Vortex pinning with bounded fields for the Ginzburg-Landau equation

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VORTEX PINNING WITH BOUNDED FIELDS FOR THE GINZBURG–LANDAU EQUATION

Nelly ANDRE^a, Patricia BAUMAN^b,1, Dan PHILLIPS^b,^∗^,2

aUniversité François Rabelais, département de mathématiques, Parc Grandmont, 37200 Tours, France bDepartment of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Received 31 January 2002

ABSTRACT. – We investigate vortex pinning in solutions to the Ginzburg–Landau equation.

The coefficient,a(x), in the Ginzburg–Landau free energy modeling non-uniform superconduc- tivity is nonnegative and is allowed to vanish at a finite number of points. For a sufficiently large applied magnetic field and for all sufficiently large values of the Ginzburg–Landau parameter κ=1/ε, we show that minimizers have nontrivial vortex structures. We also show the existence of local minimizers exhibiting arbitrary vortex patterns, pinned near the zeros ofa(x).

2003 Éditions scientifiques et médicales Elsevier SAS

RÉSUMÉ. – On étudie la localisation des vortex des solutions de l’équation de Ginzburg–

Landau. Dans l’énergie libre de Ginzburg–Landau, le coefficienta(x)modélise la supraconduc- tivité non uniforme. Ce coefficient est positif et s’annule en un nombre fini de points. On montre que, pour un champ magnétique assez grand et pour toutes les valeurs du paramètre de Ginzbug–

Landauκ=1/εassez grandes, les minimiseurs présentent des structures de vortex non triviales.

On montre aussi l’existence de minimiseurs locaux présentant une structure prescrite de vortex situés au voisinage des zéros dea(x).

2003 Éditions scientifiques et médicales Elsevier SAS

Introduction

In this paper we analyze several aspects of vortex pinning in superconductivity using the Ginzburg–Landau theory as our model. To describe these phenomena consider the energy

Jε(ψ, A)=

(∇ −iA)ψ²+ |∇ ×A−hee3|²+ 1 2ε²

a− |ψ|²²

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∗Corresponding author.

E-mail address: phillips@math.purdue.edu (D. Phillips).

1Partially supported by the National Science Foundation grant DMS-9971974.

2Partially supported by the National Science Foundation grant DMS-9971713.

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for ε >0. Here is a bounded simply connected domain in R²with a smooth (C^2,1) boundary and a:→R. The domain represents the cross-section of an infinite cylindrical body with e3as its generator. The body is subjected to an applied magnetic field,hee3wherehe0 is constant. The functionA:→R²is the magnetic potential and∇ ×A= ∇ ×(A¹, A²,0)is the induced magnetic field in the cylinder. The function ψ is complex-valued where |ψ|² =ψ^∗ψ represents the density of superconducting election pairs and

j= −i

2(ψ^∗∇ψ−ψ∇ψ^∗)− |ψ|²A (2) denotes the superconducting current density circulating in the cross-section . The parameter ε =1/κ is a positive number where κ is the Ginzburg–Landau parameter associated to the material. We analyze the smallε(largeκ) regime. It is here that vortex dominated current patterns are expected in stable equilibria for Jε. The prototypical picture of this phenomenon is that of a finite number of non-superconducting points in (at whichψ =0, called vortices), each of which is surrounded by a ring of the super currentj.

If the material is homogeneous, the function a in Jε is taken to be a constant, proportional toTc−T. HereT is the body’s temperature andTc is the material’s critical temperature. For T Tc (a0), it is easy to show that the only equilibria for Jε are completely non-superconducting and have ψ≡0, ∇ ×A≡hee3. ForT < Tc (a >0), superconducting minimizers exist if the applied field strength he is not too large. There are a number of mathematical investigations of the relationship between he and the nature of stable superconducting states for this case. In [11] Sandier and Serfaty showed that there exists a constantHc₁ proportional to|log(ε)|asε→0, such that ifheHc₁, then minimizers forJεare purely superconducting, satisfying|ψ|>0 in. In [12] they showed that forhe slightly greater thanHc₁ and such thatheε⁻², minimizers are in a mixed state having a vortex-like structure. It was shown by Giorgi and Phillips in [5]

that forheCε⁻²for some constantC, superconductivity is completely suppressed, in that all equilibria forJεhaveψ≡0.

Inhomogeneous superconducting materials can arise naturally due to material defects or the presence of grain boundaries. Inhomogeneities can be inserted intentionally, as well, by adding non-superconducting (normal) impurities to the material. (See [3]

and [4].) A consequence of having material inhomogeneities is that they tend to pin or stabilize supercurrent patterns. The classical Ginzburg–Landau theory can be modified to take normal inclusions into account. This is done by having the critical temperature, Tc, depend on position which is equivalent to havinga=a(x). (See [10].) It is possible thata(x)may vanish or change sign within the domain.

A mathematical study for the Ginzburg–Landau equations corresponding to the energy (1) with variablea(x)was done by Aftalion, Sandier, and Serfaty in [1] where the case

1

2 a(x)1 was considered. They proved among other things, that Hc₁ remains of order |log(ε)| asε→0. In this paper we consider the case where contains a finite number of point impurities,{x1, . . . , xn}, and thata(x)vanishes at these normal sites. In this instance, the strong pinning enables us to show that the transition threshold for he, denoted byHc1=Hc1(ε), separating the presence or absence of vortices, is of order 1 as ε→0. (See Corollary 4.4.) In addition, for eachheand allεsufficiently small, we show

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that there are local minimizers forJε with prescribed vortex structure about each of the xi corresponding to the homotopy classes in\ {x1, . . . , xn}. (See Theorem 4.6.) In this way we are able to pin supercurrent patterns near the zeros ofa(x). (See Theorem 4.6.) Another way of introducing inhomogeneities is by making holes (voids) in the body.

In [8,9,13]Jε was studied witha=1,he=0 but withmultiply connected by Jimbo and Morita, Jimbo and Zhai, and Rubinstein and Sternberg, respectively. In that setting, local minimizers with prescribed vortex structures associated to the homotopy classes ofwere shown to exist.

We require thata(x)satisfy the following.

Assume:a∈C¹(\{x1, . . . , xn})∩C^β() for someβ >0, √

a∈H¹(), a(x)0 for allxin, anda(x)=0 iffx∈ {x1, . . . , xn}wherex1, . . . , xnare distinct points in and n∈N. Moreover, assume that there are positive constants mi, Mi and αi so that mi|x−xi|^αⁱa(x)Mi|x−xi|^αⁱ in some neighborhoodUi ofxi for 1in.

DEFINITION. – Let ε >0 and let (ψε, Aε)∈H¹(;C)×H¹(;R²) ≡M. Then (ψε, Aε)is an equilibrium forJε if and only if(ψε, Aε)is a weak solution of the Euler–

Lagrange equations and natural boundary conditions for critical points of Jε in M, namely:

−(∇ −iAε)²ψε= 1 ε²

a− |ψε|²ψε in, (∇ −iAε)ψε·n=0 on∂,

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∇ × ∇ ×Aε= −i 2

ψ_ε^∗∇ψε−ψε∇ψ_ε^∗− |ψε|²Aε≡jε in,

∇ ×Aε=hee3 on∂.

(4) Forε=0 we set

J0(ψ, A)=

(∇ −iA)ψ²+ |∇ ×A−hee3|². (5)

Denote

H_a¹= ψ∈H¹(;C)such that|ψ| =√

aalmost everywhere. Note that H_a¹ is nonempty, since √

a ∈ H_a¹ by our assumptions on a. We prove in Section 1 (see Theorem 1.4) that each ψ ∈ H_a¹ can be written as ψ = √

ae^{iθ (x)}, where θ (x)=θ0(x)+ⁿi=1diθi(x), θ0 is a measurable function determined up to an additive constant, 2π k fork∈Z, satisfyinga|∇θ0|²<∞,D=(d1, . . . , dn)∈Zⁿ is uniquely determined, and θi(x) is the azimuthal angle about xi for 1i n(so that (cosθi(x),sinθi(x))=(x−xi)/|x−xi|for allx=xi inR²). Thusψcorresponds to a uniqueD∈Zⁿdescribing a homotopy class forψ in\{x1, . . . , xn}. We write

H_a¹=

D∈Zⁿ

H_a,D¹ .

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We note that H_a,D¹ is both open and closed in H_a¹ and that if {un} ⊂H_a,D¹ such that un% uinH¹thenu∈H_a,D¹ . (See Theorem 1.5.)

DEFINITION. – Let(ψ0, A0)∈H_a¹×H¹(;R²)≡M0. Then(ψ0, A0)is an equilib- rium forJ0if and only if(ψ0, A0)is a weak solution of the Euler–Lagrange equations and natural boundary conditions for critical points ofJ₀inM0, namely:

div

−i 2

ψ₀^∗∇ψ0−ψ0∇ψ₀^∗− |ψ0|²A0

=0 in,

−i 2

ψ₀^∗∇ψ0−ψ0∇ψ₀^∗− |ψ0|²A0

·^%n=0 on∂,

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and

∇ × ∇ ×A0=

−i 2

ψ₀^∗∇ψ0−ψ0∇ψ₀^∗− |ψ0|²A0

≡j0 in,

∇ ×A0=hee3 on∂.

(7) The functionalsJε, forε0, are gauge invariant. By this we mean that if(ψ, A)∈ M (M0) and if φ ∈ H²(), then the gauge transformation, (ψ, A) =Gφ(ψ, A) defined by

ψ≡ψe^iφ, A≡A+ ∇φ,

satisfies(ψ, A)∈M(M0),Jε(ψ, A)=Jε(ψ, A), and(ψ, A)is an equilibrium for Jε(J0)if (ψ, A)is one. In this paper we will fix a gauge by requiring (without loss of generality) thatAsatisfy

divA=0 in, A·n=0 on∂,

(8) since this can be accomplished by an appropriate gauge transformation. With this choice of gauge (the Coulomb gauge), A is determined from the value of ∇ ×A= (∂xA1−∂yA2)e3≡he3by first solving

)ξ=h in, ξ=0 on∂.

(9) From (8), (9), and the fact that is simply connected we have A= ∇^⊥ξ where (∂x, ∂y)^⊥≡(−∂y, ∂x). An important feature of the gauge choice (8) is that the boundary conditions in (3) and (6) can be replaced by

∇ψ· n=0 on∂

and, since∇ × ∇ ×A= −)A+ ∇(divA), the term∇ × ∇ ×Ain Eqs. (4) and (7) is equal to−)A.

We establish the following main results in this paper.

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THEOREM 1. – Fixhe0. For eachD∈Zⁿ,J0has an equilibrium (with our choice of gauge),(ψD, AD), inH_a,D¹ ×H¹(;R²). Moreover,(ψD, AD)is unique up to uniform rotations ofψD in,ψD→ψDe^icforc∈R. (See Theorem 3.2.)

We remark that(ψ, A)→(ψe^ic, A)is a gauge transformation inM(M0), and thus Jε(ψ, A)=Jε(ψe^ic, A)for allc∈Randε0.

THEOREM 2. – Fixhe0. Let(ψε_k, Aε_k)be an equilibrium forJε_k fork=1,2, . . . such thatεk→0⁺ and

lim inf

k→∞ Jεk(ψεk, Aεk)c <∞. (10) There exists a finite subset D = D(c, he) of Zⁿ, a subsequence {εk₊}, and (ψ⁰, A⁰)∈ H_a,D¹ ×H¹(;R²)for someD∈Dsuch that

(ψε_k+, Aε_k+) %ψ⁰, A⁰ inM.

Moreover(ψ⁰, A⁰)is an equilibrium forJ0. (See Theorem 4.1.) Note that

Jε(√

a )= ||h²_e+

|∇√

a|² forε0. (11)

Thus, givenhe, it follows from Theorem 2 that a sequence of minimizers withεk→0⁺ will satisfy (10).

THEOREM 3. – Fixhe0. Let(ψε_k, Aε_k)be a minimizer ofJε_k inMfork=1,2, . . . with εk→0+. Then a subsequence (ψε_k+, Aε_k+)→(ψD, AD) in M, where (ψD, AD) is a minimizer of J0 in M0 and (ψD, AD)∈H_a,D¹ ×H¹(;R²). Moreover, if R >0 and BR(xi) are disjoint subsets offor i=1, . . . , n, then for all +sufficiently large,

|ψε_k+|>0 outsideⁿ_i₌₁BR(xi)and the degree ofψε_k+ inBR(xi)isdi for alli∈ {1, . . . , n} whereD=(d1, . . . , dn). (See Theorem 4.2.)

We prove in Corollary 3.6 that for he 0 fixed, the set of all D in Zⁿ such that H_a,D¹ ×H¹(;R²)contains a minimizer ofJ0inM0is a nonempty finite set (depending only on,a(x), andhe), which we denote byD0=D0(he).

THEOREM 4. – Let(ψε, Aε) be a minimizer ofJε for each ε >0. Fix R >0 as in Theorem 3 and he0. There exists ε0=ε0(R, he) >0 such that for all 0< ε < ε0,

|ψε|>0 outsideⁿ_i₌₁BR(xi)and the degree of ψε inBR(xi) fori=1, . . . , n, denoted by Dε=(d1,ε, . . . , dn,ε), is inD0. Moreover, there exists he>0 (depending only on anda(x)) such that ifhe> heand 0< ε < ε0(R, he), thenDε= 0. (See Theorem 4.3.)

We remark that Theorem 4 implies that{Hc₁(ε)}is uniformly bounded inεasε→0⁺. (See Corollary 4.4.)

The equilibrium found in Theorem 1 is (by uniqueness) the minimizer for J0 in H_a,D¹ ×H¹(;R²). Since H_a,D¹ is open in H_a¹, it is also a local minimizer for J0 in M0. Given he0, let (ψD, AD)be such a solution. For local minimizers ofJε inM, we have (in contrast to Theorem 4) that all degrees inZⁿnearx1, . . . , xnare attainable:

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THEOREM 5. – Fixhe0 and anyD inZⁿ. For eachε >0 sufficiently small, there exists a local minimizer, (ψε, Aε), ofJεinMsuch that(ψε, Aε)→(ψD, AD)inMas ε→0. In addition, for anyR >0 as in Theorem 3, there existsε1(R, he) >0 such that

|ψε|>0 outsideⁿ_i₌₁BR(xi), and the degree ofψε inBR(xi)isdi for allε < ε1, where D =(d1, . . . , dn). (See Theorem 4.6.)

1. Preliminaries

It is well known that if(ψ, A)∈Mandψ=ρe^iθ, then∇θ is uniquely determined almost everywhere in{ρ >0},ρ∈W^1,2(),ρ∇θ∈L²(;R²),

(∇ −iA)ψ²= |∇ρ|²+ρ(∇θ−A)² and j≡ −i

2(ψ^∗∇ψ−ψ∇ψ^∗)− |ψ|²A=ρ²(∇θ−A)a.e. in.

(12) If(ψε, Aε)∈Mand(ψε, Aε)is an equilibrium forJε withε >0, then from (3) we can derive the equations

−div(ρε∇ρε)+ |∇ρε|²+|jε|² ρ_ε² = 1

ε²

a−ρ_ε²ρ_ε² in,

ρε∇ρε·^%n=0 on∂, divjε=0 in, and

jε·^%n=0 on∂,

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where ψε =ρεe^iθ^ε and jε =ρ_ε²(∇θε −Aε). These equations are obtained by using test functions of the form ϕ =ψ_ε^∗φ in the formulation (3) such that φ ∈L^∞() and (1+ |ψε|)|∇φ| ∈L²(). Moreover, if we definehε by∇ ×Aε=hεe3 then (4) can be rewritten as

−∇^⊥hε≡(∂y,−∂x)hε=jε in, hε−he on∂.

(14) Similarly, if(ψ⁰, A⁰)∈M0and(ψ⁰, A⁰)is an equilibrium forJ0then (6) and (7) can be rewritten as

divj0=0 in, j0·^%n=0 on∂

(15) and

−∇^⊥h0=j0 in, h0=he on∂,

(16) whereψ⁰=ρ0e^iθ⁰=√

ae^iθ⁰,h0is defined by∇ ×A⁰=h0e3, andj0=ρ₀²(∇θ0−A⁰).

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The following three results concern maximum principles and regularity for equilibria ofJε. The proofs are only a slight variation of the proofs for the case in which a≡1 in. (See [5] and [6].)

LEMMA 1.1. – If (ψε, Aε) ∈ M(M0) and (ψε, Aε) is an equilibrium for Jε(J0) whereε0, then|ψε|sup√

a.

Proof. – For(ψ0, A0)∈M0, we have|ψ0| =√

ainand hence the result is trivial in this case. Ifε >0 and(ψε, Aε)is an equilibrium forJε, the result follows by using

φε≡max 0,ψε(x)−sup

√a/ψε(x)=ρε(x)−sup

√a⁺/ρε(x)

as a test function in the weak formulation of the first two equations in (13), which yields

0

E

|∇ρε|²=

E

(−φε)·|jε|² ρ²_ε + 1

ε²

E

a−ρ_ε²ρ_ε²φε0,

whereE= {x∈: φε(x) >0}. It follows thatEhas zero measure. Thusφε0 a.e. in which proves the lemma. ✷

LEMMA 1.2. – Forε >0 equilibria are of class,C^2,β()for someβ >0.

Proof. – With our choice of gauge (8), we have∇ × ∇ ×Aε= −)Aε. The system (3) and (4) is thus uniformly elliptic and regularity follows from the classical theory. (See [6].) ✷

LEMMA 1.3. – Fixhe0. Assumeε0 and(ψε, Aε)is an equilibrium forJε. SetM=max(Jε(ψε, Aε),Jε(√

a,0), maxa). Then

Aε2,2C(M, ), (17)

ψε1,2C(M, ), (18)

and ifε >0

|∇ψε|C(M, )/ε in, (19)

where C(M, ) denotes a constant depending only on M, a(x), and , and the subscript k,2 denotes the norm inW^k,2().

Proof. – We argue for ε >0. The proofs of (17) and (18) for the case ε =0 are identical.

We write (using (12)) Jε(ψ, A)=

|∇|ψ||²+ |ψ|²|∇θ−A|²+ 1 2ε²

a− |ψ|²²+ |∇ ×A−hee3|²

.

Recall thatjε= |ψε|²(∇θε−Aε)andhε is defined by

hεe3= ∇ ×Aε. (20)

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From this and (12), we have Jε(ψε, Ae)=

|∇|ψε||²+ |ψε|⁻²|jε|²+ 1 2ε²

a− |ψε|²²+ |hε−he|²

. (21) Thusjε²2supa·Jε(ψε, Aε)C(M, ), wherejε2denotes theL²norm ofjε

in. Then from (14), we have

∇hε2C(M.).

Using this estimate together with (9) we see that∇ξ2,2C(M, ). Thus Aε2,2= ∇ξ2,2C(M, ).

Note that this implies

AεC^γ()C(M, , γ ) for eachγ ∈(0,1). (22) Now

∇ψε²2C(∇ −iAε)ψε²

2+ Aεψε²2

. So we see

∇ψε²₂C(M, ).

This proves (17) and (18) forε >0 (andε=0).

To prove (19) let y =x/ε, ε =/ε, ψε(y)=ψε(εy), andA˜ε =εAε. We have from the Ginzburg–Landau equation

)yψε−2iA˜ε· ∇yψε− | ˜Aε|²ψε=a(εy)− |ψε|²²ψε inε,

∂nψε=0 on∂ε.

Here we have used the choice of gauge. From (22) we see that | ˜Aε(y)| = |εAε(y)| εC(M, ). It follows from local elliptic estimates and Lemma 1.1 that ψε∈W^2,p(ε) forp <∞and

|∇yψε|C(M, ) inε for 0< ε1.

(Here we use that∂is of classC^2,1.) Thus|∇ψε|C(M, )/εin. ✷ The remaining results in this section are facts about

H_a¹= ψ∈H¹(;C): |ψ| =√

aa.e. in which are used later in this paper.

THEOREM 1.4. – Eachu∈H_a¹can be written as u(x)=a(x)·

n

j=1

z−zj

|z−zj| d_j

·e^iϕ(x)=a(x)·e^{iθ (x)}

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wherez=z(x)=x¹+ix²forx=(x¹, x²)in,zj =z(xj),ϕ∈H_loc¹ (\ {x1, . . . , xn}), θ (x) =ϕ(x) +ⁿj=1djθj(x), and θj(x) is the azimuthal angle of x about xj for 1j n. Moreover, for each u∈H_a¹,D≡(d1, . . . , dn)∈Zⁿ is unique,ϕ∈H_loc¹ (\ {x1, . . . , xn}) is unique up to an additive constant 2π k for k ∈ Z, and ϕ satisfies

a|∇ϕ|²C(, a, D)+|∇u|².

Proof. – Fixu∈H_a¹ and set v(x)=u(x)/√

a(x)=u(x)/|u(x)|. Then v∈H_loc¹ (\ {x1, . . . , xn};S¹) whereS¹= {z∈C: |z| =1}). It follows from Schoen and Uhlenbeck [14] that there exists a sequence{vm}such that

vm∈C²

\ n

j=1

B¹

m(xj);S¹

form=1,2,3, . . .and

vm→v inH_loc¹ \ {x1, . . . , xn}asm→ ∞.

(See also [2].) We compute the degree of eachvmnearxj; as follows:

We say that a radiusr is admissible for a givenxj andvmifBr(xj)∩ {x₁, . . . , xn} = {xj}and ∂Br(xj)⊂\ⁿi=1B¹

m(xi). For any such r, since vmis smooth and |vm| =1 in\ⁿi=1B¹

m(x1, . . . , xn), the winding number ofvmon∂Br(xj)is defined by:

dj,m= − i 2π

∂Br(xj)

v_m^∗(vm)τ (23)

whereτ=ν^⊥=(−ν2, ν1),νis the exterior unit normal on the boundary ofBr(xj), and (vm)τ is the derivative ofvmin the directionτ. It is well known from degree theory that dj,mis integer-valued and independent ofrfor all admissiblerwith respect toxj andm.

Thus if 0< r1< r2<∞andr2satisfiesBr₂(xj)⊂andBr₂(xj)∩ {x1, . . . , xm} = {xj}, then for allmsufficiently large, anyr∈ [r1, r2]is admissible forxj andvm, and we may integrate (23) to obtain

dj,m= − i 2π(r2−r1)

B_ra(xj)\Br1(xj)

v_m^∗(vm)τdx. (24)

Sincedj,mis integer-valued andvm→vinH_loc¹ (\ {x1, . . . , xn})asm→ ∞, it follows from (24) that dj,m is independent ofm for all msufficiently large. Thus there exists dj ∈Z such thatdj=dj,mfor allmsufficiently large and, lettingm→ ∞, we have

dj= − i 2π(r2−r1)

B_r

2(x_j)\B_r

1(x_j)

v^∗vτdx. (25)

We may use this to define the degree ofvnearxj, since (25) is independent ofr2> r1>0 provided that Br₂(xj)⊂ and Br₂(xj)∩ {x1, . . . , xn} = {xj} and it is clear (25) is

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independent of the particular converging sequence {vm}. In particular, we can define the degree of u inBr(xj)by (25), forv=u/|u|andr1< r2as above. (See also [7].)

Now consider the real two-dimensional vector field Fm= −

n

j=1

dj∇θj−iv^∗_m∇vm (26)

inC¹(\ⁿj=1B¹

m(xj)), whereθj(x)is the (multivalued) azimuthal angle ofxaboutxj

and thus∇θj(x)is well defined in\ {xj}for 1jn. Since∇ × ∇θj=0 in\ {xj} andvm isC²with|rm|²=vmv_m^∗ =1 in\ⁿj=1B¹

m(xj), it follows that∇ ×Fm=0 in \ⁿ_j₌₁B¹

m(xj). Thus ifmis sufficiently large so thatdj=dj,mfor 1jn, then by Stokes’ Theorem and (23),_CFm·dr=0 for any closed curve,C, in\ⁿ_j₌₁B¹

m(xj).

Moreover, there existsϕm∈C²(\ⁿj=1B¹

m(xj))such that∇ϕm=Fmform=1,2, . . .. From this and (26) we obtain

vm∇ϕm= −vm

_n

j=1

dj∇θj

−i∇vm

and hence

∇vme⁻^iϕ^m·e⁻ⁱ n

j=1djθj

=e⁻^iϕ^m·e⁻ⁱ n

j=1djθj

×

∇vm−i

vm∇ϕm+vm

n

j=1

dj∇θj

=0.

As a result (adding a constant toϕmif necessary), we have vm(x)=e^iϕ^m^(x)·eⁱ

n j=1djθj(x)

=e^iϕ^m^(x)· n

j=1

z−zj

|z−zj| d_j

.

By (26),∇ϕm= −ⁿj=1dj∇θj−iv_m^∗∇vmand sincevm→vinH_loc¹ (\ {x1, . . . , xn}), we have∇ϕm→ −ⁿj=1dj∇θj−iv^∗∇vand

e^iϕ^m≡vm· n

j=1

z−zj

|z−zj| ₋dj

→v· n

j=1

z−zj

|z−zj| ₋dj

inL²_loc(\ {x1, . . . , xm}). It follows that{ϕm}(after possibly subtracting constants 2π km

wherekm∈Z) converges inH_loc¹ (\ {x1, . . . , xn}), to someϕ∈H_loc¹ (\ {x1, . . . , xn}), andu=√

av=√ ae^i(ϕ⁺

n j=1d_jθ_j)

a.e. in. Settingθ (x)=ϕ(x)+ⁿj=1djθj(x), we have

|∇u|²= |∇√

a|²+a|∇θ|²a|∇θ|².

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Since|∇θj(x)| =c(dj)|x−xj|⁻¹anda(x)c|x−xj|^α^j whereαj>0 for 1jn,

a(x)∇θj(x)²C(, a, D) <∞ (27) whereD=(d1, . . . , dn). Thus

a|∇ϕ|²

|∇u|²+C· n

j=1

a|∇θj|²

|∇u|²+C(, a, D).

Finally, to show thatD∈Zⁿis unique andϕ∈H_loc¹ (\ {x1, . . . , xn})is unique (up to an additive constant 2π l wherel∈Z) for eachu∈H_a¹, assume thatD=(d˜1, . . . ,d˜n)∈ Zⁿ and ϕ˜ ∈H_loc¹ (\ {x1, . . . , xn}) such that u=√

aeⁱ^{[ ˜}^ϕ⁺ n

k=1d˜kθk]. Then v≡u/√ a satisfies −iv^∗∇v= ∇ ˜ϕ +ⁿ_k₌₁d˜k∇θk in \ {x1, . . . , xn}. Fixing j ∈ {1, . . . , n} and integrating overBr₂(xj)\Br₁(xj)for 0< r1< r2as in (25), we have

dj= − i 2π(r2−r1)

B_r

2(x_j)\B_r

1(x_j)

v^∗vτdx

= 1

2π(r2−r1)

Br2(x_j)\Br1(x_j)

˜ ϕτ+

n

k=1

d˜k(θk)τ

dx

= 1

2π(r2−r1)·0+ ˜dj·2π(r2−r1)

= ˜dj

where τ =τ (x) = (x −xj)^⊥/|x −xj| for all j ∈ {1, . . . , n}. Thus e^i(ϕ^{− ˜}^ϕ) =1 in \ {x1, . . . , xn}withϕ− ˜ϕinH_loc¹ (\ {x1, . . . , xn})and it follows thatϕ− ˜ϕ=2π lfor somel∈Z. ✷

For eachD∈Zⁿ; we define H_a,D¹ = u∈H_a¹: u=√

aeⁱ^[^ϕ⁺ n

j=1d_jθ_j] whereϕ∈H_loc¹ \ {x1, . . . , xn}. By Theorem 1.4, it follows that

H_a¹=

D∈Zⁿ

H_a,D¹

andH_a,D¹ ∩H_a,D¹ = ∅forD=DinZⁿ. We will need the following additional properties ofH_a,D¹ :

THEOREM 1.5. – For each D∈Zⁿ, H_a,D¹ is a nonempty, open and closed subset of H_a¹. In addition,H_a,D¹ is sequentially weakly closed inH¹(;C), i.e. if{uk} ⊂H_a,D¹ and uk→uweakly inH¹(;C), thenu∈H_a,D¹ .

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Proof. – Our hypotheses ona inensure that√

a∈H¹()and√a∇θj∈L²()for each j ∈ {1, . . . , n}(see (27)); hence √

aeⁱ^[ n

j=1d_jθ_j]∈H_a,D¹ and H_a,D¹ = ∅. To prove thatH_a,D¹ is open inH_a¹, assume thatu0=√

aeⁱ^[^ϕ⁺ n

j=1djθj]∈H_a,D¹ and let BR(u0)= u∈H_a¹:u−u0H¹(;C)< R

where R >0. Since u∈H_a¹, there existsϕ˜ ∈H_loc¹ (\ {x1, . . . , xn}) and D ∈Zⁿ such thatu=√

aeⁱ^{[ ˜}^ϕ⁺ n

j=1d˜_jθ_j]

. Setv0=u0/|u0| =u0/√

a andv=u/|u| =u/√

a. By (25), there exist positive numbersr1< r2such that for eachj ∈ {1, . . . , n},

dj= − i 2π(r2−r1)

Sj

v₀^∗(v0)τdx (28) and

d˜j= − i 2π(r2−r1)

S_j

v^∗(v)τdx

whereSj =Br₂(xj)\Br₁(xj). Sincea isC¹and |a|>0 on Sj for eachj ∈ {1, . . . , n}, we have

v₀^∗∇v0−v^∗∇v_L1(S_j)v^∗(∇v0− ∇v)_L1(S_j)+(v₀^∗−v^∗)∇v0

L¹(S_j)

C(a, r1, r2, v0)·1+ u0H¹(Sj)

· u−u0H¹(Sj). From this and (28), it follows that if R is sufficiently small (depending onr1,r2,,a, and u0), we havedj = ˜dj and u∈H_a,D¹ . ThusBR(u0)⊂H_a,D¹ forR sufficiently small and we conclude that H_a,D¹ is an open subset ofH_a¹. Now since H_a¹=D∈ZⁿH_a,D¹ and Ha,D∩H_a,D¹ = ∅forD=DinZⁿ,H_a,D¹ is also a closed subset ofH_a¹.

Finally, to prove that H_a,D¹ is weakly sequentially closed inH_a¹, assume that{uk} ⊂ H_a,D¹ and uk →u weakly in H¹(;C). By compactness, a subsequence (which we relabel as{uk}) satisfiesuk→uinL²(). Thus|u| =√

aa.e. inand henceu∈H¹

a,D for someD ∈Zⁿ. It follows from (28) (withv0replaced by uk/√

a andv replaced by u/√

aand the weak convergence ofuk touthatD=D andu∈H_a,D¹ . ✷ 2. A weighted Sobolev space

Set

V ≡

g∈H¹():

a⁻¹|∇g|²<∞

. ThenV is a Hilbert space with norm

gV =

a⁻¹|∇g|²+g²

1/2

.