Conformal reduction of boundary problems for harmonic …grudsky/Talks/Talk_14.pdf · Spectral...

Conformal reduction of boundary problemsfor harmonic functions in a plane domainwith strong singularities on the boundary

Sergei Grudsky

CINVESTAV, Mexico City, Mexico

International Workshop”Analisis, Operator Theory, and Mathematical Physics”,

Ixtapa, Mexico,23-27 January, 2012.

Sergei Grudsky (CINVESTAV, Mexico) Dirichlet problem strong singularities Mexico, January, 2012 1 / 55

This report is based on joint work withNikolai Tarkhanov.

We consider the Dirichlet problem for harmonic functions in a boundedplane domain with nonsmooth boundary. The boundary curve belongs toone of the following three classes: sectorial curves, logarithmic spirals andspirals of power type. To study the problem we apply a familiar method ofVekua-Muskhelishvili which consists in using a conformal mapping of theunit disk onto the domain to pull back the problem to a boundary problemfor harmonic functions in the disk. This latter reduces in turn to a Toeplitzoperator equation on the unit circle with symbol bearing discontinuities ofsecond kind. We develop a constructive invertibility theory for Toeplitzoperators and thus derive solvability conditions as well as explicit formulasfor solutions.

Elliptic partial differential equations are known to appear in manyapplied areas of mathematical physics. As but a few we mention

mechanics of solid medium,diffraction theory,hydrodynamics,gravity theory,quantum field theory,and many others.

In this report we focus on boundary value problems for the Laplaceequation in plane domains bounded by nonsmooth curves C. We areprimarily interested in domains whose boundaries have a finite number ofsingular points of oscillating type.

z(r) = z0 + r exp(ıϕ(r)) for r ∈ (0, r0],

wherer is the distance of z and z0,ϕ(r) is a real-valued function which tends to infinity as r → 0or is bounded while its derivative is unbounded at r = 0.

Furthermore, ϕ(r) and ϕ′(r) are allowed to tend to infinity fast enoughand our study encompasses domains with nonrectifiable boundary as well.

There is a huge literature devoted to boundary value problems forelliptic equations in domains with nonsmooth boundary.

V.A. Kondratiev.Boundary value problems for elliptic equations in domains with conicalpoints. Trudy Mosk.Mat.Obshch, 16 (1967), 209–292 pp.

V.A. Kozlov, V.G. Maz’ya, J. Rossman.Spectral Problems Associated with Corner Singularities of Solutions toElliptic Equations. AMS, Providence, R.I, 2000.

V. Maz’ya, S. Nazarov, B. Plamenevskij.Asymptotic Theory of Elliptic Boundary Value Problems in SingularlyPerturbed Domains. Vol. 1, 2, Birkhauser, Basel, 2000.V. Rabinovich.Potential type operators on curves with vorticity points. Zeitschrift furAnalysis und ihre Anwendungen, 18 (1999), No. 4, 1065-108 pp.

V. Rabinovich, B.W. Schulze and N. Tarkhanov..Boundary value problems in oscillating cuspidal wedges, RockyMountain J. Math, 34, (Fall 2004), No.3, 73 pp.

In most papers one treats piecewise smooth curves with corner points orcusps.

There have been essentially fewer works dealing with more complicatedcurves C. They mostly focus on qualitative properties, such as

existence,uniqueness andstability of solutions with respect to small perturbations.

The present report deals not only with qualitative investigations ofboundary value problems in domains whose boundaries strongly oscillate atsingular points but also with constructive solution of such problems.

We restrict ourselves to the Dirichlet problem for the Laplace equation

∆u := (∂/∂x)2u + (∂/∂y)2u = 0 (1)

in a simply connected domain D with boundary C in the plane of variables(x , y) ∈ R2. The boundary data are

u = u0 (2)

Our approach to the study of elliptic problems in domains withnonsmooth boundary goes back at least as far as

N. Muskhelishvili and I. Vekua.It consists in reducing the problem in D to a singular integral equation onthe unit circle by means of a conformal mapping of the unit disk onto D.The coefficients of the singular integral equation obtained in this way failin general to be continuous, for they are intimately connected with thederivative of boundary values of the conformal mapping. This method wassuccessfully used for solving problems in domains with piecewise smoothboundary, where the singular points are corner points or cusps. In this casethe coefficients of the mentioned singular integral equation havediscontinuities of the first kind. Since the theory of such equations is wellelaborated, one has succeeded to construct a sufficiently complete theoryof boundary value problems for a number of elliptic equations in domainswith piecewise smooth boundary.

Note that by now the theory of singular integral equations (or, in otherterms, the theory of Toeplitz operators) with oscillating coefficients is wellelaborated, too. This fact gives us some evidence to a reasonablesolvability theory for a number of problems in plane domains withoscillating boundary.

The persent report deals with Laplace problems for the Laplace equationfor three classes of boundary curves C, namely sectorial curves, logarithmicand power spirals.

Let D be a simply connected bounded domain in the plane of realvariables (x , y). The boundary of D is a closed Jordan curve which wedenote by C. Consider the Dirichlet problem (1), (2) in D with data u0 onC. As usual we introduce a complex structure in R2 by z = x + ıy andpick a conformal mapping z = c(ζ) of the unit disk D = {ζ ∈ C : |ζ| < 1}onto the domain D, cf. Riemann mapping theorem. Throught the report

c′(0) > 0. (3)

Then problem (1), (2) can be reformulated as

1|c′(ζ)|2

∆U = 0 for |ζ| < 1,

U = U0 for |ζ| = 1,(4)

where U(ζ) := u(c(ζ)) and U0(ζ) := u0(c(ζ)).

For 1 ≤ p <∞, we denote by Hp(D) the Hardy space on the unit disk.By the conformal mapping z = c(ζ) the space is transported to theso-called Hardy-Smirnov space E p(D) of functions on D. A holomorphicfunction f on D is said to belong to E p(D) if

supr∈(0,1)

∫Cr|f (z)|p |dz | <∞,

where Cr is the push-forward of the circle |ζ| = r by z = c(ζ). It is easy tosee that f ∈ E p(D) if and only if

p√c′(ζ) f (c(ζ)) ∈ Hp(D). (5)

We will also study boundary value problems in Hardy-Smirnov spaceswith weights E p(D,w). Set

w(ζ) =n∏

(1− ζ

)−µk(6)

for ζ ∈ D, where {ζ1, . . . , ζn} are pairwise different points on the unitcircle. Here, µ1, . . . , µn are real numbers in the interval (−1/q, 1/p), pand q being conjugate exponents, i.e. 1/p + 1/q = 1. A holomorphicfunction f in D is said to lie in E p(D,w) if

supr∈(0,1)

∫Cr|f (z)|p |w(c−1(z))|p |dz | <∞.

It is well known that for each harmonic function u(x , y) in D there is ananalytic function f (z) in D whose real part is u. We therefore look for asolution u of problem (1) and (2), which has the form u = <f withf ∈ E p(D,w). The boundary condition u = u0 is understood in the senseof nontangential limit values of u almost everywhere on C.

Definition

Given any Dirichlet data u0 on C of class Lp(C,w) in the sense that∫T|u0(c(ζ))|p |w(ζ)|p |c′(ζ)| |dζ| <∞,

we shall say that problem (1), (2) possesses a solution in <E p(D,w) ifthere is a harmonic function u in D, such that u = <f for somef ∈ E p(D,w) and u = u0 on C.

f (c(ζ)) = h+(ζ)

w(ζ) p√c′(ζ)

for ζ ∈ D, where h+ is an analytic function of Hardy class Hp(D).

<( h+(ζ)

w(ζ) p√c′(ζ)

)= U0(ζ)

for ζ ∈ T,andU0(ζ) = u0(c (ζ))

This problem can in turn be reformulated as

12( h+(ζ)

w(ζ) p√c′(ζ)

+h−(ζ)

w(ζ) p√c′(ζ)

)= U0(ζ)

for ζ ∈ T, whereh−(ζ) = h+(ζ) ∈ Hp(D)

Finally, we transform the Dirichlet problem to

a(ζ) h+(ζ) + h−(ζ) = g(ζ) (7)

for ζ ∈ T, where

a(ζ) = w(ζ)

w(ζ)p

√c′(ζ)

c′(ζ)=

w(ζ)exp

(− ı2

p arg c′(ζ))

and g(ζ) = 2U0(ζ)w(ζ) p√c′(ζ).

arg c′(ζ) = α(c(ζ))− arg ζ − π

for ζ ∈ T, where α(c(ζ)) is the angle at which the tangent of C at thepoint z = c(ζ) intersects the real axis. Note that g ∈ Lp(T).

Remind thath+(ζ) ∈ Hp(D), h−(ζ) ∈ Hp(D)

Let now(STg)(ζ) := 1

g(ζ ′)ζ ′ − ζ

dζ ′, ζ ∈ T,

stand for the singular Cauchy integral. If 1 < p <∞, then ST is abounded operator in Lp(T), and the operators

P±T :=12 (I ± ST)

prove to be continuous projections in Lp(T) called analytic projections.

P±T Lp(T) = Hp±,

whence P±T Hp± = Hp± and P±T Hp∓ = 0.

Hp+= Hp(D), Hp− ⊕ {c} = Hp(D)

On applying P+T to both sides of equality (7) and taking into account

that (P+T h−)(ζ) = h−(∞) and h−(∞) = h+(0) = h+(0), we get

(T (a)h+)(ζ) + h+(0) = g+(ζ) (8)

for ζ ∈ T, whereT (a) := P+

T aP+T

is a Toeplitz operator with symbol a on Lp(T) and g+(ζ) = (P+T g) (ζ).

Theorem

If u = <f with f ∈ E p(D,w) is a solution of the Dirichlet problemin D, then h+(ζ) = w(ζ) p

√c′(ζ) f (c(ζ)) is a solution of system (8)

If h+ ∈ Hp+ is a solution of (8), thenu(c(ζ)) = < (h+(ζ)/w(ζ) p

√c′(ζ)) is a solution of the Dirichlet

problem in D.

Corollary (Exat formula)

If the operator T (a) is invertible on the space Hp+ and(T (a)−11

)(0) = 1, (9)

then the Dirichlet problem in D has a unique solution of the form

u(z) = <( 1

w(ζ) p√c′(ζ)

((T (a)−1g+)(ζ)−1

2(T (a)−1g+)(0) (T (a)−11)(ζ))),

with z = c(ζ), where g+ = P+T

(2u0(c(ζ))w(ζ) p

√c′(ζ)

Remark

Condition (9) is actually fulfilled in all cases to be treated in this work.

Proof of Corollary Exat formulaApplying the operator T (a)−1 to (8) yields

h+(ζ) + T (a)−1h+(0) =(

T (a)−1g+)(ζ)

for all ζ ∈ T. Since both sides of the equality extend to holomorphicfunctions in the disk, we can set ζ = 0, obtaining

2 h+(0) =(

T (a)−1g+)(0)

by (9). Hence it follows that the solution of (8) is unique and it has theform

h+(ζ) =(

T (a)−1g+)(ζ)− 1

T (a)−1g+)(0)(T (a)−11)(ζ),

as desired.

Factorization of symbols

A bounded linear operator A on a Hilbert space H is said to be normallysolvable if its range im A is closed. A normally solvable operator is calledFredholm if its kernel and cokernel are finite dimensional. In this case theindex of A is introduced as

ind A := α(A)− β(A),

where α(A) = dim ker A and β(A) = dim coker A.

The symbol a(ζ) of a Toeplitz operator T (a) is said to admit ap -factorisation, with 1 < p <∞, if it can be represented in the form

a(ζ) = a+(ζ)ζκa−(ζ), (10)

where κ is an integer number,

a+ ∈ Hq+, a− ∈ Hp− ⊕ {c},1/a+ ∈ Hp+, 1/a− ∈ Hq− ⊕ {c}, (11)

p and q are conjugate exponents (i.e. 1/p + 1/q = 1), and

(1/a+)ST(1/a−) is a bounded operator on Lp(T). (12)

The functions a+ and a− in (10) are determined uniquely up to aconstant factor. On putting the additional condition a−(∞) = 1 onedetermines the factorisation uniquely.

Theorem (Simonenko)

An operator T (a) is Fredholm in the space Hp+ if and only if the symbola(ζ) admits a p -factorisation. If T (a) is Fredholm, then ind T (a) = −κ.

Corollary (Invertibility)

An operator T (a) is invertible on Hp+ if and only if the symbol a(ζ)admits a p -factorisation with κ = 0. In this case

(T (a))−1 = (1/a+)P+T (1/a−).

Given a nonvanishing function a ∈ C(T), we denote by inda(T)(0) thewinding number of the curve a(T) about the origin, or the index of theorigin with respect to a(T).

Theorem

Suppose a ∈ C(T). Then the operator T (a) is Fredholm on the spaceHp+ if and only if a(ζ) 6= 0 for all ζ ∈ T. Under this condition the indexof T (a) is given by

ind T (a) = −inda(T)(0).

We now introduce the concept of sectoriality which is of crucialimportance in this report.

Definition

A function a ∈ L∞(T) is called p -sectorial if ess inf |a(ζ)| > 0 and thereis a real number ϕ0 such that

supζ∈T| arg (exp(ıϕ0)a(ζ)) | <

max{p, q} (13)

for all ζ ∈ T.

DefinitionA function a ∈ L∞(T) is said to be locally p -sectorial if ess inf |a(ζ)| > 0

and for any ζ0 ∈ T there is an open arc containing ζ0, such that (13) issatisfied for all ζ in the arc with some ϕ0 ∈ R depending on ζ0.

Theorem

If a(ζ) is a p -sectorial symbol, then the operator T (a) is invertiblein the space Hp+.

If a(ζ) is a locally p -sectorial symbol, then T (a) is a Fredholmoperator in Hp+.

Suppose h ∈ GH∞+, that is h ∈ H∞+ and 1/h ∈ H∞+. Then theoperator T (h) is invertible in Hp+, (T (h))−1 = T (1/h). Analogously, ifh ∈ GH∞− then the operator T (h) is invertible in Hp+ and(T (h))−1 = T (1/h).

Theorem

Let a(ζ) = h(ζ)a0(ζ), where h ∈ GH∞± and a0 ∈ L∞(T). Thenthe operator T (a) is Fredholm in the space Hp+ if and only if theoperator T (a0) is Fredholm, in which case ind T (a) = ind T (a0).

Let a(ζ) = c(ζ)a0(ζ), where c ∈ C(T) and a0 ∈ L∞(T). ThenT (a) is Fredholm in Hp+ if and only if c(ζ) vanishes at no point of Tand T (a0) is Fredholm, in which case

ind T (a) = ind T (a0)− indc(T)(0).

We give a brief summary on Toeplitz operators with symbols havingdiscontinuities of the first kind. Let PC stand for the space of all piecewisecontinuous functions on T which have at most finitely many jumps.Suppose a(ζ) ∈ PC and ζ1, . . . , ζn are the points of discontinuity of a.Given any f ∈ C and ζ0 ∈ T, we introduce a function af,ζ0 ∈ PC by

af,ζ0(ζ) := exp(ıf arg

(− ζ

))for ζ ∈ T, where arg z ∈ (−π, π]. It is easily seen that af,ζ0 has at mostone point of discontinuity at ζ = ζ0, with jump

af,ζ0(eı0−ζ0) = exp (ıfπ) ,af,ζ0(eı0+ζ0) = exp (−ıfπ) .

If a(eı0±ζk) 6= 0 for all k = 1, . . . , n, then there are complex numbers fkwith the property that

a(eı0−ζk)

a(eı0+ζk)= exp (2ıfkπ) ,

and soa(ζ) = c(ζ)

n∏k=1

afk ,ζk (ζ), (14)

where c ∈ C(T).

Theorem (PC)

Let a(ζ) ∈ PC. Then the operator T (a) is Fredholm in Hp+ if and onlyif

a(eı0±ζ) 6= 0 for all ζ ∈ T;

there are integer numbers κk such that κk −1q < <fk < κk +

Under conditions this Theorem, the index of the operator T (a) in Hp+

is actually given by

ind T (a) = −(indc(T)(0) +

n∑k=1

κk), (15)

Sectorial curves

In this section we consider a simply connected domain D b R2 whoseboundary C is smooth away from a finite number of points. By this ismeant that C is a Jordan curve of the form

C =n⋃

k=1Ck ,

where Ck = [zk−1, zk ] is an arc with initial point zk−1 and endpoint zkwhich are located after each other in positive direction on C, and zn = z0.Moreover, (zk−1, zk) is smooth for all k.

Definition

The curve C is called p -sectorial if, for each k = 1, . . . , n, there is aneighbourhood (z−k , z

+k ) of zk on C and a real number ϕk , such that

supz∈(z−

k ,z+k )\{zk}

|α(z)− ϕk | <{

π2 , if p ≥ 2,π2 (p − 1), if 1 < p < 2, (16)

where α(z) is the angle at which the tangent of C at the point z intersectsthe real axis.

Example (1)If zk is a corner point of C, then the angle at which the tangent of C at

z intersects the real axis has jump jk < π when z passes through zk . Itfollows that (16) is fulfilled at zk with a suitable ϕk , if p ≥ 2, and isfulfilled if moreover jk < (p − 1)π, if 1 < p < 2.

If zk is a cuspidal point of C, then the angle has jump jk = π when zpasses through zk . Hence, condition (16) is violated, i.e. cuspidal pointsare prohibited for sectorial curves.

Example (2)z(eiθ) = θ + i δ|θ|α+1 sin |θ|−α, |θ| < ε, α > 0, δ ∈ R.

If p ≥ 2, this curve is p− sectorial for any α.If 1 < p < 2 and |δα| < tan π

2 (p − 1), then this curve is p− sectorial.

Example (3)z(eiθ) = θ(1 + ı sin |θ|−α), |θ| < ε, α > 0.

This curve is not p− sectorial.

Dirichlet data on sectorial curves

Theorem

Suppose C is p -sectorial for 1 < p <∞ and w(ζ) ≡ 1. Then theToeplitz operator (8) corresponding to this curve is invertible.

We shall have to proof that the Toeplitz operator with symbol

a(ζ) = exp(− ı2

p arg c′(ζ))

is invertible. Recall that arg c′(ζ) = α(c(ζ))− arg ζ − π

2 for ζ ∈ T.The idea of the proof is to represent the symbol in the form

a(ζ) = c(ζ)a0(ζ),

where a0 is p -sectorial and c ∈ C(T) is such that indc(T)(0) = 0.Sergei Grudsky (CINVESTAV, Mexico) Dirichlet problem strong singularities Mexico, January, 2012 35 / 55

Theorem (p -sectorial)

Let C be a p -sectorial curve. Then the Dirichlet problem has a uniquesolution u = <f with f ∈ E p(D) and this solution has a following form inD

u(z) = <∫T

12πı

ζ + c−1(z)ζ − c−1(z) u0(c(ζ))

for z ∈ D.

Begin proof of Theorem p -sectorialAccording to previous Theorem, a(ζ) admits of p -factorisation of the

kind a(ζ) = a+(ζ)a−(ζ). On other hand

a(ζ) =(c′(ζ)c′(ζ)

for ζ ∈ T. In the case of p -sectorial curves the angle α(z) is bounded, andso the curve C is rectifiable. By a well-known result the derivative c′(ζ)

belongs to H1+, whence p√c′(ζ) ∈ Hp+ and p

√c′(ζ) ∈ Hp− ⊕ {c}.

Comparing this with a(ζ) = a+(ζ)a−(ζ) we get

p√c′(ζ)a+(ζ) = p

√c′(ζ)(1/a−(ζ)).

End proof of Theorem p -sectorialThe left-hand side of this equality belongs to H1+ and the right-hand

side to H1− ⊕ {c}. Hence it follows that

p√c′(ζ)a+(ζ) = c,

p√c′(ζ)(1/a−(ζ)) = c

where c is a complex constant. The factorisation a(ζ) = a+(ζ)a−(ζ) with

a+(ζ) = c(

1/ p√c′(ζ)

a−(ζ)) =1c

p√c′(ζ)

and(T (a))−1 = p

√c′(ζ)P+

T (1/ p√c′(ζ)),

which is due to Corollary Invertibility.

Dirichlet data on logarithmic spirals

Let Σa be a horizontal half-strip of the formΣa = {z ∈ C : <z > 0, =z ∈ (0, a)}, a > 0.

z = `ϕ(z) := exp(−eıϕz)

of the half-strip into the complex plane Cz .The mapping z = `ϕ(z) is conformal if and only if a < 2π cosϕ.For υ ∈ [0, a], set Rυ = {z ∈ C : <z ≥ 0, =z = υ}. Then the curveSυ,ϕ := `ϕ(Rυ) is a spiral.

(exp υ

)exp (ϑ cotϕ) , (17)

Ca,ϕ := S0,ϕ ∪ Sa,ϕ ∪ ba,ϕ,

where S0,ϕ and Sa,ϕ are given by (17) and the arc ba,ϕ byz = exp (−eıϕıυ) with υ ∈ [0, a].

The conformal mapping z = ca,ϕ(ζ) of D onto Da,ϕ given by

ca,ϕ(ζ) = exp(− eıϕ a

πln(η +

√η2 − 1

))(18)

with η = η(ζ), η(ζ) =1iζ + 1ζ − 1.

arg c′a,ϕ(ζ) =

ϕ− π

2 − arg ζ − a cosϕ− aπ

sinϕ ln |η +√η2−1|,

if ζ ∈ (1, ı),ϕ− arg ζ − a

πcosϕ arccos η,

if ζ ∈ [ı,−ı],ϕ+

2 − arg ζ − aπ

sinϕ ln(η +√η2−1),if ζ ∈ (−ı, 1).

We introduce the Toeplitz operator T (aa,ϕ) with symbol

aa,ϕ(ζ) = exp(− ı2

p arg c′a,ϕ(ζ)),

Theorem (Logarithmic spiral)

The operator T (aa,ϕ) in Hp+ with 1 < p <∞ is Fredholm if and only if

p 6= aπ

cosϕ.

If p > aπ

cosϕ, then T (aa,ϕ) is invertible in the space Hp+.

If p < aπ

cosϕ, then ind T (aa,ϕ) = 1.

Begin proof of Theorem Logarithmic spiralIntroduce the function

ha,ϕ(ζ) := exp(− ı2

p(ϕ− a

πsinϕ ln(η +

√η2 − 1)

which obviously belongs to GH∞+. Indeed,

|ha,ϕ(ζ)| =

exp(− 2

p a sinϕ), if ζ ∈ (1, ı),

1, if ζ ∈ (−ı, 1),

a0(ζ) =aa,ϕ(ζ)

ha,ϕ(ζ)

End proof of Theorem Logarithmic spiralHence a0(ζ) is a PC function with discontinuity points {1, ı,−ı}. One

verifies readily that

arg a0(eı0− 1) = 2p

32π, arg a0(eı0+ 1) = 2

(π2 + a cosϕ

arg a0(eı0− ı) = 2p (π + a cosϕ), arg a0(eı0+ ı) = 2

(π2 + a cosϕ

arg a0(eı0+(−ı)) = 2p

32π, arg a0(eı0+(−ı)) = 2

We thus conclude that a0(ζ) possesses a representation (14) withζ1 = 1, ζ2 = ı, ζ3 = −ı and

f1 =1p(

1− aπ

cosϕ), f2 = f3 =

Observe that−1

q < f2 = f3 <1p .

Theorem

If p > aπ

cosϕ, then the Dirichlet problem has a unique solution in<E p(Da,ϕ) given by

u(z) = <∫T

12πı

ζ + c−1a,ϕ(z)

ζ − c−1a,ϕ(z)

u0(ca,ϕ(ζ))dζζ

for z ∈ Da,ϕ.

If p < aπ

cosϕ, then the Dirichlet problem has infinitely manysolutions in <E p(Da,ϕ) given by

u(z) = <(

c1 + c−1

a,ϕ(z)1− c−1

a,ϕ(z)−

c−1a,ϕ(z)

1− c−1a,ϕ(z)

ζ − 1ζ − c−1

a,ϕ(z)u0(ca,ϕ(ζ))

)for z ∈ Da,ϕ, where c is an arbitrary real constant.

Remark

If p = (a/π) cosϕ then the operator T (aa,ϕ) is not Fredholm. One canshow that it has zero null space and dense range in this case. What is stilllacking is an explicit description of the range.

Dirichlet data on spirals of power type

Let a > 0 and γ > 0. Consider the domain Da,γ in the plane of complexvariable θ that is bounded by curves

S0,γ = {θ ∈ C : <θ ≥ 1, =θ = 0},Sa,γ = {θ ∈ C : θ = (<θ + ıa)γ , <θ ≥ 1},ba,γ = {θ ∈ C : θ = (1 + ı=θ)γ , =θ ∈ (0, a)}.

The boundary of Da,γ is thus the composite curve Ca,γ = S0,γ ∪Sa,γ ∪ ba,γ .Given any δ > 0, we define

z = hδ(θ) :=exp(ıθ)θδ

for θ ∈ Da,γ . This function maps Da,γ onto a domain Da,γ,δ.

Consider the curve hδ(S0,γ) in the z -plane. Introduce the polarcoordinates r = |z | and ϑ = arg z .{

r = (<θ)−δ,ϑ = <θ,

and so the equation of hδ(S0,γ) reduces to

r = ϑ−δ

with ϑ ≥ 1. In this way we obtain what will be referred to as the powerspiral.

Note that the curve hδ(S0,γ) is rectifiable if and only if δ > 1. Indeed,the integral

L(hδ(S0,γ)) =

∫ ∞1

√(r(ϑ))2 + (r ′(ϑ))2 dϑ

∫ ∞1

ϑ−δ

√1 +

ϑ2 dϑ

is finite if and only if δ > 1.Sergei Grudsky (CINVESTAV, Mexico) Dirichlet problem strong singularities Mexico, January, 2012 47 / 55

Theorem

Assume γ ∈ (0, 1/2] and δ > 0. If a > 0 is small enough, thenhδ : Da,γ → Da,γ,δ is a conformal mapping.

The conformal mapping z = ca,γ,δ(ζ) of D onto Da,γ,δ given by

ca,γ,δ(ζ) =exp ı

ln(η +

√η2 − 1

)γ( aπ

ln(η +

√η2 − 1

)γδ (20)

with η = η(ζ), 2(ζ) = 1iζ + 1ζ − 1.

Introduce the function z(ζ) =aπ

ln(η +

√η2 − 1

)+ 1 of ζ ∈ D.

c′a,γ,δ(ζ) =√

2ıγ aπ(1 + ıδ (z(ζ))−γ)

(z(ζ))γ−γδ−1√1 + ζ2 (1− ζ)

exp ı(z(ζ))γ . (21)

We consider the problem in space <Ep(Da,γ,δ,w), where w = (1− ζ)−µ

Theorem (Power spiral)

Let γ ∈ (0, 1/2], δ > 0 and let ca,γ,δ(ζ) be the conformal mapping of Donto Da,γ,δ given by (20). Then operator T (aa,γ,δ) corresponding toregion Da,γ,δ is Fredholm if and only if

µ 6= 0

.If −1

q < µ < 0, then the operator T (aa,γ,δ) is invertible.

If 0 < µ <1p , then operator T (aa,γ,δ) is Fredholm and

ind T (aa,γ,δ) = −1.

Begin proof of Theorem Power spiralConsider the Toeplitz operator T (aa,γ,δ) with symbol

aa,γ,δ(ζ) = exp(− ı2

p arg c′a,γ,δ(ζ))w(ζ)

= exp(− ı2

p arg c′a,γ,δ(ζ))(1− 1/ζ)−µ

(1− ζ)−µ

= exp(− ı2

p arg c′a,γ,δ(ζ))(−ζ)µ

Setha,γ,δ(ζ) = exp

(− ı2

p (z(ζ))γ)∈ GH∞+

a0(ζ) =aa,γ,δ(ζ)

ha,γ,δ(ζ)

End proof of Theorem Power spiral

limζ→eı0−1

arg a0(ζ) − limζ→eı0+1

arg a0(ζ) = 2π(1

p + µ),

limζ→eı0−ı

arg a0(ζ) − limζ→eı0+ı

arg a0(ζ) =π

limζ→eı0−(−ı)

arg a0(ζ) − limζ→eı0+(−ı)

arg a0(ζ) =π

ζ1 = 1, ζ2 = ı, ζ3 = −ı, and

f1 =1p + µ, f2 = f3 =

Thus according to Theorem PC operator T (aa,γ,δ) is Fredholm if and onlyif µ 6= 0.

Theorem (Solution)

Let γ ∈ (0, 1/2], δ > 0 and let z = ca,γ,δ(ζ) be the conformal mappingof D onto Da,γ,δ given by (20).

If −1q < µ < 0, then the Dirichlet problem has a unique solution in

<E p(Da,γ,δ,w) given by

u(z) = <∫T

12πı

ζ + c−1a,γ,δ(z)

ζ − c−1a,γ,δ(z)

u0(ca,γ,δ(ζ))dζζ

for z ∈ Da,γ,δ.

Theorem (Solution)

If 0 < µ <1p , then the Dirichlet problem has a solution in

<E p(Da,γ,δ,w) if and only if

∫ 2π

0u0(ca,γ,δ(eıϕ)) cot ϕ2 dϕ = 0.

Under this condition, the solution of the Dirichlet problem in<E p(Da,γ,δ,w) has the form

u(z) =<∫T

√√√√ (c−1a,γ,δ)

′(z)(c−1

a,γ,δ)′(z0)

− w(c−1a,γ,δ(z))

1−c−1a,γ,δ(z)

ζ−c−1a,γ,δ(z)

· u0(ca,γ,δ(ζ))

ζ − 1dζ

c−1a,γ,δ(z)

for z ∈ Da,γ,δ, where z0 = ca,γ,δ(0).

Remark

In the case µ = 0 the operator T (aa,γ,δ) is not Fredholm. So it is reasonto introducer the weight spaces.

Conformal reduction of boundary problems for harmonic …grudsky/Talks/Talk_14.pdf · Spectral...

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