Vorticity measure of type-II superconducting thin films

JAMCJ Appl Math Comput (2011) 37:267–286DOI 10.1007/s12190-010-0433-4

Vorticity measure of type-II superconducting thin films

Kwang Ik Kim · Zuhan Liu

Received: 21 October 2009 / Published online: 9 August 2010© Korean Society for Computational and Applied Mathematics 2010

Abstract In this paper, we study the Ginzburg-Landau system of variable thicknesssuperconducting thin films, placed in an applied magnetic field hex, when hex is of theorder of the “first critical field”, i.e. of the order of | ln ε|. We examine the asymptoticbehavior of the “vorticity-measures” associated to the vortices of the solution anddescribe the distribution of vortices.

Keywords Superconducting thin films · Vortices · Ginzburg-Landau equations

Mathematics Subject Classification (2000) 35J55 · 35Q40

1 Introduction

Quantized vortices in superconductors have often been modeled by the phenomeno-logical model of Ginzburg and Landau [1, 2]. Below the critical temperature, quan-tized vortices can be nucleated in a type-II superconductor when it is subject to anapplied magnetic field above certain critical strength. The study of Abrikosov onthe vortex lattices based on the Ginzburg-Landau model was a Nobel prize winning

Partially supported by PRC Program thought the NRF of Korea (Grant No. 2009-0094070) andPOSTECH BK1.

Research of Z. Liu was partially supported by the National Natural Science Foundation of China(Nos. 10471119, 10771181).

K.I. KimDepartment of Mathematics, Pohang University of Science and Technology, Pohang 790-784,Republic of Koreae-mail: [email protected]

Z. Liu (�)Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, Chinae-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

268 K.I. Kim, Z. Liu

work that has become the highlight of superconductivity and quantum superfluidity.Consider a three-dimensional variable thickness superconducting thin film, which issymmetric with respect to the (x1, x2)-plane. The superconducting domain is givenby �δ = � × (−δa(x), δa(x)), where a(x) : � → R is a given smooth function sat-isfying a(x) ≥ a0 > 0, for any x in �, and � is a bounded smooth domain in R

2. Forthis thin structure, a reduction of the full three dimensional G-L model can be madewhich leads to a simplified model that retains the basic features of the vortex state[3–6], referred to as (G.L.):

(G.L.)

{−(∇ − iA0) · a(x)(∇u − iA0u) = a(x)u

ε2 (1 − |u|2) in �,∂u∂ν

= 0 on ∂�.

The solutions of this system are the critical points of the following Ginzburg-Landauenergy:

J (u) = 1

2

∫�

a(x)

[|∇A0u|2 + 1

2ε2

(1 − |u|2)2]. (1.1)

Here A0(x) is a given smooth magnetic potential vector, A0(x) = (A10(x),A1

0(x)),which satisfies

div(a(x)A0

)= 0, curlA0 = hex in �, (1.2)

A0(x) · ν = 0 on ∂�. (1.3)

We denote ∂j = ∂∂xj

, curlA0 = ∂1A20 − ∂2A

10 and ∇A0u = ∇u − iA0u. hex > 0 is the

applied field that is vertical to the (x1, x2)-plane, and ν denotes the outward normalvector to ∂�. κ = 1

εis the material parameter that determines the type of supercon-

ducting material. The complex superconducting order parameter u indicates the localstate of the material. |u|2 represents the density of superconducting electron pairs, sothat, |u| � 1 corresponds to the superconducting state, |u| � 0 corresponds to the nor-mal state. An isolated zero of the order parameter u, around which u has a non-zerowinding number, is called vortex.

We know that a superconductor placed in an applied magnetic field may changeits phases when the field varies. When hex remains lower than some value Hc1 of theorder | ln ε|, the material is in the superconducting phase everywhere and repels themagnetic field. When the critical value Hc1 is reached, there is a phase-transition fromthe superconducting state to the “mixed-state”, where vortices appear. The mixed-phase is defined by the coexistence of the normal and superconducting phases in thematerial. Raising hex, vortices get more and more numerous and tend to arrange in atriangular lattice. When the second critical value Hc2 (of the order of 1/ε2) is reached,there is another phase-transition from the “mixed-state” to the normal state.

It has been conjectured that for a minimizing configuration u of (1.1), the vor-tices (zeros of u) should be pinned near the minimal points of the function a, see[3–6]. Many authors have addressed this question in the regime of extreme type IIsuperconducting materials, ε → 0. For instance, the precise characterization of Hc1

was given in detail in [4]. Ding and Du [4–6] proved that the global minimizers ofthis functional had bounded number of vortices and gave the locations of the vortices

Vorticity measure of type-II superconducting thin films 269

(pinning phenomena) when the applied magnetic field hex < Hc1 + K ln | ln ε|. Herewe mention that the pinning phenomena was first rigorously discussed in [7] by Linand Du. While when hex � Hc1 , the study of global minimizers of Jε is more delicatein sense that the number of vortices does not remain bounded as ε → 0. In [8], Kimand Liu describe the repartition of vortices in the minimizers for applied fields of theorder of | ln ε|.

Here, we wish to address the question of the behavior of critical points in general,i.e. the asymptotic behavior, as ε → 0, of solutions of the Ginzburg-Landau system(G.L.), that are not necessarily global or local minimizers. Let us now describe ourassumptions and main results. In this paper, we deal mainly with intermediate fieldshex ≤ C| ln ε|. It is natural to assume that the energy blows up like |hex|2:

hex ≤ C| ln ε|, (1.4)

Jεn(uεn) ≤ Ch2ex. (1.5)

Such an assumption (1.5) is automatically satisfied for the minimizer of the energy.We can adjust the technique that was used in [9, 10] to the present needs and obtainthe following result for solutions of (G.L.):

Proposition 1.1 If hex ≤ C| ln ε|, there exists ε0 such that if ε < ε0 and uε satisfies(G.L.), (1.5), there exists a family of balls (depending on ε) (Bi)i∈Iε = (B(bi, ri))i∈Iε

satisfying: {x:∣∣uε(x)

∣∣≤ 1

2

}⊂⋃i∈Iε

B(bi, ri), (1.6)

Card Iε ≤ C| ln ε|20, (1.7)∑i∈Iε

ri ≤ ε12 and ri ≥ Cε| ln ε|4, (1.8)

1

2

∫Bi

a(x)|∇A0uε|2 ≥ πa(bi)|di || ln ε|(1 − o(1)), (1.9)

where bi is the center of the vortex and ri is the radius, di is the winding number ofu|u| on the boundary of Bi . Thus,

Nε = 2π∑i∈Iε

|di | ≤ Chex. (1.10)

Now we can define the vorticity-measure associated to u

με = 2π∑

i diδbi

hex. (1.11)

Let us now state our main result. We denote by H 11 (�) the set {f ∈ H 1(�) : f − 1 ∈

H 10 (�)}, and M(�) the space of bounded Radon measures on �.


Theorem 1.1 Let � ⊂ R2 be a bounded, smooth and simply connected domain. Let

a(x) ∈ C1(�) and 0 < min� a(x) ≤ a(x) in �. Let εn → 0, and un be correspondingcritical points of Jεn , with hypothesis (1.2)–(1.5), then, the following properties hold.

(1) ConvergenceUp to extraction of a subsequence, we have

hn

hex⇀ h∞ in H 1

1 (�), (1.12)

where hn is so chosen such that

−1

a∇⊥hn = (iun,∇A0un). (1.13)

For any (bi, di) satisfying the results of Proposition 1.1, up to extraction

μn := 2π∑

i diδbi

hex⇀ μ∞ := −div

(1

a∇h∞

)+ 1 weakly in M(�).

(1.14)

(2) h∞ is a critical point with respect to domain-diffeomorphisms for the functional

L(h) = 1

2

∫�

1

a|∇h|2 + h (1.15)

defined over H 11 (�), meaning that for any smooth compactly supported vector

field X : � → R2, the derivative at t = 0 of

L(h(x + tX(x)

))= 1

2

∫�

1

a(x + tX(x))

∣∣∇h(x + tX(x)

)∣∣2 + h(x + tX(x)

)is zero.

(3) Distribution of vorticity measureIf ∇h∞ ∈ C0(�) ∩ BV(�), then

μ∞ = 1|∇h∞|=0, (1.16)

thus μ∞ is a nonnegative L∞ function.

Remark 1.1 Such a function in (1.13) can be found by the unique solution of thefollowing problem:

−div

(1

a(x)∇hn

)= curl(iun,∇un) − curl

(A0|un|2

)in �, (1.17)

and

hn = hex on ∂�. (1.18)


There have been many works on the mathematical theory of superconductivity[2–22] and the reference therein. The work [12] by Serfaty and Sandier is closelyrelated to our present paper. Our proofs used the technique of [12]. This necessitatessome technical modifications of the arguments in [12]. Additionally, the fact that thevector potential A is given means that the field appearing in Ampere’s law is not thecurl of A. This leads to a change in the definition of the appropriate stress-tensor (see(3.1)–(3.3)). For these tensors, a “convergence in finite part” analysis is done, with alimit that is not divergence free (unlike those in [12]).

We organize the paper as follows: In Sect. 2, we derive the convergence of theinduced magnetic field except on a set of small perimeter. In Sect. 3, we study theasymptotic behavior of the stress-energy tensor associated to the solution u of the(G.L.). In Sect. 4, we study the property of the limiting vorticity-measure. In Sect. 5,we give the conclusion.

2 Convergence except on a set of small perimeter

Recall the definition of the p-capacity (p ≥ 1) for compact sets [23]:

Pcap(E) = inf

{∫�

|∇φ|p, φ ∈ C∞0 (�), φ ≥ 1 in E

}. (2.1)

Let us consider a sequence of critical points un with the hypothesis of Theorem 1.1.We write un = ρne

iϕn , with ρn = |un|, at least formally. Of course ϕn is not well-defined where un vanishes. Since un is a solution of (G.L.), this implies some a prioriestimates (see [4]):

|un| ≤ 1, ‖∇un‖L∞(�) ≤ C

εn

. (2.2)

By the definition,

−1

a∇⊥hn = ρ2

n(∇ϕn − A0) (2.3)

and this has a meaning everywhere. Then, we can rewrite the energy the followingway:

J (uεn) = 1

2

∫�

a

[∣∣∇|un|∣∣2 + |un|2|∇ϕn − A0|2 + 1

2ε2n

(1 − |un|2

)2]

= 1

2

∫�

a

[∣∣∇|un|∣∣2 + |∇hn|2

a2|un|2 + 1

2ε2n

(1 − |un|2

)2]

≥ 1

2

∫�

a

[∣∣∇|un|∣∣2 + |∇hn|2

a2+ 1

2ε2n

(1 − |un|2

)2]. (2.4)

Thus, combining this inequality with the hypothesis (1.4), (1.5), we have∥∥∥∥ |un|hex

∥∥∥∥H 1(�)

≤ C,

∥∥∥∥ hn

hex

∥∥∥∥H 1(�)

≤ C. (2.5)


Hence, up to extraction of a subsequence, we can assume that there exists a h∞ ∈H 1

1 (�) such that

hn

hex⇀ h∞ in H 1

1 (�). (2.6)

Then, defining μn by (1.11),∫�

a|μn| remains bounded in view of (1.9), we canassume that it converges weakly in the sense of measures to a limiting Radon mea-sure μ∞. Then, we will have, for any choice of (bi, di) satisfying (1.6)–(1.9),

μn = 2π∑

i∈I diδbi

hex⇀ μ∞ = −div

(1

a∇h∞

)+ 1. (2.7)

First, we will prove the following proposition:

Proposition 2.1 For all δ > 0, there exists Eδ ⊂ � with 1 − cap(Eδ) < δ and

∫�\Eδ

∣∣∣∣∇(

hn

hex− h∞

)∣∣∣∣2

→ 0 as n → ∞. (2.8)

In order to prove this proposition, we start with the following lemma.

Lemma 2.1 There exists a sequence such that the following hold

|un| = 1 in �∖ ⋃

i∈Iε

Bi, (2.9)

‖un − un‖L∞(�) → 0. (2.10)

Moreover, denoting by −∇⊥hn = a(iun,∇A0 un), then

‖hn − hn‖H 1(�) → 0 (n → ∞), (2.11)

−div

(1

a∇hn

)+ hex = 0 in �

∖⋃i

Bi . (2.12)

Proof Define a continuous function χn: (0,1) → (0,1) such that⎧⎪⎨⎪⎩

χn(x) = x, 0 ≤ x ≤ 12 ,

χn(x) = 1, x ≥ 1 − | ln εn|−4,

χn is affine between 12 and 1 − | ln εn|−4.

(2.13)

We can replace un by un := χn(|un|) un|un| . Then, un satisfies (2.9), (2.10). Since hn

satisfies {− 1

a∇⊥hn = (iun,∇un) − |un|2A0,

hn = hex.(2.14)


Therefore {−div( 1

a∇hn) + hex = 0 in � \⋃i Bi ,

hn = hex on ∂�.(2.15)

Denote ρn = |un|, ρn = |un|, un = ρneiϕ . Note that ‖A0‖L∞ ≤ Chex, ‖∇ϕn‖L2(�) ≤

Chex, hex ≤ C| ln ε|. On the other hand, (2.4) yields ‖1 − ρ2n‖L2(�) ≤ εhex. Combin-

ing these relations with (2.13), we have

‖∇hn − ∇hn‖L2(�) = a∥∥ρ2

n(∇ϕn − A0) − ρ2n(∇ϕn − A0)

∥∥L2(�)

≤ a[∥∥ρ2

n∇ϕn − ρ2n∇ϕn

∥∥L2(�)

+ ∥∥(ρ2n − ρ2

n

)A0∥∥

L2(�)

]≤ a[∥∥(ρ2

n − ρ2n

)∇ϕn

∥∥L2(�)

+ ∥∥(ρ2n − ρ2

n

)A0∥∥

L2(�)

]= o(1). (2.16)

Therefore, by Poincare inequality,

‖hn − hn‖H 1(�) → 0 (n → ∞). (2.17)

This completes the proof of Lemma 2.1. �

We now introduce gn as the solution of{−div( 1a∇gn) + 1 = 2π

hex

∑i diδbi

in �,

gn = 1 on ∂�,(2.18)

where the (bi, di) are the vortices as defined in Proposition 1.1. For simplicity, wewill denote

kn = hn

hex, kn = hn

hex. (2.19)

Lemma 2.2 For any p < 2, there exists α(p) > 0 such that∥∥∥∥−div

(1

a∇hn

)+ hex − 2π

∑i

diδbi

∥∥∥∥W−1,p(�)

≤ εα(p). (2.20)

Proof We drop the subscript n in this proof for simplicity. According to Lemma 2.1,we may assume |u| = 1 on � \⋃i Bi . Let q be such that 1

q+ 1

p= 1, 1

q+ 1

q ′ = 12 . For

any ξ ∈ W1,q

0 (�), by (1.8), we have

∣∣∣∣∫⋃

i Bi

∇⊥ξ · (iu,∇u)

∣∣∣∣ ≤ ‖∇u‖L2(�)‖∇ξ‖Lq(�)

(∣∣∣∣⋃i

Bi

∣∣∣∣)1/q ′

≤ Chex‖∇ξ‖Lq(�)

(∑i

r2i

)1/q ′


≤ Chex‖∇ξ‖Lq(�)ε2/q ′

. (2.21)

By integration by parts, we obtain∫�

ξcurl(iu,∇u) = −∫

�

∇⊥ξ(iu,∇u).

Hence, it follows from above inequality that∥∥∥∥∫

�

ξcurl(iu,∇u) +∫

�

∇⊥ξ(iu,∇u)

∥∥∥∥W−1,p(�)

≤ Cεα(p), (2.22)

where � = � \⋃i Bi . On the other hand, since |u| = 1 on �, then curl(iu,∇u) = 0.It follows from Stokes’s theorem that∫

�

∇⊥ξ(iu,∇u) =∫

∂�

ξ

(iu,

∂u

∂τ

)=∑

i

∫∂Bi∩�

ξ

(iu,

∂u

∂τ

)(2.23)

and ∣∣∣∣∑i

∫∂Bi∩�

(ξ − ξ(bi)

)(iu,

∂u

∂τ

)∣∣∣∣=∣∣∣∣∑

i

∫Bi

∇⊥ξ · (iu,∇u) + (ξ − ξ(bi))curl(iu,∇u)

∣∣∣∣. (2.24)

Note that q > 2, W 1,q

0 (�) embeds in some C0,γ with 0 < γ < 1, and |curl(iu,∇u)| ≤|∇u|2, then

∑i

∫Bi

∣∣(ξ − ξ(bi))curl(iu,∇u)

∣∣ ≤ (maxi

ri

)γ ‖ξ‖C0,γ

∑i

∫Bi

|∇u|2

≤ ‖ξ‖W

1,q0 (�)

(max

iri

)γ ‖∇u‖2L2(�)

≤ ‖ξ‖W

1,q0 (�)

(max

iri

)γ

h2ex

≤ εα(p)‖ξ‖W

1,q0 (�)

. (2.25)

Therefore we have∣∣∣∣∫

�

∇⊥ξ · (iu,∇u) − 2π∑

i

diξ(bi)

∣∣∣∣≤ εα(p)‖ξ‖W

1,q0 (�)

. (2.26)

In summary, we have proved∥∥∥∥curl(iu,∇u) − 2π∑

i

diδbi

∥∥∥∥W−1,p(�)

≤ εα(p). (2.27)


Similarly, it follows from ‖A0‖L∞ ≤ Chex that

∣∣∣∣∫

�

ξcurl((

1 − |u|2)A0)∣∣∣∣ =

∣∣∣∣∫

�

∇⊥ξ((

1 − |u|2)A0)∣∣∣∣

≤ Cεα(p)‖∇ξ‖L2(�). (2.28)

Hence,

∥∥curl((

1 − |u|2)A0)∥∥

W−1,p(�)≤ εα(p). (2.29)

Note that

−div

(1

a∇h

)+ hex = curl(iu,∇u) − curl

((1 − |u|2)A0

), (2.30)

we finish the proof of Lemma 2.2. �

Lemma 2.3 For any p < 2, there exists α(p) > 0 such that

‖kn − gn‖W 1,p(�) ≤ Cεα(p).

Moreover,

‖gn − h∞‖W 1,p(�) → 0,

and thus ∥∥∥∥ hn

hex− h∞

∥∥∥∥W 1,p(�)

→ 0.

Proof We have

−div

(1

a∇(kn − gn)

)= 1

hex

(curl(iun,∇un) − 2π

∑i

diδbi

)

+ 1

hexcurl((

1 − |un|2)A0). (2.31)

Hence, in view of the previous results,

∥∥∥∥−div

(1

a∇(kn − gn)

)∥∥∥∥W−1,p(�)

≤ Cεα(p).

By elliptic regularity, we obtain

‖kn − gn‖W 1,p(�) ≤ Cεα(p).


We treat the case of h∞ − gn.∥∥∥∥−div

(1

a∇(h∞ − gn)

)∥∥∥∥W−1,p(�)

=∥∥∥∥μ∞ − 2π

∑i diδbi

hex

∥∥∥∥W−1,p(�)

= ‖μ∞ − μεn‖W−1,p(�). (2.32)

In view of (2.27), μεn is bounded in W−1,p(�) for all p < 2. It is also bounded inmeasures, hence by Murat’s theorem [24], it is compact in W−1,p(�) for all p < 2,thus μεn → μ∞ strongly in W−1,p(�) for all p < 2. Therefore, ‖−div( 1

a∇(h∞ −

gn))‖W−1,p(�) → 0 and, similarly, we get the result for h∞ − gn. �

Lemma 2.4 There exists a sequence of sets An such that 1cap(An) → 0 and∫�\An

|∇(kn − gn)|2 → 0 as n → ∞.

Proof We split this proof into 3 steps.

Step 1. By the standard result on capacities [23], we have, for p < 2, that

Pcap({

x ∈ �: |kn − gn| > εα(p)/2})≤ Cε−pα(p)/2‖kn − gn‖p

W 1,p(�). (2.33)

Hence, by Lemma 2.3,

Pcap({

x ∈ �: |kn − gn| > εα(p)/2})≤ Cεpα(p)/2. (2.34)

Let us denote by kn − gn the function kn − gn truncated at the level εα(p)/2, andCn = {x ∈ � : |kn − gn| > εα(p)/2}, (2.34) yields

Pcap(Cn) ≤ Cεα(p)/2. (2.35)

Step 2. We prove that ∫�\⋃i Bi

|∇gn|2 ≤ C| ln ε|. (2.36)

In fact, gn can be rewritten as

gn = ξ0 + 1

hex

∑i

gi,

where gi is the solution of{−div( 1a∇gi) = 2πdiδbi

in �,

gi = 0 on ∂�,(2.37)

and ξ0 is the solution of {−div( 1a∇ξ0) + 1 = 0 in �,

ξ0 = 1 on ∂�.(2.38)


It is easy to check that ∫�\Bi

|∇gi |2 ≤ Cd2i ln

1

ri,

then

‖∇gn‖L2(�\⋃i Bi )≤ C +

∑i ‖∇gi‖L2(�\⋃i Bi )

hex≤ C + C

∑i |di |hex

| ln ri | 12 .

By (1.8), ri ≥ Cε and∑

i |di | ≤ Chex, thus∫�\⋃i Bi

|∇gn|2 ≤ C| ln ε|. (2.39)

Step 3. First, as in [12], choose a function ξ that satisfies ξ = 0 in⋃

i B(bi, ri), ξ = 1in � \⋃i B(bi,2ri), 0 ≤ ξ ≤ 1, ‖ξ‖2

H 1 ≤ Ch2ex, ‖∇ξ‖L∞ ≤ 1

mini ri� 1

ε| ln ε|4 . Now,

let us then set An = Cn ∪ (⋃

i∈I B(bi,2ri)). We have

1cap(An) ≤ 1cap(Cn) + 2∑

i

ri ≤ Pcap(Cn) + o(1)

where we have used the fact that the 1cap is dominated by the Pcap (p ≥ 1). Thus,using (2.35), we obtain

1cap(An) → 0.

On the other hand,∫�\An

1

a

∣∣∇(kn − gn)∣∣2

≤∣∣∣∣∫

�

1

aξ∇(kn − gn) · ∇(kn − gn)

∣∣∣∣≤∣∣∣∣∫

�

ξdiv

(1

a∇(kn − gn)

)(kn − gn) +

∫�

1

a(kn − gn)∇ξ · ∇(kn − gn)

∣∣∣∣. (2.40)

Since div( 1a∇(kn − gn)) is supported in

⋃i B(bi, ri), and ξ vanishes there, we have∫

�

ξdiv

(1

a∇(kn − gn)

)(kn − gn) = 0.

Therefore, by (2.34), (2.36), and the definition of ξ ,∫�\An

1

a

∣∣∇(kn − gn)∣∣2

≤ C‖kn − gn‖L∞(�)‖∇ξ‖L2(�)(‖∇ kn‖L2(�\⋃i Bi )+ ‖∇gn‖L2(�\⋃i Bi )

)

≤ Cεα(p)

2 | ln ε|11. (2.41)


Thus, ∫�\An

∣∣∇(kn − gn)∣∣2 ≤ o(1). (2.42)


Lemma 2.5 There exists a set Bn such that

1cap(Bn) → 0

and ∫�\Bn

∣∣∇(gn − h∞)∣∣2 → 0.

Proof From Lemma 2.3, we have ‖gn − h∞‖W 1,p → 0 for p < 2. Hence, from thesame theorem on capacities, denoting

Bn = {x ∈ �: |gn − h∞| > δn

},

we have

Pcap(Bn) ≤ ‖gn − h∞‖p

W 1,p

δpn

.

Therefore, we can choose a suitable δn → 0 such that

1cap(Bn) → 0. (2.43)

We still denote by gn − h∞ the function gn − h∞ truncated at the level δn. We have‖gn − h∞‖L∞(�) ≤ Cδn → 0, and∫

�\Bn

1

a

∣∣∇(gn − h∞)∣∣2 =

∫�

1

a∇(gn − h∞) · ∇(gn − h∞)

= −∫

�

div

(1

a∇(gn − h∞)

)(gn − h∞). (2.44)

By the definition of gn, div( 1a∇(gn −h∞)) remains bounded in the sense of measures,

thus above equality yields ∫�\Bn

∣∣∇(gn − h∞)∣∣2 → 0.


Proof of Proposition 2.1 For any δ > 0, we can consider Bn ∪ An, and extract a fur-ther subsequence such that ∀n, Bn ∪ An ⊂ Eδ satisfies 1cap(Eδ) < δ. Then, by (2.11),Lemma 2.4, Lemma 2.5, and the triangular inequality, we are led to the conclusion ofProposition 2.1. �


3 Passing to the limit

In this section, let u be the solution of the Ginzburg-Landau equations (G.L.), wedefine the tensor:

(Tij ) =(

hhex

2− 1

4ε2a(x)

(1 − |u|2)2)(1 0

0 1

)

+ a

2

⎛⎝ (∂

A10

1 u)2 − (∂A2

02 u)2 2(∂

A10

1 u, ∂A2

02 u)

2(∂A1

01 u, ∂

A20

2 u) (∂A2

02 u)2 − (∂

A10

1 u)2

⎞⎠ . (3.1)

Here, (·,·) the scalar product in C identified to R2, ∂

Aj0

j u = ∂ju− iAj

0u. By the directcomputation we have

∑j

∂jTij = −1

2

[|∇A0u|2 + 1

2ε2

(1 − |u|2)2]∂ia, ∀i. (3.2)

We wish to apply this result to (un,An) and pass to the limit n → ∞ in the tensor, inorder to obtain a limiting tensor Lij :

(Lij ) = h∞2

(1 00 1

)

+ 1

2a

((∂2h∞)2 − (∂1h∞)2 −2∂1h∞∂2h∞

−2∂1h∞∂2h∞ (∂1h∞)2 − (∂2h∞)2

). (3.3)

Proposition 3.1 For any δ > 0, there exists Eδ such that 1 − cap(Eδ) < δ and ∀i, j

∫�\Eδ

∣∣∣∣Tnij

h2ex

− Lij

∣∣∣∣→ 0 as n → 0. (3.4)

Let us start by rewriting the tensor. In � \⋃i Bi , since |un| ≥ 1 − o(1), u does notvanish (for n large enough), hence, as previously, we can write formally un = ρne

iϕn

with ρn = |un|. From now on, we will drop the subscript again. Then, we obtain

∂ju − iAj

0u = (∂jρ)u

|u| + iu(∂jϕ − A

j

0

)and thus ⎧⎪⎨

⎪⎩(∂

Aj0

j u)2 = (∂jρ)2 + ρ2(∂jϕ − Aj

0),

(∂A1

01 u, ∂

A20

2 u) = ∂1ρ∂2ρ + ρ2(∂1ϕ − A10)(∂2ϕ − A2

0).

(3.5)

Recall that

−∇⊥h = aρ2(∇ϕ − A0).


Hence, we have, in � \⋃i Bi , that⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(∂A1

01 u)2 = (∂1ρ)2 + 1

a2ρ2 (∂2h)2,

(∂A2

02 u)2 = (∂2ρ)2 + 1

a2ρ2 (∂1h)2,

(∂A1

01 u, ∂

A20

2 u) = ∂1ρ∂2ρ − 1a2ρ2 ∂1h∂2h.

(3.6)

Thus, Tij can be rewritten as

(Tij ) =(

hhex

2− 1

4ε2a(x)

(1 − |u|2)2)(1 0

0 1

)

+ a

2

((∂1ρ)2 − (∂2ρ)2 2∂1ρ∂2ρ

2∂1ρ∂2ρ (∂2ρ)2 − (∂1ρ)2

)

+ 1

2aρ2

((∂2h)2 − (∂1h)2 −2∂1h∂2h

−2∂1h∂2h (∂1h)2 − (∂2h)2

). (3.7)

We prove that the second term in the right-hand side of (3.7) tends to zero.

Lemma 3.1 Denoting ρn = |un|, we have∫�\⋃i B(bi ,2ri )

a

2

[|∇ρn|2 + 1

2ε2

(1 − ρ2

n

)2]→ 0 as n → ∞.

Proof For simplicity, we drop the subscripts n. Note that

−div(a∇ρ) + |∇h|2aρ3

= 1

ε2a(1 − ρ2)ρ in � \

⋃i

Bi . (3.8)

Multiplying (3.8) by ξ(1 − ρ) and integrating, we have

∫�

−ξdiv(a∇ρ)(1 − ρ) + |∇h|2aρ3

ξ(1 − ρ) = 1

ε2

∫�

aξ(1 − ρ)2ρ(1 + ρ),

where ξ was defined at the beginning of Step 3 in the proof of Lemma 2.4. Integratingby parts, noting ∂ρ

∂n= 0 on ∂�, we derive that

∫�

a|∇ρ|2ξ + a(1 − ρ)2ρ(1 + ρ)

ε2ξ =

∫�

|∇h|2aρ3

ξ(1 − ρ) + a∇ξ · ∇ρ(1 − ρ). (3.9)

Using that ξ = 0 on⋃

i Bi , we have

∣∣∣∣∫

�

|∇h|2aρ3

ξ(1 − ρ)

∣∣∣∣≤ C

∫�

|∇h|2‖1 − ρ‖L∞(�\⋃i Bi ) = o(1),


where we have used the fact that∫�

|∇h|2 ≤ Ch2ex. From (2.4), we have ‖∇ρ‖L2 ≤

O(hex) and∫�(1 − ρ)2 ≤ Cε2h2

ex. By the definition of ξ , ‖∇ξ‖L∞ ≤ 1ε| ln ε|4 . Thus,

∣∣∣∣∫

�

a∇ξ · ∇ρ(1 − ρ)

∣∣∣∣ ≤ ‖∇ρ‖L2‖1 − ρ‖L2‖∇ξ‖L∞ = o(1),

hence, the right-hand side of (3.9) tends to zero. On the other hand,

∫�

a|∇ρ|2ξ + a(1 − ρ)2ρ(1 + ρ)

ε2ξ ≥ C

∫�\⋃i Bi

a

[|∇ρ|2 + (1 − ρ2)2

ε2

]

from which we complete the proof this lemma. �

Proof of Proposition 3.1 Since⋃

i B(bi,2ri) ⊂ Eδ , above lemma yields the conver-gence to zero in L1(� \ Eδ) of the second term of the right-hand side of (3.7), aswell as that of the term a

ε2h2ex

(1 − |u|2)2δij . hhex

converges to h∞ weakly in H 1, and

strongly in L2(�), hence the term hhex2h2

exδij converges to h∞

2 δij in L2(�). For the third

term, we use ‖ 1|u|2 − 1‖L∞(Eδ) → 0 (since |u| ≥ 1 − o(1) in � \⋃i Bi ), and combine

it with the result of Proposition 2.1, we have

1

ah2ex|u|2

((∂2h)2 − (∂1h)2 −2∂1h∂2h

−2∂1h∂2h (∂1h)2 − (∂2h)2

)

→ 1

a

((∂2h∞)2 − (∂1h∞)2 −2∂1h∞∂2h∞

−2∂1h∞∂2h∞ (∂1h∞)2 − (∂2h∞)2

)(3.10)

strongly in L1(� \ Eδ). This completes the proof of Proposition 3.1. �

Proposition 3.2 We have that, ∀i,

∑j

∂jLij = 1

2|∇h∞|2∂i

(1

a

)in �,

where Lij was defined in (3.3), which imply that h∞ is a critical point with respectto domain-diffeomorphisms for L(h).

Proof For any ξ ∈ C∞0 (�), denote γt = {x ∈ �: ξ(t) = t} the level-sets of ξ . Since

∑j

∂j Tij = −1

2

[|∇A0u|2 + 1

2ε2

(1 − |u|2)2]∂ia,

we have ∫�

1

2

[|∇A0u|2 + 1

2ε2

(1 − |u2|)2]∂ia · ξ =

∫�

∑j

∂jTij · ξ.


By the co-area formula, we have

∀i,

∫�

Lij · ∇ξ =∫

t

(∫γt

Lij · ν)

dt. (3.11)

Recall that 1 − cap(Eδ) < δ, where Eδ is given in Proposition 3.1, and the 1 − capcontrols the perimeter, hence

meas{t : γt ∩ Eδ �= ∅} ≤ Cper(Eδ) < oδ(1). (3.12)

Using the co-area formula and Proposition 3.1 again, ∀i,∣∣∣∣∫

t/γt∩Eδ=∅

(∫γt

(Lij − Tij

h2ex

)· ν)

dt

∣∣∣∣=∣∣∣∣∫

x∈�/ξ(x)/∈ξ(Eδ)

(Lij − Tij

h2ex

)· ∇ξ

∣∣∣∣≤∫


∣∣∣∣Lij − Tij

h2ex

∣∣∣∣|∇ξ |

≤ ‖∇ξ‖L∞(�)

∫�\Eδ

∣∣∣∣Lij − Tij

h2ex

∣∣∣∣→ 0. (3.13)

Thus, ∀δ > 0, by Proposition 2.1 and the proof of Proposition 3.1,∫t/γt∩Eδ=∅

(∫γt

Lij · ν)

dt

=∫

t/γt∩Eδ=∅

(∫γt

Tij

h2ex

· ν)

dt + oδ(1)

=∫


Tij

h2ex

· ∇ξ + oδ(1)

= 1

h2ex

∫x∈�/ξ(x)/∈ξ(Eδ)

1

2

[|∇A0u|2 + 1

2ε2

(1 − |u|2)2]∂ia · ξ + oδ(1)

→∫


−1

2|∇h∞|2∂i

(1

a

)· ξ + oδ(1), (3.14)

where oδ(1) → 0 as δ → 0. Set f (t) = ∫γt

Lij ·ν + 12 |∇h∞|2∂i(

1a)ξ 1

|∇ξ | , then f ∈ L1

because Lij ∈ L1:∫t

∣∣f (t)∣∣dt =

∫t

∣∣∣∣∫

γt

Lij · ν + 1

2|∇h∞|2∂i

(1

a

)ξ

1

|∇ξ |∣∣∣∣

≤∫

t

∫γt

|Lij | + 1

2|∇h∞|2

∣∣∣∣∂i

(1

a

)∣∣∣∣|ξ | 1

|∇ξ |

≤∫

�

|Lij ||∇ξ | + 1

2|∇h∞|

∣∣∣∣∂i

(1

a

)∣∣∣∣|ξ | < ∞. (3.15)


By (3.14), (3.15), we obtain∫t

f (t)1t/γt∩Eδ=∅dt = 0, ∀δ > 0. (3.16)

From (3.12), 1t/γt∩Eδ=∅ → 1 almost everywhere, as δ → 0. By Lebesgue’s domi-nated convergence theorem, letting δ → 0, we get∫

t

f (t)dt = 0. (3.17)

Combining (3.11) with (3.17), we obtain∫�

Lij · ξ = −1

2

∫�

|∇h∞|2∂i

(1

a

)ξ. (3.18)

This is true for any ξ ∈ C∞0 (�), thus

∂jLij = 1

2|∇h∞|2∂i

(1

a

).

This completes the proof of Proposition 3.2. �

4 Properties for h∞ and μ∞

Recall that

L(h) = 1

2

∫�

1

a|∇h|2 + h (4.1)

over H 11 (�), we have the following result.

Proposition 4.1 h∞ is critical with respect to domain diffeomorphisms, and, if∇h∞ ∈ C0(�), [

μ∞ + 1

2∇(

1

a

)∇h∞

]|∇h∞|2 = 0.

Proof The first assertion follows from Proposition 3.2. Let us set ht (x) =h∞(x + tX(x)), at (x) = a(x + tX(x)) where X(x) ∈ C∞

0 (�,R2). If ∇h∞ is con-

tinuous, then

limt→0

h∞(x + tX(x)) − h∞(x)

t= ∇h∞(x) · X uniformly in x. (4.2)

Therefore,

ht (x) = h∞(x) + t∇h∞ · X(x) + o(t).


We now perform variations in L with this family of functions

L(ht ) − L(h) = 1

2

∫�

1

a

[|∇ht |2 − |∇h∞|2]+ [ht − h∞] +(

1

at

− 1

a

)|ht |2,

d

dt

∣∣∣∣t=0

L(ht ) =∫

�

(−div

(1

a∇h∞

)+ 1

)∇h∞ · X + 1

2|∇h∞|2∇

(1

a

)· X,

(4.3)

where we have used the fact that −div( 1a∇h∞) + 1 is a measure, ht is C0, hence

this integral and the integration by parts have a meaning. Since h∞ is stationary withrespect to domain-diffeomorphisms, d

dt|t=0L(ht ) = 0, and thus∫

�

[(−div

(1

a∇h∞

)+ 1

)∇h∞ + 1

2|∇h∞|2∇

(1

a

)]· X = 0,

∀X ∈ C∞0

(�,R

2),that is (

−div

(1

a∇h∞

)+ 1

)∇h∞ + 1

2|∇h∞|2∇

(1

a

)= 0.

Hence [(−div

(1

a∇h∞

)+ 1

)∇h∞ + 1

2|∇h∞|2∇

(1

a

)]· ∇h∞

=[μ∞ + 1

2∇(

1

a

)∇h∞

]|∇h∞|2 = 0.

The conclusion follows. �

Proposition 4.2 If ∇h∞ ∈ C0(�) ∩ BV(�), then h∞ and μ∞ satisfy the additionalproperties

μ∞ = 1|∇h∞|=0.

Here μ∞ is a positive measure, absolutely continuous with respect to the Lebesguemeasure.

Proof From Lemma 4.2 in [12], there exists sn → 0 such that sn|∂{|∇h∞| <

sn}| → 0. Consider such a sequence sn such that sn|∂{|∇h∞| < sn}| → 0, and take aξ ∈ C∞

0 (�). By Proposition 4.1,(μ∞ + 1

2∇(

1

a

)∇h∞

)1|∇h∞|>0 = 0,

hence we have ∫�

(μ∞ + 1

2∇(

1

a

)∇h∞

)ξ

=∫

|∇h∞|<sn

(μ∞ + 1

2∇(

1

a

)∇h∞

)ξ


= −∫

∂{|∇h∞|<sn}1

a

∂h∞∂n

ξ

+∫

|∇h∞|<sn

1

a∇h∞∇ξ + 1

a∇(

1

a

)∇h∞ξ + ξ. (4.4)

Note that ∣∣∣∣∫

∂{|∇h∞|<sn}1

a

∂h∞∂n

ξ

∣∣∣∣≤ C‖ξ‖L∞sn∣∣∂{|∇h∞| < sn}

∣∣→ 0 (4.5)

and ∣∣∣∣∫

|∇h∞|<sn

1

a∇h∞∇ξ

∣∣∣∣ ≤ Csn‖∇ξ‖L∞ → 0, (4.6)

∣∣∣∣∫

|∇h∞|<sn

∇(

1

a

)∇h∞ξ

∣∣∣∣ ≤ Csn‖ξ‖L∞ → 0. (4.7)

Letting n → ∞ in (4.4), we have∫�

(μ∞ + 1

2∇(

1

a

)∇h∞

)ξ =

∫|∇h∞|=0

ξ, ∀ξ ∈ C∞0 (�). (4.8)

We conclude that μ∞ = 1|∇h∞|=0, hence μ∞ is a positive measure, absolutely con-tinuous with respect to the Lebesgue measure. �

5 Conclusion

In this paper, we examine the asymptotic behavior of the vorticity-measures associ-ated to the vortices of the solution that describe the distribution of vortices in a simplyconnected domain. For a non-simply domain, up to now, one does not understand theeffect of domain topology to the vorticity-measure. This is a open question.

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Vorticity measure of type-II superconducting thin films

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