε small enough

|u|/2 ≤ |v| ≤ 2 |u|.and also that |u| ≥ c and |v| ≥ c in C \Bε/2, provided ε is small enough. 2

We will also need the following simpler :

Lemma 11.8 For any p ≥ 0 the following bound holds∫

C\B2

rp((|u| − |v|)2 + |∇(h− k)|2) ≤ cp

∫

C\B1

(|u| − |v|)2 + |∇(h− k)|2.

Proof : Outside B1 we have on one hand

−∆h + 2∇|u||u| ∇h + |u|2 h = 0,

−∆k + 2∇|v||v| ∇k + |v|2 k = 0,

and on the other hand

∆|v|+ |v|(

1− |v|2ε2

− |∇k|2|v|4

)= 0,

∆|u|+ |u|(

1− |u|2ε2

− |∇h|2|u|4

)= 0.

We define ξ = h− k and D = |u| − |v|. We have already seen that

−∆ξ + 2∇|v||v| ∇ξ + |v|2 ξ = −|v|2 div

( |u|+ |v||u| |v| D ∇h

), (11.62)

and we have now−∆D +

cε

ε2D = − 1

|u|3 ∇ξ∇(h + k). (11.63)

where

cε := |u|2 + |v|2 + |v| |u| − 1− ε2

|u|3 |v|3 |∇k|2 |u|3 − |v|3|u| − |v| .

263

Observe that the last term in cε is uniformly bounded in C \ B1 by a constanttimes ε2, thus we can state that cε ≥ 2 in C \ B1 provided ε is chosen smallenough.

Let η be a cutoff function equal to 0 in B1 and equal to 1 outside B2. Wemultiplying (11.62) by χ2 rp ξ and integrate the results C. We get∫

Cη2 rp (|∇ξ|2 + |ξ|2) ≤ c

∫

C|D| |∇(η2 rp ξ)|+ c

∫

C|ξ| |∇ξ| |∇(rp η2)|. (11.64)

As we have done in Lemma 9.8, we now use the inequality a b ≤ κ a2 +14κ

b2

to estimate all the terms in the right hand side of this inequality. For example,we estimate

c

∫

C|ξ| |∇ξ| rp η |∇η| ≤ 1

4

∫

Crp η2 |ξ|2 + c

∫

C|∇ξ|2 |∇η|2

so that the first term on the right hand side, which involves |ξ|2 can be absorbedin the left hand side of (11.64). We obtain

∫

Cη2 rp (|∇ξ|2 + |ξ|2) ≤ c

∫

C|D| η2 rp + c

∫

B2\B1

|∇ξ|2 (11.65)

Next, we multiplying (11.63) by η2 rp D and integrate the results C. We get∫

Cη2rp

(|∇D|2 +

1ε2|D|2

)≤ c

∫

C|D||∇D||∇(η2 rp)|+ c

∫

Cη2rp|D||∇ξ|.

Again, we use the inequality a b ≤ κ a2 +14κ

b2 to get

∫

Cη2 rp

(|∇D|2 +

1ε2|D|2

)≤ c

∫

B2\B1

|D|2 +12

∫

Cη2 rp |∇ξ|2. (11.66)

The result follows at once from the combination of the two inequalities (11.65)and (11.66). 2

11.3.3 The Pohozaev formula

We now use the radially symmetric solutions defined in Theorem 11.1. Andconsider the auxiliary function

w :=u

|v| .

Observe that, in opposition to what we have done in Chapter 9, we do notconsider the quatient of the two solutions u and v but rather the quotient of uand |v|. In order to simplify the equations it will be convenient to drop the εindices in the definition of Sε and Tε and simply write S and T .

264

It verifies the following equation

∆w + 2∂r|v||v| ∂rw +

w

r2+ w |v|2 (1− |w|2)

ε2− w (|A|2 − |B|2)

−2 w ∂Bθ − 2 i w Ar∂r|v||v| − 2 i ∂Aw = 0,

(11.67)

in C \ 0. Where Ar is the first component of the 1-form A in cylindricalcoordinates, namely A = Ar dr + r Aθ dθ.

The Pohozaev argument applied to this equation reads

14

∫

BR

1r∂r

(r2 |v|2) (1− |w|2)2

ε2+ 2

∫

BR

r∂r|v||v| |∂rw|2 =

− 12R

∫

∂BR

(|w|2 − 1) +R

2

∫

∂BR

(|∂τw|2 − |∂νw|2)

+R

4

∫

∂BR

|v|2 (1− |w|2)2ε2

+12

∫

BR

r ∂r|w|2 (|A|2 − |B|2)

+2∫

BR

r |w|2 ∂rΦAr∂r|v||v| +

∫

BR

r ∂r|w|2(∂Bθ − ∂AΦ)

+∫

BR

r ∂rΦ ∂A|w|2,

(11.68)

where Φ is the multivalued function on C \ 0 defined by

∇Φ =u ∧∇u

|u|2 .

In particular, we have u = |u| eiΦ away from the origin. In order to obtainthe above Pohozaev formula, we have just taken the scalar product of (11.67)with r ∂rw and integrated the result over BR \ Bs and finally let s tend to 0.The boundary term on ∂Bs vanishes here also exactly like for the correspondingPohozaev identity established in Chapter 9. Everything is very close to whatwe have already done in Chapter 9, except the derivation of the last term.

Indeed, the last term comes from the identity

2 rw ∂AΦ · ∂rw + r (2i∂Aw) · ∂rw = r ∂rΦ ∂A|w|2.In order to prove this identity, we observe that we always have

∂rw · (i∂θw) =12

(∂rΦ ∂θ|w|2 − ∂r|w| ∂θΦ

).

Using this, we obtain

2rw∂AΦ · ∂rw + r(2i∂Aw) · ∂rw = r ∂AΦ∂r|w|2 − r(2iAθ ∂θw) · ∂rw

= r ∂rΦ(Ar∂r|w|2 + Aθ∂θ|w|2)= r ∂rΦ ∂A|w|2,

265

as desired.

It will be convenient to observe that the following identities hold

∇⊥h = |u|2(∇Φ−A) and ∇⊥k = |v|2 (∇θ −B). (11.69)

These follow at once from the second equation of (11.1).

We would like to pass to the limit in the Pohozaev formula as R tends to+∞. To this aim, observe that 1 − |w|2, |∇|w|| and dh decay exponentially at∞. Since −? d?h = u∧ (du− iAu) this implies that (iu) · (du− iAu) also tendsto 0 exponentially. Hence

(iw) · (dw − iAw) =1|v|2 (iu) ∧ (du− iAu),

also tends to 0 exponentially at ∞. Using this, we see that

limR→+∞

(1

2R

∫

∂BR

|w|2 − R

2

∫

∂BR

(|∂τw|2 − |∂νw|2)

− R

4

∫

∂BR

|v|2 (1− |w|2)2ε2

)

= limR→+∞

(π − R

2

∫

∂BR

|A · τ |2 − |A · ν|2)

.

(11.70)Using the result of Corollary 11.1, we see that (11.68) becomes

14

∫

C

1r∂r

(r2 |v|2) (1− |w|2)2

ε2+ 2

∫

Cr∂r|v||v| |∂rw|2

=12

∫

Cr ∂r|w|2 (|A|2 − |B|2) + 2

∫

Cr |w|2 ∂rΦAr

∂r|v||v|

+∫

Cr ∂r|w|2(∂Bθ − ∂AΦ)−

∫

Cr ∂rΦ ∂A|w|2 +

π

2a∞ ·A⊥∞.

(11.71)

11.3.4 The end of the proof

We set

Q :=14

∫

C

1r∂r

(r2 |v|2) (1− |w|2)2

ε2+ 2

∫

Cr∂r|v||v| |∂rw|2 .

In the remaining of the proof, we will show that each term, which is on the righthand side of (11.70) is bounded by 1/100 times Q. This will imply that Q = 0and then this will imply that

u = |v| ei(θ+τ),

for some τ ∈ R. This will be the end of the proof of Theorem 11.1. To do so, itis useful to observe that there exists a constant c > 0 such that, for all ε smallenough we have

Q ≥ c

∫

C

(|u| − |v|)2ε2

+ c

∫

Bε

|∂rw|2 .

266

Let us derive two useful estimates. By definition, B = T dθ, hence B(0) = 0.Moreover, we also have A(0) = 0 thus, using estimate (11.45) (which holds alsofor B which is the Coulomb gauge), we conclude that

|A|+ |B| ≤ cα |z|α, (11.72)

for any α ∈ (0, 1). Using the fact that A(0)−B(0) = 0, since

A−B − 12π

log r ∗ ∇⊥(h− k),

is a constant vector field, we may apply the result of Proposition 7.4 to obtain,for all ν ∈ (0, 1)

2∑

j=0

∫

Cr−2(ν−j)−2

∣∣∇j(A−B)∣∣2 ≤ c

∫

Cr2−2ν |∇(h− k)|2.

Assuming from now on that ν ∈ (1−2−1/2, 1) and using the result of Lemma 11.7,we conclude that

2∑

j=0

∫

Cr−2(ν−j)−2

∣∣∇j(A−B)∣∣2 ≤ c

∫

C\Bε/2

r−2ν(|u| − |v|)2

+ c

∫

Bε

r2−2ν |∂rw|2

≤ c ε2−2ν Q.

(11.73)

Now Lemma 11.7 yields, for all ν ∈ (1− 2−1/2, 1)∫

Cr2−2ν |∇(h− k)|2

|u|2 ≤ c ε2−2ν Q. (11.74)

Combining this with Lemma 11.8, we also get, for all p ≥ 0∫

C\B2

rp (|u| − |v|)2 ≤ c ε2−2ν Q.∫

C\B2

rp |∇(h− k)|2 ≤ c ε2−2ν Q.(11.75)

In the remaining, we will assume that the constant ν ∈ (1 − 2−1/2, 1) andthat α ∈ (1− ν, 1).

Step 1. Let us decompose the first term on the right hand side of (11.71)in the following way

∫

Cr ∂r|w|2(|A|2 − |B|2) =

∫

Bγε

r ∂r|w|2(|A|2 − |B|2)

+∫

C\Bγε

r ∂r|w|2(|A|2 − |B|2) ,

(11.76)

267

where γ ∈ (1/2, 1).

Step 1.1. Let us recall that 0 < c ≤ |w| ≤ C in Bε, for all ε small enough.Using (11.72) together with Cauchy-Schwarz inequality, we can estimate

∫

Bγε

r∂r|w|2(|A|2 − |B|2) ≤ c ε1+α

(∫

Bε

|∂rw|2)1/2 (∫

Bε

|A−B|2)1/2

.

Thanks to (11.73), we get∫

Bγε

r∂r|w|2(|A|2 − |B|2) ≤ c ε3+αQ.

Hence, this quantity is bounded by Q/100, provided ε is chosen small enough.

Step 1.2. On the other hand, outside Bγε integrating by part we have∫

C\Bγε

r∂r|w|2(|A|2 − |B|2) ≤ c

∫

∂Bγε

ε||u| − |v||‖|A|2 − |B|2|

+c

∫

C\Bγε

||u| − |v||r

|∂r(r2(|A|2 − |B|2))|.(11.77)

Making use of the mean value formula, we conclude that we may choose γ insuch a way that

ε

2

∫

∂Bγε

||u| − |v||∣∣|A|2 − |B|2

∣∣ ≤∫

Bε\B ε2

||u| − |v||∣∣|A|2 − |B|2

∣∣

≤ c εα

(∫

Bε\Bε/2

(|u| − |v|)2)1/2 (∫

Bε

|A−B|2)1/2

,

where we have also used (11.72). Using (11.73) we conclude that

c

∫

∂Bγε

ε ||u| − |v||∣∣|A|2 − |B|2

∣∣ ≤ c ε3+αQ.

Hence, this term is also bounded by Q/100 for all ε small enough.

In order to estimate the second term in the right hand side of (11.77), weuse (11.72) to get

∣∣∣∣∣∫

C\Bγε

||u| − |v||r

∣∣∂r(r2(|A|2 − |B|2))∣∣∣∣∣∣∣

2

≤

c

(∫

C2\Bε/2

r2(1+ν+α)(|u| − |v|)2)

1∑

j=0

∫

Cr−2(ν−j+1)|∇j(A−B)|2

.

268

In order to evaluate the first term on the right hand side of this inequality, wedecompose this term into two integrals and use (11.75) to get

∫

C2\Bε/2

r2+2ν+2α (|u| − |v|)2 ≤∫

B1\Bε/2

(|u| − |v|)2

+∫

C2\B1

r2+2ν+2α (|u| − |v|)2

≤ c ε2Q+ c ε2−2ν Q.

Next, we make use of (11.73) to obtain

c

∣∣∣∣∣∫

C\Bγε

||u| − |v||r

∣∣∂r(r2 (|A|2 − |B|2))∣∣∣∣∣∣∣ ≤ c ε2−2ν Q.

Again, we can conclude that the second term on the right hand side of (11.77)is bounded by Q/100 for all ε small enough.

Step 2. We bound now the second term in the right hand side of (11.71).Again, we decompose the integral as the sum of the integral over Bε and theintegral over C \Bε.

Step 2.1. We use the fact that

r∂r|v||v| ,

is uniformly bounded in Bε together with Cauchy-Schwarz inequality to obtain∫

Bε

r |w|2 ∂rΦAr∂r|v||v| ≤ c

∫

Bε

∣∣∣∣r∂r|v||v|

∣∣∣∣ |∂rΦ| |A−B|

≤ c

(∫

Bε

|∂rΦ|2)1/2 (∫

Bε

|A−B|2)1/2

.

Next, observe that

|∂rΦ| = 1|u|2 |u ∧ ∂ru| ≤ c

|v|2 |u ∧ ∂ru| ≤ c |∂rw|.

Thus we may estimate, using (11.73)

∫

Bε

r |w|2 ∂rΦAr∂r|v||v| ≤ c

(∫

Bε

|∂rw|2)1/2 (∫

Bε

|A−B|2)1/2

≤ c ε2Q.

Again, this term is bounded by Q/100 for all ε small enough.

Step 2.2. Observe that (11.69) implies that

|∂rΦ|2 ≤∣∣∣∣∂τh

|u|2 −∂τk

|u|2∣∣∣∣2

+ |Ar −Br|2 ,

269

since∂τk = 0 and Br = 0.

Hence|∂rΦ| ≤ c |∇(h− k)|+ c |A−B|,

in C \Bε. Furthermore, we know that Br = 0, therefore we may bound |Ar| =|Ar − Br| ≤ |A − B|. Using these simple inequalities, together with Cauchy-Schwarz inequality, we get

∫

C\Bε

r |w|2 ∂rΦ Ar∂r|v||v| ≤ c

∫

C\Bε

r∂r|v||v| |A−B|2.

+c

(∫

C\Bε

r2ν+2|∇(h− k)|2)1/2 (∫

C\Bε

r−2−2ν |A−B|2)1/2

.

(11.78)To obtain this estimate, we have implicitly used the result of Lemma 11.1, whichasserts that

r ∂r|v||v| ,

is bounded in C, by some constant which is independent of ε. Now, the first twoterms on the right hand side of (11.78) are seen to be bounded by c ε2−2ν Q,using (11.73), (11.74) and (11.75). While for the last term we decompose it as

∫

C\Bε

r∂r|v||v| |A−B|2 =

∫

C\B1

r∂r|v||v| |A−B|2 +

∫

B1\Bε

r∂r|v||v| |A−B|2.

And, using Proposition 11.2, we get∫

C\Bε

r∂r|v||v| |A−B|2 ≤ c

∫

C\B1

ε2e−2αr|A−B|2 +∫

B1\Bε

r∂r|v||v| |A−B|2.

Finally, we use (11.73) to conclude that∫

C\Bε

r∂r|v||v| |A−B|2 ≤ c ε2−2ν Q.

To summarize, we have proved that∫

C\Bε

r |w|2 ∂rΦAr∂r|v||v| ≤ c ε2−2ν Q,

and thus is bounded by Q/100 for all ε small enough.

Step 3. The third term on the right hand side of (11.71) is decomposed in

270

the following way, using (11.69).∫

Cr ∂r|w|2 (∂Bθ − ∂AΦ) =

∫

Cr ∂r|w|2 (|B|2 − |A|2)

+∫

Cr ∂r|w|2 ∂⊥B (k − h)

|v|2

+∫

Cr ∂r|w|2 ∂⊥Bh

|u|2 (|w|2 − 1)

+∫

Cr ∂r|w|2

∇⊥B−Ah

|u|2 .

(11.79)

Let us denote by I, II, III and IV the different terms which appear on theright hand side of this identity.

In order to estimate I-IV we argue as before and decompose each of theseintegrals into the integral over Bγε and the integral over C \ Bγε, for someγ ∈ (1/2, 1).

Step 3.1.1. For the first term I, we get using (11.72), together with Cauchy-Schwarz inequality

∫

Bγε

r∂r|w|2(|B|2 − |A|2) ≤ cεα+1

(∫

Bγε

|∂rw|2)1/2 (∫

Bγε

|A−B|2)1/2

.

which, thanks to (11.73) is bounded by c ε3+αQ and hence is smaller thatQ/100,provided ε is chosen small enough.

Step 3.1.2. Integrating by part now the integral of the same quantity butover C \Bγε yields

∫

C\Bγε

r∂r|w|2 (|B|2 − |A|2) = −γ

∫

∂Bγε

ε(|w|2 − 1)(|B|2 − |A|2)

−∫

C\Bγε

(|w|2 − 1)1r∂r(r2(|B|2 − |A|2)).

Using the mean value formula, we obtain the existence of γ ∈ (1/2, 1) such that

ε

2

∫

∂Bγε

||w|2 − 1| ||B|2 − |A|2| ≤∫

Bε\B ε2

||w|2 − 1| ∣∣(|B|2 − |A|2)∣∣

≤ c εα

(∫

Bε\Bε/2

(|u| − |v|)2)1/2

×(∫

Bε

|B −A|2)1/2

,

Now we use (11.73) to conclude that

γ

∫

∂Bγε

ε(|w|2 − 1) (|B|2 − |A|2) ≤ c ε3+αQ,

271

which is also bounded by Q/100 for all ε small enough.

Finally, we have∣∣∣∣∣∫

C\Bγε

(|w|2 − 1)1r

∂r

(r2(|B|2 − |A|2))

∣∣∣∣∣

2

≤(∫

C\Bε/2

r2(ν+α)(|u| − |v|)2)

1∑

j=0

∫

C\Bε/2

r−2(ν−j+1)|∇j(A−B)|2 .

(11.80)Which, thanks to (11.75) and (11.73) is bounded by c ε2−2ν Q and hence, is lessthan Q/100, for ε small enough.

Step 3.2.1. The second integral II which appears on the right hand sideof (11.79) is also decomposed into the sum of the integral over Bγε and theintegral over C \Bγε, for some γ ∈ (1/2, 1). First we have, by Cauchy-Schwarzinequality

∫

Bγε

r∂r|w|2 ∂⊥B (k − h)|v|2 ≤

(∫

Bε

|∂rw|2)1/2 (∫

Bε

r2|B|2|v|2

|∇(h− k)|2|v|2

)1/2

.

(11.81)Thanks to (11.74), we easily conclude, as we have already done, that

∫

Bγε

r ∂r|w|2 ∂⊥B (k − h)|v|2 ≤ c ε1+αQ,

which in turn is bounded by Q/100, provided ε is small enough. Observe thatwe have used the fact that, thanks to (11.72), we know that

r2 |B|2|v|2 ≤ c ε2 r2α,

in Bγε and we have used the fact that α > 1− ν.

Step 3.2.2. As usual, we integrate by part∫

C\Bγε

r ∂r|w|2 ∂⊥B (k − h)|v|2 = −γ ε

∫

∂Bγε

∂r|w|2 ∂⊥B (k − h)|v|2

−∫

C\Bγε

(|w|2 − 1)1r

∂r

(r2 ∂⊥A (k − h)

|v|2)

.

(11.82)The mean value formula allows us to state that

∣∣∣∣∣ε

2

∫

∂Bγε

r ∂r|w|2 ∂⊥B (k − h)|v|2

∣∣∣∣∣ ≤ c

(∫

Bε\Bε/2

|∂rw|2)1/2

×(∫

Bε\Bε/2

|B|2 |∇(h− k)|2)1/2

.

(11.83)

272

which, thanks to (11.72) and (11.74), is bounded by c εαQ and hence, is lessthat Q/100 provided ε is small enough.

The second integral on the right hand side of (11.82) is decomposed in thefollowing way

∫

C\Bγε

(|w|2 − 1)1r

∂r

(r2 ∂⊥B (k − h)

|v|2)

= 2∫

C\Bγε

(|w|2 − 1)∂⊥B (k − h)

|v|2

+∫

C\Bγε

(|w|2 − 1) r ∂rB∇⊥(k − h)

|v|2

−2∫

C\Bγε

(|w|2 − 1)∂r|v||v|3 r ∂⊥B (k − h)

+∫

C\Bγε

(|w|2 − 1)rB

|v|2 ∂r∇⊥(k − h).

(11.84)All the terms which appear in this equality can be handled as before to showthat they are bounded by Q/100, provided ε is small enough. For example,using (11.72), we have

∣∣∣∣∣∫

C\Bγε

(|w|2 − 1)∂⊥B (k − h)

|v|2

∣∣∣∣∣

≤∣∣∣∣∣∫

C\B1

(|w|2 − 1)∂⊥B (k − h)

|v|2

∣∣∣∣∣ +

∣∣∣∣∣∫

B1\Bγε

(|w|2 − 1)∂⊥B (k − h)

|v|2

∣∣∣∣∣

≤(∫

C\B1

r2α (|u| − |v|)2)1/2 (∫

C\B1

|∇(k − h)|2)1/2

+

(∫

B1\Bγε

r2(α+ν−1)(|u| − |v|)2)1/2 (∫

C\B1

r2−2ν |∇(k − h)|2)1/2

.

And using (11.75) together with (11.74), we see that this term is bounded byc ε2−2ν Q. The same arguments work for the next two terms. Let us now focuson the last term ∫

C\Bγε

(|w|2 − 1)rB

|v|2 ∂r∇⊥(k − h).

Obviously this term can be estimated by∫

C\Bγε

(|w|2 − 1)rA

|v|2 ∂r∇⊥(k − h) ≤ c

∫

C\Bε/2

rα+1 ||u| − |v|| |∇2(h− k)|,

where we have used (11.72). We claim that∫

C\Bγε

rα+1 ||u| − |v|| |∇2(h− k)| ≤ c ε1−νQ. (11.85)

Assuming that the claim is already proved, we conclude, using (11.75) that∫

C\Bγε

(|w|2 − 1)rA

|v|2 ∂r∇⊥(k − h) ≤ c ε(3−3ν)/2Q,

273

and hence is bounded by Q/100, provided ε is small enough.

It remains to prove the claim. To begin with, standard elliptic estimatesyield for all r > 0

∫

B2r\Br

|∇2(h− k)|2 ≤ c

∫

B4r\Br/2

|∆(h− k)|2 + c r−4

∫

B4r\Br/2

|h− k|2.

Hence, we obtain∫

C\Bε/2

r4−2ν |∇2(h− k)|2 ≤ c

∫

C\Bε/4

r4−2ν |∆(h− k)|2

+ c

∫

C\Bε/4

r−2ν |h− k|2.

However, we know that h− k satisfies the equation

−∆(h− k) + 2∇|v||v| ∇(h− k) + |v|2 (h− k) + 2

∇(|u| − |v|)|u| ∇h

+2|u|

∇|v||v| (|v| − |u|) + (|u|2 − |v|2)h = 0.

Now, we use the fact that

r|∇|v||v| ≤ c,

and also that

r|∇h||u| ≤ c,

to conclude that we can estimate∫

C\Bε/2

r4−2ν |∇2(h− k)|2 ≤ c

∫

C\Bε/4

r2−2ν |∇(h− k)|2

+ c

∫

C\Bε/4

(r4−2ν + r−2ν) |h− k|2

+ c

∫

C\Bε/4

r2−2ν |∇(|u| − |v|)|2

+ c

∫

C\Bε/4

r−2ν (|u| − |v|)2

+ c

∫

C\Bε/4

r4−2ν (|u| − |v|)2 |h|2.

Thanks to (11.73) we can evaluate the first term by c ε2−2ν Q. The fourthterm is easily seen to be bounded by c ε2−2ν Q too.

Let us now evaluate the last term. Thanks to Proposition 11.2, we knowthat h is bounded (an even decays exponentially fast to 0) in C \B1. Moreover,

274

it follows from Lemma 11.6 that |∇h| ≤ c r−1 in C \ Bε/4. Therefore we canbound

∫

C\Bε/4

r4−2ν (|u| − |v|)2 |h|2 ≤ c

∫

C\B1

r4−2ν (|u| − |v|)2

+ c

∫

C\Bε/4

r4−2ν (1− log r) (|u| − |v|)2.

Finally, applying (11.75), we see that this term is bounded by c ε2−2ν Q.

We now turn to the evaluation of the third term, which involves∫

C\Bε/4

r2−2ν |∇(|u| − |v|)|2.

We write, as in the proof of Lemma 11.8, D := |u| − |v|. Using the result ofLemma 11.6, it is an easy exercise to see that

|∆D| ≤ c

( |D|ε2

+1r

|∇(h− k)||u|

). (11.86)

in C \Bε/8.

To obtain the relevant estimate in C \ BRε, we use a strategy close to theone used in the proof of Lemma 11.8. Le us recall that D := |u| − |v| satisfies

−∆D +cε

ε2D = − 1

|u|3 ∇ξ∇(h + k).

where cε ≥ 2 in C\BRε provided ε is chosen small enough and R is chosen largeenough, but fixed.

Let χ be a cutoff function equal to 0 in BRε/2 and equal to 1 outside BRε.Further assume that |∇χ| ≤ c (R ε)−1, for some constant c independent of ε andR. We multiply the above equation by χ2 r2−2ν D and integrate the results onC

∫

Cr2−2νχ2|∇D|2 +

2ε2

∫

Cχ2r2−2ν |D|2 ≤ c

∫

Cχ2r1−2ν |∇(h− k)||D|

+ c

∫

Cr1−2νχ2|D||∇D|

+ c

∫

Cr2−2νχ|∇χ||D| |∇D|.

Making use of the inequality a b ≤ κ a2 +14κ

b2, we can bound

c

∫

Cr1−2ν χ2 |D| |∇D| ≤ 1

2

∫

Cr2−2ν χ2 |∇D|2 + c

∫

Cr−2 χ2 |D|2.

275

Hence, for R chosen large enough, we conclude that

c

∫

Cr1−2ν χ2 |D| |∇D| ≤ 1

2

∫

Cr2−2ν χ2 |∇D|2 +

12ε2

∫

Cχ2 |D|2.

Similarly,

c

∫

Cr2−2ν χ |∇χ| |D| |∇D| ≤ 1

2

∫

Cr2−2ν χ2 |∇D|2

+c

R2ε2

∫

BRε\BRε/2

r2−2ν |D|2.

Hence we obtain∫

Cr2−2ν χ2 |∇D|2 +

2ε2

∫

Cχ2 r2−2ν |D|2 ≤ c

c

R2ε2

∫

BRε\BRε/2

r2−2ν |D|2

+c

(∫

C\BRε/2

r2−2ν |∇(h− k)|2)1/2 (∫

C\BRε/2

r−2 |D|2)1/2

.

It is now an easy exercise, using (11.73) to show that the right hand side of thelast inequality is bounded by c ε1−ν Q.

Let R > 0 be chosen large enough, but fixed. We can use Sobolev embeddingtogether with classical elliptic estimates to show that, for all r > 0

∫

B2r\Br/2

|∇D|2 ≤ c r2

∫

B4r\Br/2

|∆D|2 + c r−2

∫

B4r\Br/2

|D|2.

Multiplying by r2−2ν , we get easily∫

BRε\Bε/4

r2−2ν |∇D|2 ≤ c

∫

B2Rε\Bε/8

r4−2ν |∆D|2 + c

∫

B2Rε\Bε/8

r−2ν |D|2.

Using (11.86), we conclude that∫

BRε/2\Bε/4

r2−2ν |∇(|u| − |v|)|2 ≤ c

∫

B2Rε\Bε/8

ε−2ν (|u| − |v|)2

+ c

∫

B2Rε\Bε/8

r2−2ν |∇(h− k)|2.

Finally, we use (11.74) to evaluate the second term and we get∫

BRε/2\Bε/4

r2−2ν |∇(|u| − |v|)|2 ≤ c ε2−2ν Q.

Hence, we have proved that

c

∫

C\Bε/4

r2−2ν |∇(|u| − |v|)|2 ≤ c ε1−ν Q.

276

Finally, it remains to estimate∫

C\Bε/4

(r−2ν + r4−2ν) |h− k|2.

Let us recall that ξ := h− k satisfies

−div(∇ξ

|v|2)

+ ξ = −div( |u|+ |v||u| |v| D∇h

),

where, as above D := |u| − |v|.First we consider a cutoff function χ identically equal to 1 outside B1 and

equal to 0 in B1/2. We multiply the above equation by r4−2ν χ2 ξ and integratethe result over C. Using the fact that |u| and |v| are uniformly bounded frombelow in C \B1/2 we get

∫

Cχ2r4−2ν |∇ξ|2 +

∫

Cχ2r4−2ν |ξ|2 ≤ c

∫

C|D| r−1 |∇(χ2 r4−2ν ξ)|

+ c

∫

C|∇(χ2 r4−2ν)| |ξ| |∇ξ|.

Arguing as above, we easily conclude that∫

C\B1

r4−2ν |h− k|2 ≤ c

∫

C\B1/2

r2−2ν |∇(h− k)|2

+ c

(∫

C\B1/2

|D|2 r4−2ν

)1/2 (∫

C\B1/2

r2−2ν |∇(h− k)|2)1/2

+ c

∫

C\B1/2

r2−2ν |∇(h− k)|2 + c

∫

C\B1/2

r−2ν |D|2.

In order to conclude, we use (11.74) and (11.75), and obtain∫

C\B1

r4−2ν |h− k|2 ≤ c ε2−2ν Q.

It remains to estimate∫

B1\Bε/4

r−2ν |h− k|2.

However, since by construction h− k = ? d(A−B), it immediately follows from(11.74) that ∫

B1\Bε/4

r−2ν |h− k|2 ≤ c ε2−2ν .

This ends the proof of the claim.

277

Step 3.3.1. Now we need to bound the third term III on the right handside of (11.79). Again we decompose this integral as the sum of the integralover Bγε and C \Bγε

∣∣∣∣∣∫

Bγε

r ∂r|w|2 ∂⊥Bh

|u|2 (|w|2 − 1)

∣∣∣∣∣

2

≤ c

(∫

Bγε

r2α (1− |w|)2)(∫

Bγε

|∂rw|2)

,

(11.87)where we have use (11.72) together with the fact that

|∇h| ≤ c r−1 |u|2,

in Bγε. Indeed, we know for (11.1) that

|∇h| ≤ |u| |∇u− iA|.

Now, we use the fact that |u| ≥ c r/ε in Bγε together with the fact that |∇u| ≤c ε−1 and since |A| ≤ c rα, to get the desired inequality.

We claim that ∫

Bε

r2α|1− |w||2 ≤ c ε2αQ, (11.88)

for all ε small enough. In order to prove this identity, we use the fact that

(|w|(r, θ)− |w|(γε, θ))2 =

(∫ γε

r

∂r|w|)2

≤ c log r

∫

Bε

|∂rw|2.

for any γ ∈ [1/2, 1].Integrating this inequality over Bγε, we already conclude that

∫

Bε

r2α (w(r, θ)− w(γε, θ))2 ≤ c ε2+2α log εQ. (11.89)

In addition, the mean value formula yields the existence of γ ∈ (1/2, 1) suchthat ∫

Bε

r2α |1− |w|(γε, θ)|2 ≤ c ε1+2α

∫

∂Bγε

|1− |w||2

≤ c ε2α

∫

Bε\Bε/2

|1− |w||2.

The claim then follows at once from this last inequality and from (11.89).

It is now an easy exercise to show that∫

Bγε

r ∂r|w|2 ∂⊥Bh

|u|2 (|w|2 − 1) ≤ c εαQ,

278

and hence is bounded by Q/100 provided ε is small enough.

Step 3.3.2. Next, we integrate by part to get∫

C\Bγε

r∂r|w|2 ∂⊥Bh

|u|2 (|w|2 − 1) = −∫

C\Bγε

r

2∂⊥Bh

|u|2 ∂r(|w|2 − 1)2

= −∫

∂Bγε

ε∂⊥Bh

|u|2 (|w|2 − 1)2

−∫

C\Bγε

1r∂r

(r2

2∂⊥Bh

|u|2)

(|w|2 − 1)2.

To begin with, we have, thanks to Lemma 11.6∣∣∣∣∣∫

∂Bγε

ε∂⊥Bh

|u|2 (|w|2 − 1)2∣∣∣∣∣ ≤ c ε2+2α

∫

∂Bγε

(|w|2 − 1)2

ε2.

Using the mean value formula, we conclude as we have already done∣∣∣∣∣∫

∂Bγε

ε∂⊥Bh

|u|2 (|w|2 − 1)2∣∣∣∣∣ ≤ c ε1+2αQ.

Now, in view of Lemma 11.6, it should be clear that,

supC\Bε/2

1r

∂r

(r2

2A.∇⊥h

|u|2)

,

is bounded by some constant independent of ε. Hence we have∣∣∣∣∣∫

C\Bγε

1r

∂r

(r2

2∂⊥Bh

|u|2)

(|w|2 − 1)2∣∣∣∣∣ ≤ c ε2Q.

We conclude that the third term on the right hand side of (11.79) is also boundedby Q/100, provided ε is small enough.

Step 3.4.1 We now deal with IV the fourth term in (11.79). Using Lemma 11.6,together with Cauchy-Schwarz inequality, we get

∫

Bγε

r ∂r|w|2 (B −A)∇⊥h

|u|2 ≤ c

(∫

Bγε

r2ν+1 ε |∂rw|2)1/2

×(∫

Bγε

r−2−2ν |A−B|2)1/2

.

which, thanks to (11.73) is bounded by c ε2Q and hence is smaller than Q/100,provided ε is chosen small enough.

279

Step 3.4.2. Integrating by part now the integral of the same quantity butover C \Bγε yields

∫

C\Bγε

r∂r|w|2(B −A)∇⊥h

|u|2 = −γ

∫

∂Bγε

ε(|w|2 − 1)(B −A)∇⊥h

|u|2

−∫

C\Bγε

(|w|2 − 1)1r∂r

(r2(B −A)

∇⊥h

|u|2)

.

Using the mean value formula, we obtain the existence of γ ∈ (1/2, 1) such that

ε

2

∫

∂Bγε

||w|2 − 1|∣∣∣∣ (B −A)

∇⊥h

|u|2∣∣∣∣ ≤

∫

Bε\B ε2

1r||u| − |v|| |(B −A)|

≤ c ε−1

(∫

Bε\Bε/2

(|u| − |v|)2)1/2 (∫

Bε\Bε/2

|B −A|2)1/2

,

Now we use (11.73) to conclude that

γ

∫

∂Bγε

ε(|w|2 − 1) (B −A)∇⊥h

|u|2 ≤ c ε2Q,

which is also bounded by Q/100 for all ε small enough.

Finally, we have(∫

C\Bγε

(|w|2 − 1)1r

∂r

(r2(B −A)

∇⊥h

|u|2))2

≤(∫

C\Bε/2

r2ν ||u| − |v||2)

1∑

j=0

∫

C\Bε/2

r−2(ν−j)−2|∇j(A−B)|2 ,

(11.90)Which, thanks to (11.75) and (11.73) is bounded by c ε2−2ν Q and hence, is lessthan Q/100, for ε small enough. Observe that, to obtain the last estimate, wehave used the result of Lemma 11.6 which implies that

∣∣∣∣∂r

(r2 ∇h

|u|2)∣∣∣∣ ≤ c,

in C \Bε/2.

Step 4. Let us bound now the fourth term on the right hand side of (11.71).Once again we write this integral as the sum of the integral over Bγε and theintegral over C \Bγε, for some γ ∈ (1/2, 1).

Step 4.1. Clearly, we have

∫

Bγε

r ∂rΦ∂A|w|2 ≤ c

(∫

Bγε

|∂rw|2)1/2 (∫

Bγε

r2+2α |∇|w||2)1/2

, (11.91)

280

where we have used the fact that ∂rΦ| ≤ c|∂rw| in Bγε and also (11.72).

However, we have

∇|w| = ∇(|u| − |v|)|v| +

|∇v||v|2 (|v| − |u|),

This together with Lemma 11.1 implies that (11.91) becomes∣∣∣∣∣∫

Bγε

r∂rΦ∂A|w|2∣∣∣∣∣

2

≤ c

(∫

Bγε

|∂rw|2)(∫

Bγε

r2α(|u| − |v|)2)

≤ c

(∫

Bγε

|∂rw|2)(∫

Bγε

r2αε2|∇(|u| − |v|)|2)

.

Using (11.75) we see that the first term on the right hand side of this inequalityis bounded by c ε1−αQ.

It remains to estimate∫

Bγε

r2α ε2 |∇(|u| − |v|)|2.

We write, as in the proof of Lemma 11.8, D := |u| − |v|. Using the result ofLemma 11.6, it is an easy exercise to see that

|∆D| ≤ c

( |D|r2

+1r

|∇(h− k)|u|

), (11.92)

in B2ε. Now, we can use Sobolev embedding together with classical ellipticestimates to show that, for all r > 0

∫

B2r\Br/2

|∇D|2 ≤ c r2

∫

B4r\Br/2

|∆D|2 + c r−2

∫

B4r\Br/2

|D|2.

Multiplying by r2α, we get easily∫

Bγε

r2α |∇D|2 ≤ c

∫

B2γε

r2α+2 |∆D|2 + c

∫

B2γε

r2α−2 |D|2.

Using (11.92), we conclude that∫

Bγε

r2α ε2 |∇(|u| − |v|)|2 ≤ c

∫

B2γε

r2α−2 ε2 (|u| − |v|)2

+ c

∫

B2γε

r2α ε2 |∇(h− k)|2|u|2 .

Finally, we use (11.74) to evaluate the second term and (11.88) to evaluate thefirst term and we get

∫

Bγε

r2α ε2 |∇(|u| − |v|)|2 ≤ c ε2αQ.

281

Therfore, we have proved that∫

Bγε

r ∂rΦ∂A|w|2 ≤ c (εα + ε1−α)Q,

and in particular, is bounded by Q/100 provided ε is chosen small enough.

Step 4.2. We now turn to the evaluation of the same quantity Bγε. Again,we integrate by parts

∫

C\Bγε

r ∂rΦ∂A|w|2 = −∫

∂Bγε

γ ε ∂rΦ(|w|2 − 1) Ar

−∫

C\Bγε

∂A (r∂rΦ) (|w|2 − 1).

This first term is bounded as usual using the mean value formula and we findeasily, using (11.72) that

∣∣∣∣∣∫

∂Bγε

γ ε ∂rΦ(|w|2 − 1)Ar

∣∣∣∣∣ ≤ c ε2+α−ν Q.

In order to estimate the second term, we recall that

∂rΦ = (Ar −Br) +∂τ (h− k)|u|2 .

And hence we can bound∫

C\Bγε

∂A (r∂rΦ) (|w|2 − 1) ≤ c

∫

C\Bγε

||u| − |v|| rα |∇(h− k)|

+c

∫

C\Bγε

||u| − |v|| rα+1 (|A−B|+ r|∇(A−B)|)

+c

∫

C\Bγε

||u| − |v|| rα+1|∇2(h− k)|.

We use Cauchy-Schwarz inequality, (11.73)-(11.75) and (11.85) to conclude that∫

C\Bγε

∂A (r∂rΦ) (|w|2 − 1) ≤ c ε(3−3ν)/2Q.

Again this quantity is bounded by Q/100 if ε is chosen small enough.

Step 5. So it just remains to bound the last term in the right hand side of(11.71). First observe that we can write

π

2|a∞A⊥∞| =

∣∣∣∣(∫

Clog r∇h

)·(∫

Cz′ h(z′)

)∣∣∣∣ .

However, since ∫

Cz′ k(z′) = 0, and B∞ = 0,

282

we can also write

π

2|a∞A⊥∞| =

∣∣∣∣(∫

Clog r∇(h− k)

)·(∫

Cz′ (h(z′)− k(z′))

)∣∣∣∣ .

We can bound the first term on the right hand side as follows. First, using(11.74) we get

∣∣∣∣∫

B1

log r∇(h− k)∣∣∣∣2

≤(∫

B1

r2ν−2 log2 r|u|2) (∫

B1

r2−2ν |∇(h− k)|2|u|2

)

≤ c ε2−2ν Q,

(provided 2ν > 1, which is true) and then, using (11.75), we obtain

∣∣∣∣∣∫

C\B1

log r∇(h− k)

∣∣∣∣∣

2

≤(∫

C\B1

1r3

log2 r

)(∫

C\B1

r3|∇(h− k)|2)

≤ c ε2−2ν Q.

The second term can be bounded as follows. First we have, using an inte-gration by parts ∣∣∣∣

∫

Cz′ (h− k)

∣∣∣∣ ≤ c

∫

Cr2 |∇(h− k)|.

Next we bound, using (11.74)

∣∣∣∣∫

B1

r2 |∇(h− k)|∣∣∣∣2

≤(∫

B1

r2ν+2 |u|2) (∫

B1

r2−2ν |∇(h− k)|2|u|2

)

≤ c ε2−2ν Q,

and then, using (11.75)

∣∣∣∣∣∫

C\B1

log r∇(h− k)

∣∣∣∣∣

2

≤(∫

C\B1

1r3

) (∫

C\B1

r7|∇(h− k)|2)

≤ c ε2−2ν Q.

Once more we conclude that π/2 a∞A⊥∞ is bounded by Q/100 provided ε issmall enough. And this finishes the proof.

283

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290

Index

Sets :As, 24, 87Ωσ, 24Oσ, 87

Parameters :δ, 106τ0j , 106

δ±j , 170γ±j , 27, 50λj , 26l1, 159l2, 159l3, 161

rj , 36, 106θi, 36, 106

X⊥, 199∂X , 196, 199∂ν , 199∇⊥, 199dAu, 2, 288z ∧ z′, 199z · z′, 55, 199Area, 165?, 288

Dirichlet to Neumann Operators :DN , 176DN j

int,ε, 136DNext,0, 142DNext,ε, 141DNint,0, 138DNint,ε, 141DN , 186

Functionals :Eε, 4Gκ, 1, 20Q, 69Wg, 7

Nonlinear oerators :M, 155MI , 161MII , 161N , 15, 55Nε, 73, 120Q1, 156Q2, 156Q3, 157

Linearized operators :L, 55L1, 55Lε, 73Lε,τ , 73L′ε, 129L1, 55Lε, 73, 128Lε,τ , 73Lε,j , 106Lε, 128Lε, 120, 128

Function spaces :Ck,α

ν (Ω \ Σ), 24Ck,α

ν,D(Ω \ Σ), 25Ck,α

µ (O \ Σ), 88Ek,α

µ (O \ Σ), 88L2

ν , 169Hk

ν , 170

291

E , 154F , 154

Norms and seminorms :[w]′k,α,s, 88[w]k,α,s, 24, 87‖w‖Ck,α

ν (Ω\Σ), 24, 87‖w‖Ck,α

ν (Ω\Σ), 88‖w‖L2

ν, 169

‖w‖Hkν, 170

‖w‖Ek,αµ

88

Radially symmetric solutions :Sε, 200S, 11, 53S∗, 64Sε, 72Sj , 106T , 54Sε, 290Tε, 290

Jacobi fields :Φ0

1, 56Φ+1

1 , 56Φ−1

1 , 56Φ0

ε, 73Φ+1

ε , 73Φ−1

ε , 73Φ0

ε,?, 131Φ+1

ε,?, 131Φ−1

ε,?, 131Ψk

ε,j , 154Ψ0

1, 71Ψ+1

1 , 66, 71Ψ+2

1 , 72Ψ−1

1 , 66, 71Ψ−2

1 , 72Ψ0

ε, 73Ψ+1

ε , 73Ψ−1

ε , 73Φ0

ε,j , 128Φ+1

ε,j , 128Φ−1

ε,j , 128

Weight function :ρ, 181, 186, 187, 246

Cut off functions :ηt, 118ηε,j , 154ηt,j , 118ψd0 , 119, 154ϕd±1 , 119, 154

Functions :Hε, 264Hj , 106J±n , 59K, 261φ?, 129φj , 26φ, 119ψu, 243ψvε , 264ξ, 113h, 288wj

n, 57

Approximate solutions :u, 119ud0,d±1 , 119u(d0, d±1, w), 154u1, 53u?, 6, 8, 106uε,τ , 73

292

Index

A. Jaffe and C. Taubes conjectures,3, 20, 287

Approximate solution, 105Attractive case, 3

Bessel functions, 58Bethuel-Brezis-Helein’s conjecture, 12,

284

Catenoid, 46Conformal Killing vector field, 196Conformal vector field, 18Conjugate operator, 55Conservation laws, 204Coulomb gauge, 290, 298, 305Critical points at infinity, 52Current, 290

Dirichlet to Neumann map, 41, 42,136, 176

Domain decomposition method, 36,41, 127

Eigenfrequencices, 64Eigenfrequencies (Higher), 76Eigenfrequencies (Lower), 83Eigenfunction, 26η-compactness Lemma, 230, 295Excess of phase, 239External magnetic field, 1

Gauge invariance, 4, 20, 287Gauge theory, 1Ginzburg-Landau functional, 1Ginzburg-Landau parameter, 1Gross-Pitaevskii equation, 18

Harmonic map, 6

Hodge operator, 288

Indicial root, 27, 170Induced magnetic field, 1, 288, 290Injectivity range, 30Integrable case, 2

Jacobi fields, 15, 47, 52, 56

Liapunov-Schmidt decomposition method,52

Limit profil, 10Linearized Ginzburg-Landau opera-

tor, 53, 55Liouville equation, 51

Maximum principle, 31, 40Maximum principle in weighted spaces,

75Minimal surface, 46Minimizer, 2, 9, 10, 19, 20

Nondegenerate critical point, 14, 19,143

p-energy, 9Pohozaev formula, 18, 195, 230, 292,

316

Renormalized energy, 7, 106Repulsive case, 3Rescaled Schauder estimates, 28ρ-conformal vector field, 18, 202

Scalar curvature, 15Self dual case, 2Semilinear elliptic equation, 196, 217Surjectivity range, 39

293

Vortices, 2Vorticity, 3

Weighted Holder space, 9, 16, 23, 87Weighted Lebesgue space, 174Weighted Sobolev space, 18, 169

Yamabe problem, 15Yang-Mills-Higgs equations, 20

294