GINZBURG{LANDAU VORTICES: DYNAMICS, PINNING, AND HYSTERESIS · GINZBURG{LANDAU VORTICES: DYNAMICS,...

GINZBURG–LANDAU VORTICES: DYNAMICS, PINNING, ANDHYSTERESIS ∗

FANG-HUA LIN† AND QIANG DU‡

SIAM J. MATH. ANAL. c© 1997 Society for Industrial and Applied MathematicsVol. 28, No. 6, pp. 1265–1293, November 1997 001

Abstract. In this paper, we consider three problems related to the mathematical study ofvortex phenomena in superconductivity based on the G–L models. First, we study the long-timebehavior of the solutions of the time-dependent Ginzburg–Landau equations. Then we describeresults concerning the pinning effect of thin regions in a variable thickness thin film. Finally, weprove the existence of vortex-like solutions to the steady state Ginzburg–Landau equations andstudy the hysteresis phenomenon near the lower critical field.

Key words. superconductivity, Ginzburg–Landau equations, vortex solution, long-time behav-ior, vortex pinning, hysteresis

AMS subject classifications. 35B40, 35A20, 35J65, 35K99, 82D55

PII. S0036141096298060

1. Introduction. Below the critical temperature Tc, the response of a super-conducting material to an externally imposed magnetic field is most convenientlydescribed by the diagram given in Figure 1, which shows the minimum energy stateof the superconductor as a function of H0, the applied magnetic field, and the di-mensionless material parameter κ (known as the Ginzburg–Landau parameter). Theparameter κ determines the type of superconducting material [10]. For κ < 1√

2, type-I

superconductors, there is a critical magnetic field HC below which the material willbe the superconducting Meissner state but above which it will be in the normal state.For κ > 1√

2, type-II superconductors, there is a third state known as the mixed (or

vortex) state. The vortex state consists of many normal filaments embedded in asuperconducting matrix. Each of these filaments carries with it a quantized amountof magnetic flux and is circled by a vortex of superconducting current. Thus, thesefilaments are often know as vortex lines. One of the most challenging problems tomathematicians working on the superconductivity models is to understand vortexphenomena in type-II superconductors, which include the recently discovered high-temperature superconductors.

The transition from the normal state to the vortex state takes place by a bifur-cation as the magnetic field is lowered through some critical value HC2 . The criticalfield HC1

, on the other hand, is calculated so that at this field the energy of the whollysuperconducting solution becomes equal to the energy of the single vortex filamentsolution for infinite superconductors.

The vortex structures have been studied extensively on the mezoscale by using thewell-known Ginzburg–Landau (G–L) models of superconductivity [10, 13, 14]. Theexistence of vortex-like solutions for the full nonlinear G–L equations has been inves-tigated by researchers using methods ranging from asymptotic analysis to numerical

∗ Received by the editors January 31, 1996; accepted for publication (in revised form) August 28,1996.

http://www.siam.org/journals/sima/28-6/29806.html† Courant Institute, New York University, New York, NY 10012 ([email protected]). The research

of this author was partially supported by NSF grant MS-9401546.‡ Department of Mathematics, Iowa State University, Ames, IA 50011-2064. Current address:

Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay,Kowloon, Hong Kong ([email protected]). The research of this author was partially supported byNSF grant MS-9500718.

1265

1266 FANG-HUA LIN AND QIANG DU

H0

TMeissner

Vortex

Superconducting

H0

T

H c

TcTc

Normal

Normal

c1

c2H

H

κ < 1√2

κ > 1√2

Fig. 1. The various states of superconductors.

simulations; however, it has not been justified by rigorous mathematical analysis.Much progress has been made in recent years [3, 20, 21] to establish a mathematicalframework for a rigorous description of both the static and dynamic properties ofthe vortex solutions; in particular, as the coherence length tends to zero (κ goes toinfinity), various results have been obtained. From a technological point of view, thisis of interest since recently discovered high critical temperature superconductors areknown to have large values of κ, say κ in excess of 50.

Vortex lines may move as a result of internal interactions between these filamentsand external forces (due to applied fields or thermal fluctuations) acting on them.Unfortunately, such vortex motion in an applied magnetic field induces an effectiveelectrical resistance in the material and, thus, a loss of superconductivity. Therefore,it is crucial to understand the dynamic of these vortex lines. At the same time, one isinterested in studying mechanisms that can pin the vortices at a fixed location, i.e.,prevent their motion. Various such mechanisms have been advanced by physicists,engineers, and material scientists. For example, normal (nonsuperconducting) impu-rities in an otherwise superconducting material sample are believed to provide sitesat which vortices are pinned. Likewise, regions of the sample that are thin relative toother regions are also believed to provide pinning sites. These mechanisms have beenintroduced into the general G–L framework to derive various variants of the originalG–L models of superconductivity. Numerical simulation clearly suggests the pinningeffect.

In this paper, we deal with three independent problems, yet all of them are relatedto the study of vortex phenomena. First we study the long-time behavior of thesolutions of the time-dependent G–L equations and the main result is Theorem 2.1.Then we describe results (Theorems 3.1–3.5) concerning the pinning effect of thinregions in a variable thickness thin film, based on the models developed in [6, 12],and finally, we prove the existence of vortex-like solutions to the steady state G–Lequations (Lemma 4.1) and study the hysteresis phenomenon near the lower criticalfield.

Before addressing the above problems, we introduce the notation and the modelsthat will be used in the paper. The starting point of our study is the phenomenological

GINZBURG–LANDAU VORTICES 1267

model due to Ginzburg and Landau for superconductivity in isotropic, homogeneousmaterial samples. Let Ω be a smooth bounded domain in RI 3, occupied by the su-perconducting material. By ignoring the effect of the region exterior to the sample,the steady state model can be stated as a minimization problem of the free energyfunctional

G(ψ,A) =

∫Ω

fn + a|ψ|2 +

b

2|ψ|4 +

1

2ms

∣∣∣(ih∇+esc

A)ψ∣∣∣2

+µs8π

h · (h− 2H0)

dΩ ,(1)

where fn is the free energy density of the nonsuperconducting state in the absence ofa magnetic field, ψ is the (complex-valued) superconducting order parameter, A is themagnetic vector potential, h = (1/µs)curlA is the magnetic field, H0 is the appliedmagnetic field, a and b are constants whose values depend on the temperature andsuch that b > 0, es is the mass of the superconducting charge carriers which is twicethe electronic charge e, c is the speed of light, µs is the permeability, and 2πh isPlanck’s constant. It can be rewritten in nondimensionalized form:

G(ψ,A) =

∫Ω

(1

2(1− |ψ|2)2 +

∣∣∣∣(i

κ∇+ A

)ψ

∣∣∣∣2

+ |curlA−H0|2)dx,(2)

where κ is the so-called G–L parameter.The functional G(ψ,A) has an interesting gauge invariance property and the min-

imization of G in appropriate functional spaces gives the following system of nonlineardifferential equations that are named the G–L equations:

(i

κ∇+ A

)2

ψ − ψ + |ψ|2ψ = 0 in Ω,(3)

curl curlA =i

2κ(ψ∇ψ∗ − ψ∗∇ψ)− |ψ|2A in Ω,(4)

along with natural boundary conditions

curlA ∧ n = H0 ∧ n on ∂Ω(5)

and (i

κ∇ψ + Aψ

)· n = 0 on ∂Ω,(6)

where n is the exterior normal to the boundary ∂Ω. The G–L vortices are representedby the zeros of the complex order parameter ψ. In section 4, we will prove the existenceof vortex solutions to the above system when κ is large and the applied field is nearthe lower critical field.

Equations (3)–(4) are the steady state G–L equations. The time-dependent G–Lmodel is often described by the Gorkov–Eliashberg evolution equation [17]:

η∂ψ

∂t+ i η κΦψ +

(i

κ∇+ A

)2

ψ − ψ + |ψ|2 ψ = 0,

∂A

∂t+∇Φ + curl curl A = − i

2κ(ψ∗∇ψ − ψ∇ψ∗)−A|ψ|2.

(7)


Here, Φ denotes the (real) scalar electric potential, η is a relaxation parameter, andψ∗ denotes the complex conjugate of ψ. For simplicity, we take η = 1 in the rest ofthe paper.

The system is supplemented by the initial and boundary conditions

ψ (x, 0) = ψ0(x), A(x, 0) = A0(x), x ∈ Ω;(8)

(i

κ∇+ A

)ψ · n = 0,

curl A ∧ n = H0 ∧ n,(∂A

∂t+∇Φ

)· n = ~E · n = 0 on ∂Ω.

(9)

Note that (7) and (9) are gauge invariant [11], in the sense that if (ψ, A, Φ) isa solution, then so is (ψχ, Aχ, Φχ), where

ψχ = ψ ei κ χ, Aχ = A +∇χ, Φχ = Φ− ∂χ

∂t.

The dynamics of vortices can be determined from the solutions of the time-dependentequations (7)–(9). The long-time asymptotic behavior of solutions of equations (7)–(9)as t→∞ will be studied later in section 2.

For type-II superconductors, the minimizers of G are believed to exhibit vortexstructures. Numerical experiments show that for large values of κ and moderate fieldstrength, the number of vortices could be exceedingly large even for a small samplesize in actual physical scale. Thus, resolving the vortex phenomenon by using the fullG–L equations remains computationally intensive.

Various simplifications have been made to reduce the complexity. For thin filmsof superconducting material, a two-dimensional model has been developed [6, 12] thatcan account for thickness variations through an averaging process. The model is givenby the following minimization problem:

Gaε (ψ) =

∫Ω

a(x)

(|(∇− iA0)ψ|2 +

1

2ε2(1− |ψ|2)2

)dx,(10)

where Ω denotes the platform of the film, a(x) measures the relative thickness of thefilm, and A0 is a prescribed vector potential due to the normal (to film) componentof the applied field. The role of the Ginzburg–Landau parameter κ is assumed by theparameter ε(∝ 1/κ).

It was proved that for fixed ε, the minimizers of the above problem, along withthe prescribed vector potential A0, provide the leading order approximation to thesolution of the three-dimensional problem [6]. The creation and interaction of vorticesbased on the above model is connected to the prescribed magnetic potential. Thenumber of vortices cannot be prescribed a priori, independently of A0. To simplify theanalysis further, a simpler problem, in which the number of vortices is prescribed andthe magnetic potential is ignored, can be studied. By rescaling the spatial variables,one may consider the minimization of the functional

Faε (ψ) =

∫Ω

a(x)

(|∇ψ|2 +

1

2ε2(1− |ψ|2)2

)dx(11)


with the boundary condition

ψ(x) = g(x) for x ∈ ∂Ω,(12)

where g is smooth with |g(x)| = 1,x ∈ ∂Ω. This can be viewed as a generalization ofthe problem studied in [3, 20] in which a(x) ≡ 1, i.e.,

Fε(ψ) =

∫Ω

(|∇ψ|2 +

1

2ε2(1− |ψ|2)2

)dx.(13)

The pinning effect of the variable thickness will be studied in section 3 by exam-ining the properties of minimizers of (10) and (11).

The rest of the paper is devoted to the problems we have discussed above.

2. The uniqueness of the asymptotic limit. The global existence and unique-ness (up to gauge transformations) of classical solutions of (7)–(9) have been studiedby various authors, e.g., [7, 11, 28]. The dynamics of vortices (including the simplercase which ignores the magnetic field) have been studied by [15, 16, 23, 24, 25] andrecently proved in [22]. In [28], the long-time behavior, in particular the existence ofthe global attractor, is also investigated. Here, we shall sketch the proof of the asymp-totic stability result, which shows that, as t → ∞, (7)–(9) has a unique asymptoticlimit up to gauge transformations.

As in [11], one may choose the so-called zero electric potential gauge for system(7)–(9). To do so, one must solve

∂χ∂t

= Φ(14)

and, at t = 0,

∆χ = −divA in Ω with ∇χ · n = −A · n on ∂Ω.(15)

Thus, in this gauge, Φ ≡ 0 and system (1) reduces to the gradient flow of theenergy functional:

E (ψ, A) = 12

∫Ω

[ ∣∣∣∣(i

κ∇+ A

)ψ

∣∣∣∣2

+ 12

(|ψ|2 − 1)2

+ |curlA−H0|2]dx.(16)

The initial condition satisfies divA(0) = 0 and A(0) · n = 0. Moreover, we makea physically meaningful assumption that ‖ψ(0)‖∞ ≤ 1. Let v = (ψ, A). As shown in[11] and [28], the flow

dv

dt= − grad E(v), v(0) = v0,(17)

has a global classical solution. Let V = H2(Ω) ×H2(Ω), where H2(Ω) and H2(Ω)are spaces of functions whose components (or real and imaginary parts) are in thestandard Sobolev space H2(Ω).

Our main result concerning (17) now follows.Theorem 2.1.

V∞ = limt→∞ v(t) exists in V .


To describe the idea, we start with the ODEdx

dt= − grad f(x), x ∈ RI N

x(0) = x0 .(18)

We assume f ∈ C2 ( RI N ), ∇f(0) = 0, and x0 is close to 0.Case (i). If A = ∇2 f(0) is positive definite, then x(t) → 0 (at an exponential

rate) as t→ +∞.This result is standard. One can calculate

d

dt|x|2 = −2〈 ∇2f(x) · x, x 〉 ≤ −2λ |x|2 .

Here we shall assume |x|(t) ≤ δ0, and (∇2f(x)) ≥ λ I whenever |x| ≤ δ0. Thus|x(t)| ≤ |x(0)| e−λt ∀t ≥ 0 and |x(t)| ≤ |x0| + 1

λ |x(0)| = |x0| + 1λ |∇f(x0)|. We

shall always assume x0 is so close to the origin that |x0|+ 1λ |∇f(x0)| < δ0. Then the

assumption |x(t)| ≤ δ0 is true for all t > 0 and, thus, x(t) → 0 at an exponential rateas t→ +∞.

Case (ii). det (A) 6= 0. Then one has that

− d

dt(f(x)− f(0))

1/2=

1

2

|∇f(x)| |x|(f(x)− f(0))1/2

, if f(x) > f(0),(19)

d

dt(f(0)− f(x))

1/2=

1

2

|∇f(x)| |x||f(x)− f(0)|1/2 , if f(x) < f(0).(20)

We obtain the following: either there is a T ∈ (0,∞) such that

f(x)(T ) ≤ f(0)− δ0 (for some δ0 > 0)

or

limt→+∞ x(t) = x∞ exists.

Indeed, if x is close to 0, then

|∇f(x)||f(x)− f(0)|1/2 ≥ λmin√

2λmax

= C(A) > 0.

Here λmin and λmax are the minimum and maximum, respectively, eigenvalues of(AT A)1/2. Therefore,

∫ ∞

0

|x(t)| dt ≤ 2δ1/20

C(A)

by (19)–(20) whenever f(x)(t) ≥ f(0)−δ0 for all t ≥ 0. In such a case the conclusionlimt→+∞ x(t) = x∞ follows (in fact x∞ = 0 ) as for Case (i).

Case (iii). det (A) = 0 and f(x) is real analytic in B1. We have the followingwell-known estimate (cf. [26, 27]). There are two positive constants θ0, σ0 ∈ (0, 1)depending on f such that

|f(x)− f(0)|θ0 ≤ | grad f(x)| whenever x ∈ Bσ0(0),(21)


∇f(0) = 0. Then, as for Case (ii), one has that either there is a T ∈ (0,∞) such that

f(T ) ≤ f(0)− δ0

or

limt→∞x(t) = x∞ exists.

In [27], Simon considered the case

E(u) =

∫M

F (x, u,∇u) dx,(22)

where M is a compact manifold without boundary andu = − grad E(u) ≡ M(u),u(0) = u0 ' 0, M(0) = 0.

(23)

Here, F is assumed to be analytic in both u and ∇u for u, ∇u near 0.Suppose L is elliptic, Lv = d

ds |s=0 M(sv). Then either there is a T ∈ (0,∞) suchthat

E(u(T )) ≤ E (0)− δ0 for some δ0 > 0

or

u∞ = limt→∞u(t) exists.

To apply the above idea to the time-dependent G–L equations, we need to usethe following estimate given in [11] (also see [28] for similar results).

Lemma 2.2. Let v = (ψ,A) be the solution of the time-dependent G–L equations(17). Then

∫ T

t

[(ψ(s), ψ(s)) + (A(s), A(s))

]ds+ E(ψ(T ),A(T )) = E(ψ(t),A(t))(24)

for any T > t > 0.The following lemma will enable us to apply the above conclusion of Simon.Lemma 2.3. Let (ψ,A) satisfy

curlA ∧ n = H0 ∧ n , A · n = 0, and ∇ψ · n = 0 on ∂Ω.

Let E(ψ,A) be defined by (16) and (ψ∗,A∗) be a steady state solution of the G–Lequations in the gauge divA∗ = 0 in Ω and A∗ · n = 0 on ∂Ω. Then there existconstants θ0, σ0 ∈ (0, 1) such that

|E(ψ,A)− E(ψ∗,A∗)|θ0 ≤ ‖ grad E(ψ,A)‖L2(Ω)(25)

for any ‖(ψ,A)− (ψ∗,A∗)‖C2(Ω) ≤ σ0.Proof. Let

G (ψ, A) = E (ψ, A) + 12

∫Ω

[|divA|2 ] dx.(26)


By assumption, since (ψ∗,A∗) is a steady state solution of the G–L equations in thegauge divA∗ = 0 in Ω and A∗ · n = 0 on ∂Ω, then

G (ψ∗, A∗) = E (ψ∗, A∗)(27)

and (ψ∗,A∗) remains a critical point of G. By the ellipticity of grad G(ψ,A), see[7, 11, 28], we may apply the result of [27] to conclude that there exist constantsθ0, σ1 ∈ (0, 1) such that

|G(ψ, A)−G(ψ∗,A∗)|θ0 ≤ ‖ grad G (ψ, A)‖L2(Ω)(28)

for any ‖(ψ,A)− (ψ∗,A∗)‖C2(Ω) ≤ σ1.

Let χ be a gauge transformation function and ψ = eiκχψ with A = A +∇χ. Letus choose χ such that div A = 0. Simple calculation shows that

∂G

∂ψ= eiκχ

∂E

∂ψand

∂G

∂A=∂E

∂A.

So,

‖ grad E(ψ,A)‖L2(Ω) = ‖ grad G(ψ, A)‖L2(Ω) .

Since divA∗ = 0 for small enough σ0, if ‖(ψ,A)− (ψ∗,A∗)‖C2(Ω) ≤ σ0, we have

‖(ψ, A)− (ψ∗,A∗)‖C2(Ω) ≤ σ1. Thus,

|E(ψ,A)− E(ψ∗,A∗)|θ0 = |E(ψ, A)−G(ψ∗,A∗)|θ0= |G(ψ, A)−G(ψ∗,A∗)|θ0≤ ‖ grad G (ψ, A)‖L2(Ω)

= ‖ grad E(ψ,A)‖L2(Ω).

This proves the lemma.We now turn to the proof of Theorem 2.1. It is important to observe that both

the energy functional and inequality in Lemma 2.3 are gauge invariant.By Lemma 2.2, given any ε > 0, there exists a sufficiently large time tn such that

‖ψ(tn)‖L2(Ω) < ε,(29)

‖A(tn)‖L2(Ω) < ε,(30)

and ∫ T

tn

[‖ψ(s)‖2L2(Ω) + ‖A(s)‖2L2(Ω)

]ds < ε ∀ T > tn.(31)

By gauge invariance, one may define a gauge transformation function χ(tn) suchthat ψ(tn) = eiκχ(tn)ψ(tn) and A(tn) = A(tn) +∇χ(tn) and

div A(tn) = 0 in Ω,

A(tn) · n = 0 on ∂Ω .


We note that ψ(t) = eiκχ(tn)ψ(t) and A(t) = A(t)+∇χ(tn) are again solutions ofthe time-dependent G–L equations in the zero electric potential gauge with properlymodified initial conditions and equation (24) still holds. Also,

‖ ˙ψ(tn)‖L2(Ω) < ε and ‖ ˙A(tn)‖L2(Ω) < ε

and ∫ T

tn

[‖ ˙ψ(s)‖2L2(Ω) + ‖ ˙A(s)‖2L2(Ω)

]ds < ε ∀ T > tn

remain valid. To apply Lemma 2.3, we must construct solutions that are C2(Ω) closeto some steady state for some time tn ∈ [tn, tn + 1]. To get the C2 closeness, we useanother gauge transformation ψ(t) = eiκχ(t)ψ(t) and A(t) = A(t) +∇χ(t), where χis defined by

∂χ

∂t−∆χ = div A in Ω

with boundary condition

∂χ

∂n= −A · n = 0 on ∂Ω

and initial condition at t = tn: χ(tn) = 0.Note that from the G–L equations (7)–(9), we may get [11]

div ˙A = − i2κ

[ψ∂ψ∗

∂t− ψ∗

∂ψ

∂t

].

Thus, we have ∫ tn + 1

tn

[‖ ˙A(s)‖2L2(Ω)

]ds < cε

for some generic constant c. From this, we may get∫ tn + 1

tn

[‖∇ ˙χ(s)‖2L2(Ω)

]ds < cε.

This further implies that∫ tn + 1

tn

[‖∆χ(s)‖2L2(Ω) + ‖ ˙χ(s)‖2L2(Ω)

]ds < cε.

It follows that (ψ, A) also satisfies estimates similar to those in equations (29)–(31).Notice that (ψ, A) is a solution of the time-dependent G–L equations in the gaugeΦ = −divA and it satisfies a parabolic system:

∂ψ

∂t− i κ div A ψ +

(i

κ∇+ A

)2

ψ − ψ + |ψ|2 ψ = 0,

∂A

∂t−∆A = − i

2κ(ψ∗∇ψ − ψ∇ψ∗)− A|ψ|2

(32)


with boundary conditions

curl A ∧ n = H0 ∧ n , A · n = 0, and ∇ψ · n = 0 on ∂Ω.

Using standard parabolic regularity and estimates like (29)–(31) for (ψ, A), wemay find a small δ0 > 0 such that for all δ0 < δ < 1, we have

‖ ˙ψ(tn + δ)‖C2(Ω) < cε and ‖ ˙A(tn + δ)‖C2(Ω) < cε

and

‖div A(tn + δ)‖C1(Ω) < cε

for some constant c. Thus, we may conclude that in C2(Ω), (ψ(tn + δ), A(tn + δ)) isclose to some solution (ψ∞, A∞) of the steady state G–L equations with div A∞ = 0.Now, by Lemma 2.3, we get

∣∣E(ψ(tn + δ), A(tn + δ))− E(ψ∞, A∞)∣∣θ0

≤ ‖ grad E(ψ(tn + δ), A(tn + δ))‖L2(Ω) .

Using the gauge invariance, however, this also implies that

∣∣E(ψ(tn + δ),A(tn + δ))− E(ψ∞, A∞)∣∣θ0

≤ ‖ grad E(ψ(tn + δ),A(tn + δ))‖L2(Ω) .

Since (ψ(tn + δ), A(tn + δ)) is C2 close to (ψ∞, A∞), we have∣∣E(ψ(tn + δ),A(tn + δ))− E(ψ∞, A∞)∣∣

=∣∣E(ψ(tn + δ), A(tn + δ))− E(ψ∞, A∞)

∣∣≤ cε .

By the monotonicity of E(ψ(t),A(t)), we have that for any T , there exists tn largeenough such that

E(ψ(T ),A(T ))− E(ψ∞, A∞)

≥ E(ψ(tn + δ),A(tn + δ))− E(ψ∞, A∞)

≥ cε

for some generic constant c. Using the ideas given in Case (iii) for ODEs, we concludethat ∫ ∞

T

[‖ ˙ψ(s)‖L2(Ω) + ‖ ˙A(s)‖L2(Ω)

]ds < cε.

Thus, we must have that

V∞ = limt→∞ v(t) exists in V.

This concludes the proof of the Theorem 2.1.


3. The pinning effect of variable thickness in thin films. In the previoussection, we were concerned with the dynamic properties of solutions to the time-dependent G–L equations. In this section, we focus our attention on the static case. Inparticular, we illustrate that the vortices may be pinned by inhomogeneities inside thematerial. The inhomogeneities we consider here are introduced due to the variationin thickness of the sample. During our writing, we also became aware of independentworks [9, 1, 18] on the same subject; thus, we only give a brief outline here of theapproach we have used.

3.1. The Dirichlet case. To present the main idea, we first ignore the magneticpotential and consider

minFaε (ψ) , ψ|∂Ω = g

= min

∫Ω

a(x)

(1

2|∇ψ|2 +

1

4ε2(1− |ψ|2)2

)dx , ψ|∂Ω = g

.(33)

For convenience, we assume that Ω is a bounded, smooth (say Lipschitz) domainin RI 2. We may also view ψ as a map from Ω to RI 2 and g as a smooth map from ∂Ω toS1 with deg(g, ∂Ω) = d(≥ 0). The coefficient a is a smooth (say Lipschitz or Holder)function from Ω to RI R with 0 < m ≤ a(x) ≤ M for all x ∈ Ω. The minimizers offunctional Fa

ε satisfy

div (a(x)∇ψ(x)) +a(x)

ε2(1− |ψ(x)|2)ψ(x) = 0 in Ω.(34)

First of all, let us consider the case where, in Ω, there are at least d distinct pointsb1, . . . ,bd, . . . ,bk with a(bj) = m, j = 1, 2, . . . , k. Moreover, we assume that each bjis a strict minimum, that is, a(x) > m for any x 6= bj in a small neighborhood of bj .We define

δ0 =1

2min|bj − bi|, dist(bj , ∂Ω); j 6= i , i, j = 1, . . . , k > 0.(35)

Theorem 3.1. Let ψεn , εn 0, be a sequence of minimizers of (33). Then

ψεn(x) → ψ∗(x) in C1,αloc

(Ω/b1, . . . ,bd

),(36)

where

ψ∗(x) =d∏

j=1

x− bj|x− bj |e

ih∗(x) in Ω(37)

and b1, . . . ,bd are d distinct points in Ω with

a(bj) = m = minx∈Ωa(x) .(38)

Moreover, if we write ψ∗(x) = ei(Θ(x)+h∗(x)), then h∗ ∈ H1(Ω) ∩ Cα(Ω), ψ∗ = g on∂Ω and

div [a(x)(∇Θ +∇h∗)] = 0 in Ω/b1, . . . ,bd.(39)


The proof of the above theorem is based on a series of estimates as similarlypresented in an earlier work [20]. One first has the energy upper bound

min Faε (ψ), ψ|∂Ω = g ≤ mπd log

1

ε+ C(a, g,Ω)

and energy lower bound

min

∫B

a(x)

(1

2|∇ψ|2 +

1

4ε2(1− |ψ|2)2

)dx, ψ|∂B = g , |g| ≤ 1

≥ mπd log1

ε− C(K),(40)

where m = minx∈B a(x) if deg(g, ∂B) 6= 0. Now, the class Sg(λ,K) may be definedas in [20].

Definition 3.2. Let Ω, g be given as before. We say a map u : Ω → RI 2 belongsto the class Sg(λ,K) if

(i) Fε(u) ≤ mπd log 1ε +K.

(ii) for any x ∈ x ∈ Ω, |u(x)| ≤ 12, Bλε(x) ∩ Ω ⊂ |u(x)| ≤ 3

4.Then one can show that if ψ minimizes (33), there exists some positive constants

λ,K such that ψ ∈ Sg(λ,K).One may then show that there are exactly d balls, say B1, . . . , Bd, with xε1, . . . ,x

εd

their centers, such that the corresponding dj = deg(u, ∂Bj) = 1 and

min|xεj − xεk|, dist(xεj , ∂Ω); j, k = 1, . . . , d, j 6= k ≥ δ1(λ,K) > 0.

After extracting a subsequence, we have

xεnj → x∗j as εn → 0.

The limits x∗j are all different points; moreover, a(x∗j ) = m, j = 1, 2, . . . , d. Com-bining with estimates away from vortices, the proof now follows similarly to that in[3, 20].

In the following, we briefly discuss the case when the number of minima is lessthan the degree of the boundary data. For simplicity, we focus on the case whereΩ is the unit disc B in RI 2 and g(θ) = eidθ for some positive integer d ≥ 2. Leta(x, y) = 1 + r2, r2 = x2 + y2. We consider

EB = min

Faε,B(ψ) =

∫B

(a2|∇ψ|2 +

a

4ε2(1− |ψ|2)2

)dxdy,

ψ|∂Ω = g

.(41)

Theorem 3.3. Let ψε be a sequence of minimizers of (41) as ε → 0. Then allvortices of the ψ must be separated and of degree +1 for small enough ε > 0 and theygo to the origin as ε→ 0.

Proof. Indeed, for any small parameter δ > 0 (say δ < 1/d), one may chooseε ≤ ε(δ) and ψ with ψ|∂Ω = g and vortices placed along ∂Bδ/2 such that

EB ≤ Faε,B(ψ) ≤ πd(1 + δ2) log

δ

ε+ πd2 log

1

δ+ c0d

2

= πd(1 + δ2) log1

ε+ πd2 − πd(1 + δ2) log

1

δ+ c0d

2 .


On the other hand, if there are either vortices of degree no less than 2 or vorticesoutside B√δ(0), then by [3],

Faε,B(ψ) ≥ π(d+ dδ) log

1

ε+ c(δ, d) .

For δ > 0 and ε → 0, since π(d + dδ) log 1ε + c(δ, d), one sees that all vortices of the

minimizer must go to the origin and be of degree +1.The next question is for given d (say 2 ≤ d ≤ 5) and ε > 0 (but small): How close

must these vortices be to the origin? We want to show that they cannot be too close.Indeed, for a small generic constant β, if all vortices of ψ are in the β-neighborhood

of the origin, then∫Bβ(0)

(1 + r2)

(1

2|∇ψ|2 +

1

4ε2(1− |ψ|2)2

)dxdy ≥ πd log

β

ε− c0d

2

and ∫B\Bβ(0)

(1 + r2)

(1

2|∇ψ|2 +

1

4ε2(1− |ψ|2)2

)dxdy ≥ πd2 log

1

β.

So,

Faε,B(ψ) ≥ πd log

1

ε+ πd(d− 1) log

1

β− c0d

2.

On the other hand, by placing vortices along ∂Bδ/2(0), one may construct a map withenergy

Faε,B(ψ) ≤ πd(1 + δ2) log

1

ε+ πd(d− 1− δ2) log

1

δ+ c0d

2 .

Comparing the two bounds, we see that not all vortices are in the δ-neighborhoodof the origin if we have

πd(d− 1) logδ

β≥ 2c0d

2 + πdδ2 log1

ε.

By taking β ≤ δ2, this means

πd(d− 1) log1

δ− πdδ2 log

1

ε≥ 2c0d

2 .

Again, letting log 1δ ≥ 2c0, it is sufficient to have

d− 1

2log

1

δ≥ δ2 log

1

ε,

or

ε ≥ δd−1

2δ2 .

Thus, for small but positive ε, δ cannot be too small.We have performed a series of numerical experiments to illustrate the pinning

effect even for modest values of ε. The numerical methods used here are similar to


Fig. 2. Contour plots of the magnitude of the order parameter for a model with constantthickness. The left-hand figure is for ε = .1. The right-hand figure is for ε = .02.

Fig. 3. Contour plots of the magnitude of the order parameter for a model with variablethickness a = a(x, y). The left-hand figure is for ε = .1. The right-hand figure is for ε = .02.

those discussed in [13, 14]. For computational convenience, the unit square [0, 1]2 istaken to be our sample Ω. In the first experiment, we choose the thickness functiona ≡ 1. We solve for the minimizer of (33) by using the Dirichlet boundary conditionswith |ψ| = 1 on the boundary. ψ|∂Ω has a winding number 4. The contour plots ofthe magnitude of the order parameter ψ are given in Figure 2.

Next, we choose the thickness function a = a(x, y) such that it has four minima at(0.25, 0.25), (0.75, 0.35), (0.25, 0.65), (0.75, 0.75) with the same minimum value. Thecontour plots are given in Figure 3. We see that as ε gets smaller, the vortices get“pinned” at the minima of a. This is the case illustrated by Theorem 3.1.

Now, we continue the numerical experiments again using the Dirichlet boundaryconditions with |ψ| = 1 on the boundary. However, we impose the boundary conditionsuch that ψ|∂Ω has a winding number 3. We first choose the thickness function a =a(x, y) such that it has four minima at (0.25, 0.25), (0.25, 0.65), (0.75, 0.35), (0.75, 0.75)with the same minimum value. The contour plots are given in Figure 4. Since thenumber of minima is larger than the winding number, each vortex gets pinned to adifferent minimum point.

Then, we change the thickness function a = a(x, y) such that it has two minimaat (0.25, 0.65), (0.75, 0.35) with the same minimum value. The contour plots are givenin Figure 5. Since the number of minima is less than the winding number, one vortexgets pinned at one minimum but the other two get attracted to another minimumwith some distance still between them.




For comparison purposes, we also give the pictures when choosing the thicknessfunction a ≡ 1. The contour plots are given in Figure 6.

Finally, we present an experiment on the co-existence of vortex–antivortex solu-tions to the minimizers of (33) with constant thickness functions. This question hasbeen rigorously studied in [22]. We again use the Dirichlet boundary conditions with|ψ| = 1 on the boundary. ψ|∂Ω has a winding number 0. Here, we start with a vortexstate that has a vortex with winding number +1 on one side of the domain and a vor-tex with winding number −1 on the other side of the domain. We numerically followthe gradient flow of (33). The solution reaches steady state which again consists oftwo vortices with opposite winding numbers. The contour plots of the magnitude ofthe steady state solution ψ are given in Figure 7.

3.2. The Neumann problem. We now study the minimization problem withNeumann-type boundary conditions. Let a(x) have a strict local minimum at pointsb1, . . . ,bd with a(bj) = m, j = 1, . . . , d. Recall that

δ0 =1

2min|bj − bk|, dist(bj , ∂Ω); j 6= k, j, k = 1, . . . , d > 0.

Let r0 < δ0, Ω0 = Ω \⋃di=1Br0(bi) and define

V =

u ∈ H1(Ω) | |u| ≥ 1

2in Ω0,


Fig. 6. Contour plots of the magnitude of the order parameter for a model with constantthickness. The left-hand figure is for ε = .08. The right-hand figure is for ε = .04.

Fig. 7. Contour plots of the magnitude of the order parameter. The top figure is for ε = .08.The bottom figure is for ε = .03.

deg

(u

|u| , ∂Br0(bk)

)= 1 , 1 ≤ k ≤ d.

.(42)

We consider

min Faε (ψ) , ψ ∈ V

= min

∫Ω

a(x)

(1

2|∇ψ|2 +

1

4ε2(1− |ψ|2)2

)dx , ψ ∈ V

.(43)

It is easy to see that V is a weakly closed subset of H1(Ω). In fact, it is also connected.For given ε > 0, (43) has at least one minimizer, denoted by ψε.

Theorem 3.4. Let ψεn , εn 0, be a sequence of minimizers of (43). Then,


(Ω/b1, . . . ,bd

),


where

ψ∗(x) =d∏

j=1

x− bj|x− bj |e

ih∗(x) in Ω.

Moreover, if we write ψ∗(x) = ei(Θ(x)+h∗(x)), then h∗ ∈ H1(Ω) ∩ Cα(Ω), ∂φ∗

∂n = 0 on∂Ω and

div [a(x)(∇Θ +∇h∗)] = 0 in Ω/b1, . . . ,bd.

The proof follows from constructions similar to those outlined in section 3.1.

3.3. Problems with prescribed magnetic potential. Unlike bulk material,in the thin film limit, superconductors of type-I and type-II display vortex-like struc-ture. It has been shown that the G–L functional takes the special form

F(ψ) =

∫Ω

a(x)

(1

2|i∇ψ +A0(x)ψ|2 + (1− |ψ|2)2

)dx,

where a(x) represents the relative thickness distribution of the thin film. Since, toleading order, the magnetic field penetrates the film uniformly, we get A0 to be agiven magnetic potential that can be prescribed by setting curlA0(x) = H.

Numerical simulation in [6, 14] suggests that the thickness variation provides apinning mechanism for vortices. Using techniques similar to those in section 3.1, wenow provide a rigorous analysis for such phenomena in the case where κ is large whilethe magnetic field is weak (H ≈ κ−1). By a proper scaling of the free energy, we geta functional of the form

Faε (ψ) =

∫Ω

a(x)

(1

2|∇ψ − iA0(x)ψ|2 +

1

4ε2(1− |ψ|2)2

)dx.(44)

In the present form, ε is a small parameter measuring the relative penetration depthand the sample size. Hence, we may consider problems similar to those in section 3.1with the functional defined above. We first consider the Dirichlet problem

min Faε (ψ) , ψ|∂Ω = g = min

∫Ω

a(x)

(1

2|∇ψ −A0(x)ψ|2

+1

4ε2(1− |ψ|2)2

)dx , ψ|∂Ω = g

.(45)

For simplicity, we assume that |g| = 1 on ∂Ω, deg(g, ∂Ω) = d, and a(x) has d distinctstrict minimums at points b1,b2, . . . ,bd. With a proper choice of gauge, we maydefine

A0(x, y) =1

2(Hy,−Hx)T ,

where H is the scaled applied magnetic field. We then have the following theorem.



(Ω/b1, . . . ,bd

),


where

ψ∗(x) =d∏

j=1

x− bj|x− bj |e

ih∗(x) in Ω.

Moreover, if we write ψ∗(x) = ei(Θ(x)+h∗(x)), then h∗ ∈ H1(Ω) ∩ Cα(Ω), ψ∗|∂Ω = gon ∂Ω, and

div [a(x)(∇Θ +∇h∗)] = 0 in Ω/b1, . . . ,bd.

Next, we may also prove similar results for the Neumann-type problems. Let Vbe the space defined in (42). We consider

min Faε (ψ) , ψ ∈ V = min

∫Ω

a(x)

(1

2|∇ψ −A0(x)ψ|2

+1

4ε2(1− |ψ|2)2

)dx , ψ ∈ V

.(46)



(Ω/b1, . . . ,bd

),

where

ψ∗(x) =d∏

j=1

x− bj|x− bj |e

ih∗(x) in Ω.

Moreover, if we write ψ∗(x) = ei(Θ(x)+h∗(x)) = ei(φ∗(x)), then h∗ ∈ H1(Ω) ∩ Cα(Ω),∂φ∗

∂n = A0 · n on ∂Ω (here, n is the outward normal of ∂Ω), and

div [a(x)(∇Θ +∇h∗)] = 0 in Ω/b1, . . . ,bd.

Again, the proofs of Theorems 3.3 and 3.4 are omitted due to their similarities toour earlier discussions.

4. The renormalized energy and the vortex solution of the full G–Lmodel with applied magnetic field. With proper scaling, we focus on the fol-lowing form of the G–L functional:

G(ψ,A) =

∫Ω

(1

4ε2(1− |ψ|2)2 +

1

2|(∇− iA)ψ|2 +

1

2|curlA−H0|2

)dx.

Let Ω ∈ RI 2 be a bounded Lipshitz domain and H0 be a constant field. In thisnondimensionalization, one may view ε as proportional to 1

κ and H0 as proportionalto κ times the (nondimensionalized) applied field. Studies of the densely packed vortexstate when the nondimensionalized field is near the upper critical field may be found in[4]. In the low field case, variational problems concerning the G–L functional have beenconsidered in [5] but with boundary conditions that are not completely physical. Ourdiscussion here is valid for the natural (physically meaningful) boundary conditionsand we will borrow many useful results from [5] and [20].


4.1. The renormalized energy. Following the discussions in [20, 21, 5], wenow formulate the renormalized energy: Let

ψ = eiφb(x) =

d∏j=1

x− bj|x− bj |e

ih

for some points b = (b1,b2, . . . ,bd) ∈ Ωd and ∂φb∂n = 0 on ∂Ω. Let

Bρ =d⋃

j=1

Bρ(bj) .

Choose the gauge divA = 0 in Ω and A · n = 0 on ∂Ω. We may define ζ suchthat

A = ∇⊥ζ in Ω,

ζ = 0 on ∂Ω.

Now, consider

Gρ =

∫Ω\Bρ

(1

2

∣∣∇φb −∇⊥ζ∣∣2 + |∆ζ −H0|2)dΩ

=

∫Ω\Bρ

(1

2|∇φb|2 +

1

2|∇ζ|2 −∇φb · ∇⊥ζ +

1

2|∆ζ −H0|2

)dΩ

:= dπ log1

ρ+WΩ(b, H0) +O(ρ),

where the last equality may be taken as the definition of the renormalized energyWΩ(b, H0). Note

−∫

Ω\Bρ)

∇φb · ∇⊥ζdΩ =

∫Ω\Bρ

div (ζ · ∇⊥φb)dΩ

=d∑

j=1

2πζ(bj) +O(ρ).

So,

WΩ(b, H0) =

∫Ω\Bρ

1

2|∇φb|2dΩ− dπ log

1

ρ+

d∑j=1

2πζ(bj)

+

∫Ω\Bρ

1

2

(|∇ζ|2 + |∆ζ −H0|2)dΩ +O(ρ)

=

∫Ω\Bρ

1


1

ρ+ 2π

d∑j=1

ζ(bj)

+

∫Ω\Bρ

1

2

(|∇ζ|2 + |∆ζ|2) dΩ +1

2H2

0 |Ω|

−∫

Ω\Bρ

H0∆ζdΩ +O(ρ).


Minimizing the term involving φb, we see that φb is a multivalued harmonic func-tion on Ω \ b1,b2, . . . ,bd and we may define the function gΩ(b) by∫

Ω\Bρ

1


1

ρ:= gΩ(b) +O(ρ).

Minimizing the terms involving ζ, we can choose ζ to satisfy

−∆2ζ + ∆ζ = 2πd∑

j=1

δbj in Ω,

where δbj is the Dirac–Delta measure with boundary conditions

ζ = 0 on ∂Ω,

∆ζ = H0 on ∂Ω.

So,

2π

d∑j=1

ζ(bj) = −∫

Ω

(|∇ζ|2 + |∆ζ|2) dΩ +H0

∫∂Ω

∂ζ

∂ndΓ .

Because H0 is a constant, we get∫Ω\Bρ

H0∆ζdΩ = H0

∫∂Ω

∂ζ

∂ndΓ +H0O(ρ) .

Thus,

WΩ(b, H0) =1

2H2

0 |Ω| −1

2

∫Ω

(|∇ζ|2 + |∆ζ|2) dΩ +H0O(ρ) + gΩ(b) .

Now, let us define ζ = ζb + ζH0 , where

−∆2ζb + ∆ζb = 2πd∑

j=1

δbj in Ω,

ζb = 0 on ∂Ω,

∆ζb = 0 on ∂Ω,

and ζH0= H0ζ1 with

−∆2ζ1 + ∆ζ1 = 0 in Ω,

ζ1 = 0 on ∂Ω,

∆ζ1 = 1 on ∂Ω.


Then ∫Ω

|∇ζ|2dΩ = H20

∫Ω

|∇ζ1|2dΩ +

∫Ω

|∇ζb|2dΩ− 2H0

∫Ω

∆ζ1ζbdΩ,

∫Ω

|∆ζ|2dΩ = H20

∫Ω

|∆ζ1|2dΩ +

∫Ω

|∆ζb|2dΩ + 2H0

∫Ω

∆ζ1∆ζbdΩ,

and ∫Ω

∆ζ1(∆ζb − ζb)dΩ =

∫Ω

ζ1∆(∆ζb − ζb)dΩ

= −2πd∑

j=1

ζ1(bj) .

So, ∫Ω

(|∇ζ|2 + |∆ζ|2) dΩ = H20

∫Ω

(|∇ζ1|2 + |∆ζ1|2)dΩ

+

∫Ω

(|∇ζb|2 + |∆ζb|2)dΩ− 2π

d∑j=1

ζ1(bj) .

Therefore,

WΩ(b, H0) =1

2H2

0C(Ω) + 2πH0

d∑j=1

ζ1(bj) + gΩ(b) +O(ρ),(47)

where C(Ω) is a constant and gΩ(b) has the property

gΩ(b) =

+∞ , bi = bj for some i 6= j ,−∞ , b ∈ ∂Ωd,

otherwise it is a smooth function in Ωd.Lemma 4.1. WΩ(b, H0) has a local minimum inside Ωd whenever H0 ≥ H0(Ω)

for some constant H0(Ω).Proof. Choose a small enough positive constant δ0 and let

Ωδ0 = x ∈ Ω | δ0 ≤ dist(x, ∂Ω) ≤ 2δ0 .If dist(bj , ∂Ω) ≥ δ0 for all j, then gΩ(b) ≥ −M(δ0), independently of H0. On theother hand, we can choose d distinct points b1,b2, . . . ,bd ∈ BR(x0), where x0 satisfiesζ1(x0) = minx∈Ω ζ1(x) = −m0 < 0 such that BR(x0) ⊂ x : dist(x, ∂Ω) ≥ 2δ0 andsuch that

gΩ(b) ≤ C(R)

for some constant C(R) when R is small. Moreover,

d∑j=1

ζ1(bj) ≤ −dm0 + C1R


for some constant C1. So,

WΩ(b, H0) ≤ −2πH0dm0 + C(R) + 2πH0C1R+1

2H2

0C(Ω) .

If, however, at least one bj satisfies bj ∈ Ωδ0 , then

WΩ(b, H0) ≥ −M(δ0)− 2πH0C2δ0 − 2πH0(d− 1)m0 +1

2H2

0C(Ω)

≥ −2πH0dm0 + C(R) + 2πH0C1R+1

2H2

0C(Ω)

for some constant C2 if R, δ0 are small enough and H0 is large enough. Thus,WΩ(b, H0) has a local minimum inside Ωd whenever H0 > H0(Ω).

4.2. The existence of vortex solutions. Using the renormalized energy, wenow study the solutions of the G–L equations (3)–(6).

Theorem 4.2. If H0 ≥ H0(Ω) and WΩ(b, H0) has a nondegenerate local mini-mum for some b ∈ Ωd, then, for small ε, there are solutions (ψε,Aε) to the full steadystate G–L equations with the gauge choice Aε = ∇⊥ζε in Ω, Aε · n = 0 on ∂Ω suchthat

ψε(x) → ψ∗(x) in C1,αloc

(Ω/b1,b2, . . . ,bd

),

ζε(x) → ζ∗(x) in H2(Ω),

where

−∆2ζ∗ + ∆ζ∗ = 2πd∑

j=1

δ(bj) in Ω

with

ζ∗ = 0 on ∂Ω,

∆ζ∗ = H0 on ∂Ω,

and

ψ∗(x) =d∏

j=1

x− bj|x− bj |e

ih∗(x) in Ω.

Moreover, if we write ψ∗(x) = eiθb(x)+ih∗(x) = eiφb , then φb is a multivalued harmonicfunction on Ω \ b1,b2, . . . ,bd with ∂φb

∂n = 0 on ∂Ω.To complete the proof, we use the approach in [21]. We consider the solution of

the following system:

∂ψ

∂t− (∇− iA)2ψ − 1

ε2ψ(1− |ψ|2) = 0 in Ω,(48)

∂A

∂t−∆A +

i

2(ψ∗∇ψ − ψ∇ψ∗) + |ψ|2A = 0 in Ω,(49)


and

curlA = H0 on ∂Ω,(50)

(∇− iA)ψ · n = 0 on ∂Ω,(51)

A · n = 0 on ∂Ω(52)

with properly defined initial conditions.Let

Eε(ψ,A) =1

4ε2(1− |ψ|2)2 +

1

2|(∇− iA)ψ|2 +

1

2|curlA−H0|2

and

Eε(ψ,A) = Eε(ψ,A) +1

2|divA|2 .

It is easy to see that (see [11], for example)∫Ω

Eε(ψ(t),A(t))dx ≤∫

Ω

Eε(ψ(0),A(0))dx .

Assuming that the initial condition satisfies

divA(0) = 0,

this in turn implies ∫Ω

Eε(ψ(t),A(t))dx ≤∫

Ω

Eε(ψ(0),A(0))dx.

Let ρ0 be small enough such that

δ0 = mindist(∂Ω, ∂B2ρ0(bj)), |bj − bi|, i 6= j, i, j = 1, 2, . . . , d > 0

and

minx∈∂Bρ0 (b)

W (x) ≥ C(ρ0) +W (b) .

Let us construct initial conditions. Let A(0) = ∇⊥ζb in Ω with ζb defined as

before. For small ρ, let Bρ =⋃dj=1Bρ(bj), and let ψ(0) = eiφb in Ω \Bρ and extend

ψ(0) inside each ball Bρ(bj) by the minimizer of

Iρ(ψ,A) = min

∫Bρ(bj)

(1

2|∇ψ − iAψ|2 +

1

4ε2(1− |ψ|2)2

+1

2|curlA−H0|2

)dx | divA = 0 in Ω, A · n = 0 on ∂Ω,

|ψ| |∂Bρ(bj)= 1 , (iψ, (∇− iA)ψ · τ) = g on ∂Bρ(bj),

and deg(ψ, ∂Bρ(bj)) = 1 , j = 1, 2, . . . , d

,


where τ is the unit tangent vector and g = ∇(θb − ζb) · τ on ∂Bρ(bj).Using the argument in [5] and the gauge invariance, one may show that

Iρ(ψ,A) = πd log(ρε

)+ γd+ o(1) +O(ρ).

Thus, we may assume

Eε(ψ(0),A(0)) ≤ dπ log1

ε+WΩ(b, H0) + γd+ o(1) .

Lemma 4.3. The solution of (48)–(52) has the property, for all sufficiently smallε and for all t ≥ 0, that the set x ∈ Ω, |ψε(x)| ≤ 1

2 is contained in a union of disjointdiscs Bj , j = 1, 2 . . . , d, where

(i) Bj = Bεα(xεj), xεj ∈ Bρ0(bj), for some small α.

(ii) εα∫∂Bj

Eε(ψε,Aε)dx ≤ C(K0).

Thus, dj = deg(∂Bj , ψ) is well defined and dj = 1, j = 1, 2, . . . , d.Proof. Let us define

P = t ≥ 0 | (ψε,Aε) s.t. (i)− (ii) .Clearly, P is a closed set and by choosing the initial conditions properly, we have0 ∈ P . We now show that P is open; thus P = [0,∞). First, we have the followingclaim.

Claim. For any t ∈ P , there exists a ρ∗ ∈ (0, ρ0), independent of sufficiently smallε, such that mindist(xε, ∂Bρ0

(b)), ≥ ρ∗.Assume that the claim is true. Then for any t′ > t but sufficiently close to t,

deg(∂Bρ∗(xεj), ψε(·, t′)) = 1.

We may then follow the ideas in [21] and [5] to construct, for ψε(·, t′), a new disc Bj

which satisfies (i)–(ii). Again, the center of the disc can be shown to satisfy the aboveclaim. Thus, the lemma is proved.

Now we verify the claim. Suppose the claim is not true, there exists a sequenceεn 0, tn t∗ ∈ P , such that xεni → bi ∈ Bρ0(bi) for i = 1, 2 . . . , d, and bj ∈∂Bρ0

(bj) for some j.We will follow constructions similar to those in [21] and [5] to show that one may

replace (ψε,Aε) by (ψε, Aε) to satisfy

E(ψε, Aε) ≥ πd log1

ε+WΩ(b, H0) + γd+ o(1).

First, let us do gauge transformation such that Aε = ∇⊥ζε with ζε = 0 on ∂Ω. Letus consider

min

∫Ω\⋃d

j=1Bεα (xε

j)

Eε(ψ, A)dx | A = ∇⊥ζ in Ω , deg(ψ, ∂Bεα(xεj)) = 1,

|ψ||∂Bεα (xεj) = |ψε||∂Bεα (xε

j) , and (iψ, (∇− iA)ψ · τ) = gε on ∂Bεα(xεj)

,

where gε = (iψε, (∇ − iAε)ψε · τ). Note that the boundary conditions are the samegauge invariant boundary conditions considered in [5] and α satisfies property (ii)listed above.


Let (ψε, Aε) denote a minimizer. Let ψε = ρεeiφε and ψε = ρεe

iφε on the boundary

of⋃dj=1Bεα(xεj). To extend it inside the εα balls, we define a gauge transformation

by

∆2χ = 0 ind⋃

j=1

Bεα(xεj)


κχ = φε − φε and∂χ

∂n= 0 on

d⋃j=1

∂Bεα(xεj) .

Note that χ is well defined on⋃dj=1 ∂Bεα(xεj). Thus, inside

⋃dj=1Bεα(xεj), we define

ψε = ψεeiκχ and Aε = Aε +∇χ .

By the gauge invariance and the choice of the small α, we have the energy lowerbound inside the εα balls:∫

⋃d

j=1Bεα (xε

j)

Eε(ψε, Aε)dx =

∫⋃d

j=1Bεα (xε

j)

Eε(ψε,Aε)dx

≥ πd log

(εα

ε

)− C

for small enough ε. So, we have the energy upper bound outside:∫Ω\⋃d

j=1Bεα (xε

j)

Eε(ψε, Aε)dx ≤ πdα log

(1

ε

)+ C .

Using arguments similar to those in [21] and [5], we have

|ψε| ≥ 1

2in Ω \

d⋃j=1

Bεα(xεj).

Moreover, we may modify the arguments of [21] and [5] to show that we have strongconvergence of

ψε → eiφb

and

Aε → ∇⊥ζboutside any small neighborhood of bj , j = 1, 2, . . . , d. Indeed, using the fact thatH = curl Aε satisfies the equation

div

(1

|ψε|2∇H

)= H in Ω \

d⋃j=1

Bεα(xεj) ,

we note that ρ = |ψε| > 1/2 in Ω \⋃dj=1Bεα(xεj) (in fact, ρ is arbitrarily close to 1 if

we allow ε small). Therefore, we obtain from elliptic estimates that the W 1,p norm


and the Cγ norm of H are bounded locally for some p > 2 and γ > 0. On the otherhand, from Lemma 4.1 in [21] and arguments in [21] and [5], one deduces the localstrong convergence of ψε. Then the assertion on the convergence of Aε follows. Weomit the details.

Next, for small δ, by strong convergence,∫Ω\⋃d

j=1Bδ(xεj)

Eε(ψε, Aε)dx

≥ πd log

(1

δ

)+WΩ(b, H0) + o(1) +O(δ).(53)

We can also get deg(ψε, ∂Bδ(bj)) = 1 for any j.Let us write ψε = |ψε|eiφε = ρeiθ+ih, where θ is the angle function inside each

Bδ(xεj) so that

eiθ(x) =x− bj|x− bj |

in Bδ(xεj) ∀ j.

We have, by gauge invariance, the energy upper bound outside the εα balls:∫⋃d

j=1Bδ(xεj)\Bεα (xε

j)

(|∇ρ|2 + ρ2|∇θ −∇⊥ζ|2 + |∆ζ|2) dx ≤ C log

(1

ε

)+K,(54)

where ∇⊥ζ = Aε is the magnetic potential after a gauge transformation.Thus, we have∫

⋃d


j)

(1− ρ2)|∇θ −∇⊥ζ|2dx

≤∫⋃d


j)

[2(1− ρ2)|∇⊥ζ|2 + 2(1− ρ2)|∇θ|2] dx

≤∫⋃d


j)

(1

ε(1− ρ2)2dx + ε|∇⊥ζ|4

)dx + o(1)

≤ Cε log

(1

ε

)+ ε

(∫⋃d


j)

|∆ζ|2dx)2

+ o(1)

≤ Cε log

(1

ε

)+ Cε

(log

(1

ε

))2

+ o(1)

≤ o(1),

where C is some generic constant.This implies in particular that

∫⋃d


j)

E(ψε, Aε)dx

≥∫⋃d


j)

|∇θ −∇⊥ζ|2dx + o(1)

≥ πd log

(δ

εα

)+ o(1).


Therefore, by arguments in [5] and the convergence of (ψε, Aε), we have

d∑j=1

∫Bδ(xεj)

Eε(ψε, Aε) dx ≥ πd log

(δ

ε

)+ γd+ o(1) +O(δ).(55)

Thus, for small ε and small δ,

∫Ω

Eε(ψε, Aε) dx ≥ πd log

(1

ε

)+WΩ(b, H0) + γd+ o(1) .

This leads to a contradiction of the energy upper bound obtained from the energydissipation from the energy of the initial condition since b is the nondegenerate localminimum of W (b, H0). Hence, the claim is true.

To complete the proof of the theorem, let us use the uniform bound on the solution(ψε,Aε) as t → ∞. We get a subsequence tn such that (ψε(tn),Aε(tn)) → (ψε,Aε)as t → ∞. One may then easily check that (ψε,Aε) is the critical point of the G–Lfunctional. This proves the theorem.

Remark.

(1) One may prove, using arguments given in [5], that as ε→ 0, the zeros of theψε go to the local minimum of the renormalized energy.

(2) By a more careful analysis, one may replace the assumption that the renor-malized energy has a nondegenerate local minimum by simply the existence of a localminimum.

(3) One can also prove, via [19], the existence of general critical points (saddlepoints) of the G–L functional under similar conditions.

(4) According to the nondimensionalization used here, our theorem describes thephenomenon that the G–L system has a vortex solution for an applied field of strengthon the order of 1

κ . Recall that the standard estimates for the lower critical field arelog(κ)κ . That is, we can prove the existence of a stable vortex state below Hc1.

4.3. The weak hysteresis near the lower critical field Hc for type-IIsuperconductors. We have just shown that when we decrease the applied magneticfield, there may be stable vortex states (even local energy minimizing states) evenwhen the field strength is below the lower critical field Hc1(≈ logκ/κ), say, c0/κ forsome constant c0. On the other hand, it is rather straightforward to check from thedefinition of WΩ(b, H0) that there is no local minimum of the function WΩ(b, H0)when H0 is sufficiently small. That is, we have shown the existence of a subcoolingfield 0 < Hsc < Hc1.

Let us consider another case where we gradually increase the applied field. Westart, say, with a perfect superconducting state (or the Meissner state). It is quitepossible that the Meissner state exists even when the applied field is much larger thanthe lower critical field Hc1. Indeed, if one looks for a solution to the full steady stateG–L equations (ψ,A) with |ψ| 6= 0, then one may write ψ = feiχ for some f 6= 0.

Using gauge invariance, we define Q = A− 1κ∇χ, then (f,Q) remains a solution

of the steady state G–L equations which can now be written in the following form:

1κ2 ∆f = f3 − f + f |Q|2−curl2Q = f2Q

in Ω(56)


H

Meissner

Vortex

sc0H

Hshc1H

Fig. 8. The weak hysteresis diagram for type-II superconductors near Hc1.


∂f∂n = 0Q · n = 0H = curlQ = H0

on ∂Ω.(57)

From the equation for Q, we obtain

− ∂H∂x2

= f2Q1

∂H∂x1

= f2Q2

in Ω,

or, equivalently,

div

(∇Hf2

)= H in Ω.

As in [2], we let κ→∞ to obtain

f = 1− |Q|2.

So, |∇H|2 = f4(1− f2). Let f = ρ(|∇H|). Then the equation for H becomes

div (ρ(∇H)∇H) = H in Ω,H = H0 on ∂Ω .

(58)

For H0 > 0 small enough (but obviously larger than limκ→∞ log κκ = 0), one can

show the existence of a solution to the above equation. Moreover, such a solution is alinearly stable wholly superconducting state (Meissner state) for 0 < H0 ≤ H∗

0 . Thenew field strength H∗

0 is called the superheating field; see [2] for details.

Combining this latter result with ours given in section 4.2, a weak-hysteresisdiagram of type-II superconductors near the lower critical field Hc1 is completed (seeFigure 8).


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