Dos Mickaël

Santos
email: dossantos@math.univ-lyon1.fr

Oleksandr Misiats
email: misiats@math.psu.edu

Ginzburg-Landau model with small pinning domains

Keywords: AMS Subject Classification: 49K20, 35J66, 35J50. 1 . . . . . . .

We consider a Ginzburg-Landau type energy with a piecewise constant pinning term a in the potential (a 2 -|u| 2 ) 2 . The function a is different from 1 only on finitely many disjoint domains, called the pinning domains. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit ε → 0; here, ε is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain Ω ⊂ C with Dirichlet boundary condition g on ∂Ω, with topological degree deg ∂Ω (g) = d > 0. Our main result is that, for small ε, minimizers have d distinct zeros (vortices) which are inside the pinning domains and they have a degree equal to 1. The question of finding the locations of the pinning domains with vortices is reduced to a discrete minimization problem for a finite-dimensional functional of renormalized energy. We also find the position of the vortices inside the pinning domains and show that, asymptotically, this position is determined by local renormalized energy which does not depend on the external boundary conditions.

Introduction and main results

In this work we study the minimizers of the Ginzburg-Landau type functional

E ε,δ (u) = 1 2 Ω |∇u| 2 + 1 2ε 2 (a 2 δ -|u| 2 ) 2 , (1)
where Ω ⊂ C is a bounded, smooth, simply connected domain, ε is a positive parameter (the inverse of the Ginzburg-Landau parameter κ = 1/ε), δ = δ(ε) > 0 is a geometric parameter and u is a complex-valued map. In order to define the function a δ , we need to introduce the notion of a pinning domain.

Fix M ∈ N * points a 1 , ..., a M ∈ Ω. Let ω be an open subset such that ω ⊂ B(0, 1) and 0 ∈ ω. For 1 ≤ i ≤ M and for all δ > 0 denote ω i δ := a i + δ • ω, i.e. the set ω scaled by δ and centered at a i . Definition. The set ω δ := ∪ M i=1 ω i δ is called a pinning domain. For example, if ω = B(0, 1 2 ), then the pinning domain is ω δ = ∪ M i=1 B(a i , δ 2 ).

We now define a δ : Ω → {b, 1}, b ∈ (0, 1) as:

a δ (x) = b if x ∈ ω δ 1 if x ∈ Ω \ ω δ . (2)
The functionals of this type arise in models of superconductivity for composite superconductors. The experimental pictures suggest nearly 2D structure of parallel vortex tubes ( [START_REF] Newton | Vortex lattice theory: A particle interaction perspective[END_REF], Fig I .4). Therefore, the domain Ω can be viewed as a cross-section of a multifilamentary wire with a number of thin superconducting filaments. Such multifilamentary wires are widely used in industry, including magnetic energy-storing devices, transformers and power generators [START_REF] Larbalestier | High-T c superconducting material for electric power applications[END_REF], [START_REF] Glowacki | Superconducting-magnetic heterostructures: a method of decreasing AC losses and improving critical current density in multifilamentary conductors[END_REF].

Another important practical issue in modeling superconductivity is to decrease the energy dissipation in superconductors. Here, the dissipation occurs due to currents associated with the motion of vortices ( [START_REF] Lin | Ginzburg-Landau vortices, dynamics, pinning and hysteresis[END_REF], [START_REF] Bardeen | Theory of the Motion of Vortices in Superconductors[END_REF]). This dissipation as well the thermomagnetic stability can be improved by pinning ("fixing the positions") of vortices. This, in turn, can be done by introducing impurities or inclusions in the superconductor. In the functional [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF] the set ω δ models the set of small impurities in a homogeneous superconductor. The size of the impurities in our model is characterized by the geometric parameter δ which goes to zero together with the material parameter ε. We assume henceforth that

| ln δ(ε)| 3 | ln ε| → 0. (H)
Essentially, this condition means that δ is much larger than ε on the logarithmic scale. For example, if ε = 2 -j and δ(ε) = 2 -k(j) , then (H) implies that k(j) 3 j → 0.

Notation. In what follows:

• We consider a sequence ε n ↓ 0 and we write ε instead of ε n ; the dependence of ε on n is implicit.

• We simply write δ (instead of δ(ε)); the dependence of δ on ε is implicit.

We study the minimization problem for the functional [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF] in the class

H 1 g := {u ∈ H 1 (Ω, C) | tr ∂Ω u = g}, (3)
where g ∈ C ∞ (∂Ω, S 1 ) is such that deg ∂Ω (g) = d > 0. Recall that the degree (winding number) of g is defined as

deg ∂Ω (g) := 1 2π ∂Ω g × ∂ τ g dτ.
Here "×" stands for the vectorial product in C, i.e. z 1 × z 2 = Im(z 1 z 2 ), z 1 , z 2 ∈ C, and ∂ τ is the tangential derivative. The degree is an integer, and the condition deg ∂Ω (u) = d > 0, u ∈ H 1 (Ω, C) implies that u must have at least d zeros (counting multiplicity) inside Ω. The properties of the topological degree can be found, e.g., in [START_REF] Brezis | New questions related to the topological degree[END_REF] or [START_REF] Berlyand | Ginzburg-Landau minimizers in perforated domains with prescribed degrees[END_REF].

Minimization problems for Ginzburg-Landau type functionals have been extensively studied by a variety of authors. The pioneering work on modeling Ginzburg-Landau vortices is the work of Bethuel, Brezis and Hélein [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF]. In this work the authors suggested to consider a simplified Ginzburg-Landau model [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF] with a ≡ 1 in Ω (i.e. without pinning term), in which the physical source of vortices, the external magnetic field, is modeled via a Dirichlet boundary condition with a positive degree on the boundary [START_REF] André | Vortex pinning with bounded fields for the Ginzburg-Landau equation[END_REF]. The analysis of full Ginzburg-Landau functional, with induced and applied magnetic fields, was later performed by Sandier and Serfaty in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF].

The functional [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF] with non-constant a(x) was proposed by Rubinstein in [START_REF] Rubinstein | On the equilibrium position of Ginzburg Landau vortices[END_REF] as a model of pinning vortices for Ginzburg-Landau minimizers. Shortly after, André and Shafrir [START_REF] André | Asymptotic behavior of minimizers for the Ginzburg-Landau functional with weight. I[END_REF] studied the asymptotics of minimizers for a smooth (say C 1 ) a. One of the first works to consider a discontinuous pinning term, which models a composite two-phase superconductor, was [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF]. In this work, a single inclusion described by a pinning term independent of the parameter ε was considered for a simplified Ginzburg-Landau functional with Dirichlet boundary condition g on ∂Ω. Namely the pinning term is

a(x) = 1 if x ∈ Ω \ ω b if x ∈ ω ,
here ω is a simply connected open set s.t. ω ⊂ Ω. The main objective of [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF] was to establish that the vortices are attracted (pinned) by the inclusion ω, and their location inside ω can be obtained via minimization of certain finite-dimensional functional of renormalized energy. Full Ginzburg-Landau model with discontinuous pinning term was later considered by Aydi and Kachmar [START_REF] Aydi | Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint[END_REF]. An ε-dependent but continuous pinning term a ε (x) was studied by Aftalion, Sandier and Serfaty in [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF]. The work [START_REF] André | Vortex pinning with bounded fields for the Ginzburg-Landau equation[END_REF] studies the case of the smooth a with finite number of isolated zeros, and in [START_REF] Alama | Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions[END_REF] the pinning term a takes negative values in some regions of the domain Ω. The other works related to Ginzburg-Landau functional with pinning term include, e.g., [START_REF] Lin | Ginzburg-Landau vortices, dynamics, pinning and hysteresis[END_REF], [START_REF] Sigal | Pinning of magnetic vortices by an external potential[END_REF].

In this work, we consider the minimization problem (1)-( 3) with a discontinuous pinning term given by [START_REF] Alama | Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions[END_REF]. We prove that despite the fact that a ε → 1 a.e. as ε → 0, i.e. the pinning term disappears in the limit, the pinning domains ω δ capture the vortices of Ginzburg-Landau minimizers of (1) for small ε.

The main difficulty in the analysis of this problem stems in the fact that the a priori Pohozhaev type estimate 1 -|v| 2 2 L 2 (Ω) ≤ Cε 2 for the minimizer v (on which the analysis in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] and [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF] is based) does not hold. Therefore, we develop a different strategy of reducing the study of the minimizers of (1) to the analysis of S 1 -valued maps via the uniform estimates on the modulus of minimizers away from the pinning domains (see Proposition 5 below).

Following [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], let U ε be the unique global minimizer of

E ε in H 1 with U ε ≡ 1 on ∂Ω. This U ε satisfies b ≤ U ε ≤ 1. For v ∈ H 1 g we define F ε (v) = F ε (v, Ω) := 1 2 Ω U 2 ε |∇v| 2 + 1 2ε 2 U 4 ε (1 -|v| 2 ) 2 dx.
Using the Substitution Lemma of [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], we have that for v ∈ H 1 g ,

E ε (U ε v) = E ε (U ε ) + F ε (v). ( 4
)
From the decomposition (4), we can reduce the minimization problem ( 1)-( 3) to the minimization problem for F ε in H 1 g , namely, the minimizer v ε of F ε in H 1 g has the same vorticity structure as the original minimizer u ε of ( 1)- [START_REF] André | Vortex pinning with bounded fields for the Ginzburg-Landau equation[END_REF].

Depending on the relation between M (number of inclusions), and d (number of vortices), we distinguish two cases: Case I: M ≥ d (more inclusions than vortices), Case II: M < d (more vortices than inclusions).

For example, we are going to show that for the minimizer v ε :

• if M = 3 and d = 2 (Case I), we have two distinct inclusions containing exactly one zero each,

• if M = 2 and d = 3 (Case II), we have one zero inside one inclusion and two distinct zeros inside the other inclusion.

Generally speaking, outside a fixed neighborhood of d ′ = min {d, M } inclusions (centered at a = (a i 1 , ..., a i d ′ )), the minimizer v ε is almost an S 1 -valued map. Moreover, by minimality of v ε , the selection of centers of inclusion containing its zeros and the distribution of degrees of v ε around the a i 's are related to the minimization of the Bethuel-Brezis-Hélein renormalized energy W g . In other words, we reduce the problem of finding vortices of the minimizers v ε to a two-step procedure. As the first step, we determine the inclusions with vortices, which is a discrete minimization problem for W g and is significantly simpler then the minimization of this renormalized energy functional over Ω d ′ . Secondly, we determine the locations of the zeros (vortices) locally inside each inclusion and show that their positions depend only on b, on the geometry of ω and on the relation between d and M , but not on the external Dirichlet boundary condition g (see Theorem 4 below).

Our main result in Case I is the following:

Theorem 1. Assume that M ≥ d. Let v ε be a minimizer of F ε in H 1 g (Ω)
. For any sequence ε n ↓ 0, possibly after passing to a subsequence, there are

d distinct points {a i 1 , ..., a i d } ⊂ {a i , 1 ≤ i ≤ M } and a function v * ∈ H 1 loc (Ω \ {a i 1 , ..., a i d }, S 1
) such that:

1. v * is a harmonic map, i.e.

-∆v

* = v * |∇v * | 2 in Ω \ {a i 1 , ..., a i d } v * = g on ∂Ω .
(5)

We have

v εn → v * strongly in H 1 loc (Ω \ {a i 1 , ..., a i d }) and v εn → v * in C ∞ loc (Ω \ {a 1 , ..., a M }). 3. v εn has d distinct vortices x n 1 , ..., x n d such that x n m is inside ω im δ , m = 1, ..., d and for small fixed ρ, deg ∂B(x n i ,ρ) (v εn ) = 1.

The following expansion holds

F ε (v ε ) = πdb 2 | ln ε| + π(1 -b 2 )d| ln δ| + W g ((a i 1 , 1), ..., (a i d , 1)) + W + o ε (1). ( 6
)
Here W > 0 is a local renormalized energy depending only on d, b and ω. Moreover, the dsubset {a i 1 , ..., a i d } ⊂ {a 1 , ..., a M } minimizes the Bethuel-Brezis-Hélein renormalized energy W g among the d-subsets of {a 1 , ..., a M }.

Remark 1. Here, W g denotes the renormalized energy given by Theorem I.7 in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] (with the degrees equal to 1 and the boundary data g). Its definition is recalled in Section 5.3.

The main result in Case II is

Theorem 2. Assume that M < d. Let v ε be a minimizer of F ε in H 1 g (Ω)
. For any sequence ε n ↓ 0, possibly after passing to a subsequence, there is v * ∈ H 1 loc (Ω \ {a 1 , ..., a M }, S 1 ) which satisfies (5) in Ω \ {a 1 , ..., a M }, such that:

1. v εn → v * strongly in H 1 loc (Ω \ {a 1 , ..., a M }) and v εn → v * in C ∞ loc (Ω \ {a 1 , ..., a M }).
2. For ρ > 0 small, v εn has exactly

d i := deg ∂B(a i ,ρ) (v εn ) zeros in B(a i , ρ).
They are isolated, lie inside ω i δ and they have a degree equal to 1.

3. d M ≤ d i ≤ d M + 1, where d M is the integer part of d M . ( 7
)
Moreover, if d M = m 0 ∈ N, then d i ≡ m 0 , 1 ≤ i ≤ M .
Otherwise, the configuration {(a 1 , d 1 ), ...,(a M , d M )} minimizes the renormalized energy W g among the configurations {(a 1 , d1 ), ..., (a M , dM )}. Here {a i | 1 ≤ i ≤ M } are fixed and di ∈ Z are the subjects to the constraints [START_REF] Bauman | On the zeros of solutions to Ginzburg-Landau type systems[END_REF] and M i=1 di = d.

The following expansion holds when

ε → 0 inf H 1 g F ε = πdb 2 | ln ε| + π( M i=1 d 2 i -db 2 )| ln δ| + W g ({a, d}) + W + o ε (1). ( 8
)
Here, {a, d} = {(a 1 , d 1 ), ..., (a M , d M )} is a configuration given by the previous assertion and W is local renormalized energy which depends only on ω, b, d and M .

In both cases, we prove that the asymptotic location of the vortices inside a pinning domain depends only on b, ω and on the number of zeros inside the inclusion (see Theorem 4): this location is independent of the boundary data g on ∂Ω.

Main tools

In this section we establish:

• Estimates for U ε ,
• Upper bounds for the energy of minimizers in Case I and Case II,

• An η-ellipticity estimate for minimizers.

Properties of U

ε Proposition 1 (Maximum principle for U ε , [18] Proposition 1). The special solution U ε satisfies b ≤ U ε ≤ 1 in Ω.
Proposition 2. There are C, c > 0 (independent of ε) s.t. for any R > 0 we have

|a ε -U ε | ≤ Ce -cR ε in V R := {x ∈ Ω | dist(x, ∂ω δ ) ≥ R}, (9)
|∇U ε | ≤ Ce -cR ε ε in V R . ( 10
)
The proof of the Proposition 2 is presented in the Appendix A.

Upper Bounds

Proposition 3. Let ξ = ε δ .

Upper bound in Case I: M ≥ d

There is a constant C depending only on g, ω and Ω s.t. we have inf

H 1 g (Ω) F ε (•, Ω) ≤ πdb 2 | ln ξ| + πd| ln δ| + C. ( 11
)

Upper bound in Case II: M < d

There is a constant C depending only on g, ω and Ω s.t. for all d 1 , ..., d M ∈ N s.t.

d i = d we have inf H 1 g (Ω) F ε (•, Ω) ≤ πdb 2 | ln ξ| + π i d 2 i | ln δ| + C. (12)
The proof of Proposition 3 is given in Appendix B.

Identifying bad discs

Lemma 1. Let g ε , g 0 ∈ C ∞ (∂Ω, C) be s.t. 0 ≤ 1 -|g ε | ≤ ε and g ε → g 0 in C 1 (∂Ω). Let also α ε , β ε ∈ L ∞ (Ω, [b, 1]).
Consider the weighted Ginzburg-Landau functional

F w ε (v) = 1 2 Ω α ε |∇v| 2 + β ε ε 2 (1 -|v| 2 ) 2 .
Denote v ε a minimizer of F w ε in H 1 gε . Then the following results hold:

1. Let χ = χ ε ∈ (0, 1) be s.t. χ → 0. There are ε 0 > 0, C > 0 and C 1 > 0 depending only on b, χ, Ω, g 0 C 1 (∂Ω) s.t for ε < ε 0 , if F w ε (v ε , B(x, ε 1/4 ) ∩ Ω) ≤ χ 2 | ln ε| -C 1 , then |v ε | ≥ 1 -Cχ in B(x, ε 1/2 ) ∩ Ω.
2. Let µ ∈ (0, 1). Then there are ε 0 , C > 0 depending only on b, µ, Ω,

g 0 C 1 (∂Ω) s.t. for ε < ε 0 , if F w ε (v ε , B(x, ε 1/4 ) ∩ Ω) ≤ C| ln ε|, then |v ε | ≥ µ in B(x, ε 1/2 ) ∩ Ω.
Lemma 1 is proved in Appendix C.

A model problem: one inclusion

By combining the results of Section 2, the proofs of both Theorem 1 and Theorem 2 are based on the analysis of two distinct problems:

1. A minimization problem of the Dirichlet functional among S 1 -valued map defined on a perforated domain.

2. The study of the minimizers v ε around an inclusion.

This section focuses on the second problem. More precisely, we fix ρ > 0 and study the minimization problem of F ε (•, B(a i , ρ)) with variable boundary conditions. Fix ρ > 0 and let f ε , f 0 ∈ C ∞ (∂B(0, ρ)) be s.t. f 0 is S 1 -valued and s.t.

f ε -f 0 C 1 (∂B(0,ρ)) → 0 ( 13
)
and

|f ε | -1 L 2 (∂B(0,ρ)) ≤ Cε 2 . ( 14
)
Assume also that deg

∂B(0,ρ) (f ε ) = deg ∂B(0,ρ) (f 0 ) = d 0 > 0.
For i ∈ {1, ..., M } consider the minimization problem

F ε (v, B(a i , ρ)) := 1 2 B(a i ,ρ) U 2 ε |∇v| 2 + 1 2ε 2 U 4 ε (1 -|v| 2 ) 2 dx ( 15
)
in the class

H 1 fε,i := {v ∈ H 1 (B(a i , ρ), C) | tr ∂B(a i ,ρ) v(x) = f ε (x -a i )}. ( 16
)
Without loss of generality assume a i = 0. Let v ε be a minimizer of [START_REF] Santos | The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part I: the zero degree case[END_REF] in [START_REF] Glowacki | Superconducting-magnetic heterostructures: a method of decreasing AC losses and improving critical current density in multifilamentary conductors[END_REF]. Performing the change of variables x = x δ in (15), we have

F ε (v ε , B(0, ρ)) = Fξ (v ε , B(0, ρ δ )) := 1 2 B(0, ρ δ ) Û 2 ε |∇v| 2 + 1 2ξ 2 Û 4 ε (1 -|v| 2 ) 2 dx. ( 17
)
Here, for a map w ∈ H 1 (B(0, ρ)), we denote ŵ(x) := w(δx) and ξ = ε δ . The class [START_REF] Glowacki | Superconducting-magnetic heterostructures: a method of decreasing AC losses and improving critical current density in multifilamentary conductors[END_REF] under this change of variables becomes

Ĥ1 fε := v ∈ H 1 (B(0, ρ δ ), C) | tr ∂B(0, ρ δ ) v(•) = f ε (δ•) . (18)
Note that the above rescaling enables us to fix the pinning domain independently of ε.

The asymptotic behavior of vε will be obtained in several steps:

• We first establish a bound for |v ε | (Proposition 5). This bound will allow us to localize (roughly) the vortices of v ε near the inclusion.

• We next establish sharp energy estimates (Proposition 6) and use them to obtain the uniform convergence of solutions away from the inclusion (Proposition 7 and Corollary 2). We establish the strong H 1 convergence of solutions away from the "vortices" (Proposition 8) and derive the equation satisfied by the limiting map (Proposition 10).

• The last step is the location and quantization of the vorticity: for small ε, the minimizers admits exactly d 0 zeros, and all the zeros lie in the inclusion and have a degree equal to 1 (Propositions 8 and 11).

Following the same lines as for Proposition 3, one may prove Proposition 4. Let vε be a minimizer of Fξ in [START_REF] Glowacki | Superconducting-magnetic heterostructures: a method of decreasing AC losses and improving critical current density in multifilamentary conductors[END_REF]. Then there is a constant C independent of ε s.t. we have

F ε (v ε , B(0, ρ)) = Fξ (v ε , B(0, ρ δ )) ≤ πd 0 b 2 | ln ξ| + πd 2 0 | ln δ| + C. (19)
3.1 Uniform convergence of |v ε | to 1 away from inclusions Proposition 5. Let K ⊂ R 2 be a compact set such that ω ⊂ K and dist(∂K, ω) > 0. Then there is C > 0 independent of ε s.t. for sufficiently small ε we have

|v ε | ≥ 1 -C| ln ε| -1/3 in B ρ δ \ K.
Proof. Using Lemma 1 with χ = | ln ε| -1/3 , we find that there exist

C, C 1 > 0 s.t. for ε > 0 small, if F ε (v ε , B(x, ε 1/4 )) < | ln ε| 1 3 -C 1 then |v ε | ≥ 1 -Cχ in B(x, ε 1/2
). We argue by contradiction. Assume that there exists a compact K containing ω s.t. dist(∂K, ω) > 0 and s.t., up to a subsequence, there is a sequence of points xε ∈ B(0

, ρ δ ) \ K s.t. |v ε (x ε )| < 1 -C| ln ε| -1/3 with C given by Lemma 1. Note that xε ∈ B(0, ρ δ ) \ K corresponds to x ε ∈ B(0, ρ) \ (δ • K). From Lemma 1 and Proposition 2 1 2 B(xε,ε 1/4 ) |∇v ε | 2 + 1 2ε 2 (1 -|v ε | 2 ) 2 ≥ | ln ε| 1/3 -O(1). (20)
We claim that due to the conditions ( 13), ( 14), we may extend v ε (keeping the same notation for the extension) to a smooth map, still denoted v ε , s.t.

         v ε (x) = x d 0 /|x| d 0 in B(0, 3ρ) \ B(0, 2ρ) B(0,3ρ)\B(0,ρ) (1 -|v ε | 2 ) 2 ≤ Cε 2 |∇v ε | ≤ C with C > 0 is independent of ε . (21)
To make the above extension explicit, choose

ζ ∈ C ∞ (R + , [0, 1]) s.t. ζ ≡ 0 in [0, ρ] and ζ ≡ 1 in [2ρ, 3ρ
] and take

v ε (se ıθ ) = ζ(s) + (1 -ζ(s))|f ε (ρe ıθ )| e ı[d 0 θ+(1-ζ(s))φε(ρe ıθ )] .
Here x = se ıθ , s > 0 and φ ε ∈ C ∞ (∂B(0, ρ), R) is s.t. f ε (ρe ıθ ) = |f ε |e ı(d 0 θ+φε) . Consequently, as follows from [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] and [START_REF] Lin | Ginzburg-Landau vortices, dynamics, pinning and hysteresis[END_REF], this map satisfies

1 2 B(0,3ρ) |∇v ε | 2 + 1 2ε 2 (1 -|v ε | 2 ) 2 ≤ C| ln ε|.
Therefore, the map v ε in B(0, 3ρ) fulfills the conditions of Theorem 4.1 in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF]. This theorem guarantees that:

• we may cover the set {x ∈ B(0, 3ρ-ε/b) | |v ε (x)| < 1-(ε/b) 1/8 } with a finite collection of disjoint balls B ε := {B ε j };
• the radius of B ε , rad(B ε ), which is defined as the sum of the radii of the balls B ε j , rad(B ε ) := j rad(B ε j ), satisfies rad(B ε ) ≤ 10 -2 δ • dist(ω, ∂K);

• denoting d j = deg ∂B ε j (v ε ) if B ε j ⊂ B(0, 3ρ -ε/b) and d j = 0 otherwise; we have 1 2 B ε |∇v ε | 2 + b 2 2ε 2 (1 -|v ε | 2 ) 2 ≥ π j |d j | ln δ ε -C. ( 22
)
Note that, by the construction of

v ε in B(0, 3ρ) \ B(0, ρ), if we have deg ∂B ε j (v ε ) = 0 then B ε j ⊂ B(0, 5ρ/2). Thus d j = deg ∂B ε j (v ε ) for all j.
In order to obtain a lower bound for F ε we use the identity

F ε (v ε , B(x ε , ε 1/4 ) ∪ B ε ) = b 2 2 B(xε,ε 1/4 )∪B ε |∇v ε | 2 + b 2 2ε 2 (1 -|v ε | 2 ) 2 (23) + 1 2 B(xε,ε 1/4 )∪B ε (U 2 ε -b 2 )|∇v ε | 2 + 1 2ε 2 (U 4 ε -b 4 )(1 -|v ε | 2 ) 2 .
The first integral in ( 23) is estimate via ( 22):

b 2 2 B(xε,ε 1/4 )∪B ε |∇v ε | 2 + b 2 2ε 2 (1 -|v ε | 2 ) 2 ≥ πb 2 j |deg ∂B j (v ε )| ln δ ε -C ≥ πb 2 d 0 ln δ ε -C 0 . (24)
By combining [START_REF] Lefter | On the Ginzburg-Landau energy with weight[END_REF] and Proposition 2, we have for small ε

1 2 B(xε,ε 1/4 )∪B ε (U 2 ε -b 2 )|∇v ε | 2 + 1 2ε 2 (U 4 ε -b 4 )(1 -|v ε | 2 ) 2 ≥ 1 -b 2 2 B(xε,ε 1/4 ) |∇v ε | 2 + 1 2ε 2 (1 -|v ε | 2 ) 2 -C ≥ (1 -b 2 )| ln ε| 1/3 -C ′ ; (25)
here we rely on the assumption (H) on the behavior of δ(ε) as ε → 0. Substituting the bounds ( 24) and ( 25) in ( 23) we obtain a contradiction with [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF]. This completes the proof of Proposition 5.

Distribution of Energy in

B(0, ρ δ ) Proposition 6.
The following estimates hold:

1 2 B(0,ρ/δ)\B(0,1) Û 2 ε |∇v ε | 2 = πd 2 0 | ln δ| + O(1), (26)
and

(recall that ξ = ε δ ) Fξ (v ε , B(0, 1)) = πd 0 b 2 | ln ξ| + O(1). (27)
Proof. We start by proving that

Fξ (v ε , B(0, 1)) ≥ πd 0 b 2 | ln ξ| -O(1). (28)
As before, we use Theorem 4.1 in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF]: for 0 < r < r 0 := 10 -2 • dist(ω, ∂B(0, 1)), there are C > 0 and a finite covering by disjoint balls B ε 1 , ..., B ε N (with the sum of radii at most r) of the set

{x ∈ B(0, 1 -ξ/b) | 1 -|v ε (x)| ≥ (ξ/b) 1/8 } s.t. 1 2 ∪ j B ε j |∇v ε | 2 + b 2 2ξ 2 (1 -|v ε | 2 ) 2 ≥ πD| ln ξ| -C, (29)
with D = j |d j | and

d j = deg ∂B ε j (v ε ) if B ε j ⊂ B(0, 1 -ξ/b) 0 otherwise . From Proposition 5, for ε small, if deg ∂B ε j (v ε ) = 0 then B ε j ⊂ B(0, 1 -r 0 ) ⊂ B(0, 1 -ξ/b).
It follows that D ≥ d 0 and then ( 28) is a direct consequence of (29) and the bound Ûε ≥ b.

We next prove that there is C > 0 s.t.

1 2 B(0, ρ δ )\B(0,1) Û 2 ε |∇v ε | 2 ≥ πd 2 0 | ln δ| -C. (30)
By Proposition 5,

|v ε | ≥ 1/2 in B(0, ρ δ ) \ B(0, 1), therefore, ŵε := vε |vε| is well-defined in this domain. Observe that 1 2 B(0, ρ δ )\B(0,1) |∇ ŵε | 2 ≥ 1 2 B(0, ρ δ )\B(0,1) ∇ z d 0 |z| d 0 2 = πd 2 0 ln ρ δ . (31)
We claim that (30) holds with C = πd 2 0 | ln ρ| + 1 (for small ε). By contradiction, assume (30) does not hold. Then, up to a subsequence, we have

1 2 B(0, ρ δ )\B(0,1) Û 2 ε |∇v ε | 2 < πd 2 0 ln ρ δ -1. ( 32
)
On the other hand, we have

|∇v ε | 2 = |v ε | 2 |∇ ŵε | 2 + |∇|v ε || 2
and therefore

B(0, ρ δ )\B(0,1) |∇v ε | 2 ≥ B(0, ρ δ )\B(0,1) |∇ ŵε | 2 - B(0, ρ δ )\B(0,1) (1 -|v ε | 2 )|∇ ŵε | 2 . ( 33
) Since |v ε | ≥ 1 2 in B(0, ρ δ ) \ B(0, 1) we have |∇ ŵε | ≤ 2|∇v ε |.
Therefore, by (32), Proposition 5 and (H) we estimate the last term in (33):

B(0, ρ δ )\B(0,1) (1 -|v ε | 2 )|∇ ŵε | 2 ≤ C 2 | ln ε| -1 3 B(0, ρ δ )\B(0,1) |∇v ε | 2 ≤ C 3 | ln δ| | ln ε| 1 3 → 0. ( 34
)
Combining (31), ( 33) and (34), we find that

B(0, ρ δ )\B(0,1) |∇v ε | 2 ≥ πd 2 0 ln ρ δ -o ε (1). Since | Ûε -1| ≤ Cξ 4 in B ρ δ \ B(0, 1) (see Proposition 2)
, we obtain a contradiction with (32), and (30) follows. Comparing the lower bounds ( 28) and (30) with the upper bound in Proposition 4, the Proposition 6 follows.

Using exactly the same techniques as in the proof of Proposition 6, one may easily prove the following estimate.

Corollary 1. For any R 2 > R 1 ≥ 1 F ξ (v ε , B(0, R 2 ) \ B(0, R 1 )) = O(1). 3.3 Convergence in C ∞ (K) for a compact K s.t. K ∩ ω = ∅ Proposition 7. Let K ⊂ R 2 \ ω be a smooth compact set. Then we have vε is bounded in C k (K) for all k ≥ 0 ( 35
)
and there is

C K > 0 s.t. |v ε | ≥ 1 -C K ξ 2 in K. ( 36
)
Proof. From Proposition 2

E ξ ( Ûε , K) = 1 2 K |∇ Ûε | 2 + 1 2ξ 2 (1 -Û 2 ε ) 2 = O(1). (37)
As in [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], the following expansion holds

E ξ ( Ûε vε , K) = E ξ ( Ûε , K) + Fξ (v ε , K) + ∂K (|v ε | 2 -1) Ûε ∂ ν Ûε . (38)
Using [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF], we have

∂K (|v ε | 2 -1) Ûε ∂ ν Ûε = o ε (1).
With (37) and ( 38), we conclude that E ξ ( Ûε vε , K) = O(1). Since Ûε and Ûε vε satisfy the Ginzburg-Landau equation -∆u

= 1 ξ 2 u(1 -|u| 2 ) in K, as well as | Ûε | ≤ 1 and | Ûε vε | ≤ 1. Theorem 1 in [23] implies that
Ûε and Ûε vε are bounded in C k (K) for all k ≥ 0.

It follows that vε is bounded in C k (K) for each k ≥ 0. On the other hand, using the fact that vε is bounded in C k (K) together with the equation of vε , we find that 1

-|v ε | 2 ≤ C K ξ 2 in K. Corollary 2. For K ⊂ R 2 \ ω, up to a subsequence, there is some v 0 ∈ C ∞ (K, S 1 ) s.t. vε → v 0 in C ∞ (K).
We are now in position to bound the potential part of the energy.

Corollary 3. There exists C > 0 independent of ε s.t.

1 ε 2 B(0,ρ) (1 -|v ε | 2 ) 2 = 1 ξ 2 B(0,ρ/δ) (1 -|v ε | 2 ) 2 ≤ C. ( 39
)
Proof. Note that from Propositions 4 and 6, we find that there is C > 0 s.t.

1 ξ 2 B(0,ρ/δ)\B(0,1) (1 -|v ε | 2 ) 2 ≤ C.
Thus it remains to prove the estimate in B(0, 1) for small ε. Using (35), tr ∂B(0,1) vε is bounded in C 1 (∂B(0, 1)) and 1 -|v ε | 2 ≤ Cξ 2 on ∂B(0, 1) (for small ε). These properties, allow us to construct a smooth extension ṽε of tr ∂B(0,1) vε into

B(0, 2) \ B(0, 1), s.t. h = tr ∂B(0,2) ṽε is S 1 -valued and independent of ε, 1 -|ṽ ε | 2 ≤ Cξ 2 in B(0, 2) \ B(0, 1)
and

B(0,2)\B(0,1) |∇ṽ ε | 2 + 1 2ξ 2 (1 -|ṽ ε | 2 ) 2 ≤ C 0 . (40)
(For example, this construction is performed by mimicking ( 21)) Define w ε as w ε = vε in B(0, 1) and

w ε = ṽε in B(0, 2) \ B(0, 1). Clearly, w ε ∈ H 1 h (B(0, 2)), w ε is bounded in L 2 (B(0, 2)
) and, thanks to Proposition 6 and (40),

1 2 B(0,2) |∇w ε | 2 + b 2 2ξ 2 (1 -|w ε | 2 ) 2 ≤ πd 0 | ln ξ| + C 0 .
We may now apply Proposition 0.1 in [START_REF] Del Pino | On the basic concentration estimate for the Ginzburg-Landau equation[END_REF] to

w ε in B(0, 2) to conclude that 1 ξ 2 B(0,2) (1 -|w ε | 2 ) 2 ≤ C 1 .
Therefore the bound (39) holds.

The bad discs

Consider a family of discs (B(x i , ε 1/4 )) i∈I such that (here I depends on ε)

for all i ∈ I we have x i ∈ Ω, B(x i , ε 1/4 /4) ∩ B(x i , ε 1/4 /4) = ∅ if i = j, ∪ i∈I B(x i , ε 1/4 ) ⊃ Ω.
For µ ∈ (1/2, 1), let C = C(µ), ε 0 = ε 0 (µ) be defined as in the second part of Lemma 1. For ε < ε 0 , we say that B(

x i , ε 1/4 ) is µ-good disc if F ε (v ε , B(x i , ε 1/4 ) ∩ Ω) ≤ C(µ)| ln ε| and B(x i , ε 1/4 ) is µ-bad disc if F ε (v ε , B(x i , ε 1/4 ) ∩ Ω) > C(µ)| ln ε|. ( 41
) Let J ε = J := {i ∈ I | B(x i , ε 1/4 ) is a µ-bad disc}.
Lemma 2. There is an integer N , which depends only on g and µ, s.t.

Card J ≤ N.

Proof. Since each point of Ω is covered by at most 16 discs B(x i , ε 1/4 ), we have

i∈I F ε (v ε , B(x i , ε 1/4 ) ∩ Ω) ≤ 16F ε (v ε , Ω).
The previous assertion implies that Card J ≤ 16C 0 C 1 (µ) .

The next result is a straightforward variant of Theorem IV.1 in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF].

Lemma 3. Possibly after passing to a subsequence and relabeling I, we may choose J ′ ⊂ J and a constant λ ≥ 1 (independently of ε) s.t.

J ′ = {1, ..., N ′ }, N ′ = Cst, |x i -x j | ≥ 8λε 1/4 for i, j ∈ J ′ , i = j and ∪ i∈J B(x i , ε 1/4 ) ⊂ ∪ i∈J ′ B(x i , λε 1/4 ).
We will say that, for i ∈ J ′ , B(x i , λε 1/4 ) are separated µ-bad discs. From now on, we work with separated µ-bad discs. Denote xi =

x i δ . By Proposition 5 we know that for small ε, we have xi ∈ B 1 . Clearly, up to a subsequence,

   there are α 1 , ..., α κ , κ distinct points in B 1 {Λ 1 , ..., Λ κ } a partition (in non empty sets) of J ′ s.t. for i ∈ J ′ , if i ∈ Λ k then xi → α k . (42)
Note that for i ∈ J ′ , we have

y ∈ {α 1 , ..., α κ } ⇐⇒ ∀ η > 0, for small ε,
there is a µ-bad disc inside B(y, η).

(43)

3.5 Convergence in H 1 loc (R 2 \ {α 1 , ..., α κ })
We have the following theorem.

Proposition 8. Let α 1 , ..., α κ be defined by (42). Then we have:

1. The points α 1 , ..., α κ belong to ω.

There exists

v 0 ∈ H 1 loc (R 2 \ {α 1 , ..., α κ }, S 1 ) s.t. (possibly after extraction) vε → v 0 in H 1 loc (R 2 \ {α 1 , ..., α κ }) (44) vε → v 0 in C 0 loc (R 2 \ {α 1 , ..., α κ }). ( 45
)
3. There exists η 0 > 0 s.t. for all 0 < η < η 0 and for sufficiently small ε we have

deg ∂Bη(α k ) (v ε /|v ε |) = deg ∂Bη 0 (α k ) (v 0 ) = 1. 4. κ = d 0 . Proof. Step 1: vε ⇀ v 0 in H 1 loc (R 2 \ {α 1 , ..., α κ }), v 0 ∈ H 1 loc (R 2 \ {α 1 , ..., α κ }, S 1 ) and α k ∈ ω Proposition 5 guarantees that α 1 , ..., α κ ∈ ω. Let η 0 = 10 -2 • min k =k ′ |α k -α k ′ | if κ > 1 1 if κ = 1 . (46)
Applying Theorem 4.1 in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] we have for all 0 < η < η 0 and for small ε

1 2 ∪ k∈{1,...,κ} B(α k ,η) |∇v ε | 2 + b 2 2ξ 2 (1 -|v ε | 2 ) 2 ≥ πd 0 ln η ξ -C (47)
with C independent of ε and η. Combining (47) with [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] and Corollary 1 we obtain that vε is bounded in H 1 (K); here K ⊂ R 2 \ {α 1 , ..., α κ } is an arbitrary compact set. Therefore, there exists

v 0 ∈ H 1 loc (R 2 \ {α 1 , ..., α κ }) s.t. we have vε ⇀ v 0 in H 1 loc (R 2 \ {α 1 , ..., α κ }) (possibly passing to a subsequence). Since 1 -|v ε | L 2 (K) → 0 for all compact sets K ⊂ R 2 \ {α 1 , ..., α κ }, we find that v 0 is S 1 -valued.
Following the proof of Step 7 in Theorem C in [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], we can prove that α 1 , ..., α κ / ∈ ∂ω, thus α 1 , ..., α κ ∈ ω, and the first assertion follows.

Step 2: Proof of 2. Adapting the techniques of [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF] (Theorem 2, Step 1), we establish (44) and (45) in a ball

B = B(y, R 0 ) s.t. B ⊂ R 2 \ {α 1 , ..., α κ }. Let y ∈ R 2 and let R ′ > R > 0 be s.t. B(y, R ′ ) ⊂ R 2 \ {α 1 , ..., α κ }. Since Fξ (v ε , B(y, R ′ )) is bounded independently on ε, there is R 0 ∈ (R, R ′ ) (independent of ε) s.t.
, passing to a further subsequence if necessary we have

∂B(y,R 0 ) |∂ τ vε | 2 + 1 ξ 2 (1 -|v ε | 2 ) 2 ≤ C with C independent of ε. (48)
Indeed, for r ∈ (R, R ′ ) denote

I ε (r) = ∂B(y,r) |∇v ε | 2 + 1 ξ 2 (1 -|v ε | 2 ) 2 .
Using the Fubini theorem and the Fatou Lemma we have

0 ≤ R ′ R lim inf ε I ε (r) dr ≤ lim inf ε R ′ R I ε (r) dr ≤ C ′ . Consequently, lim inf ε I ε (r) < ∞ for almost all r ∈ (R, R ′ ), so that (48) holds with C = C ′ R ′ -R . Let g ε = tr ∂B vε . Since |v ε | ≥ 1/2 in B = B(y, R 0 ), we have deg ∂B (g ε ) = 0.
The bound (48) implies that, up to choose a subsequence, g ε is weakly convergent in H 1 (∂B). Consequently there is h ∈ H 1 (∂B, S 1 ), h = e ıϕ , ϕ ∈ H 1 (∂B, R) s.t.

g ε → h uniformly on ∂B, (49)
g ε → h in H 1/2 (∂B). ( 50
)
Let η ε : B → R + be the minimizer of

B |∇η| 2 + 1 ξ 2 (1 -η) 2 in H 1 |gε| (B, R). Then η ε satisfies -ξ 2 ∆η ε + η ε = 1 in B η ε = |g ε | on ∂B .
It follows from [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF] that

B |∇η ε | 2 + 1 ξ 2 (1 -η ε ) 2 ≤ Cξ. (51)
Using (49), there is

ϕ ε ∈ H 1 (∂B, R), s.t. g ε = |g ε |e ıϕε and ϕ ε → ϕ uniformly on ∂B. Following [10], denote by ψ ε ∈ H 1 ϕε (B, R) the unique solution of -div(a 2 ∇ψ ε ) = 0. (Here a = b in ω and a = 1 in R 2 \ ω.) From (50), ψ ε → ψ in H 1 (B) where ψ ∈ H 1 ϕ (B, R) is the unique solution of -div(a 2 ∇ψ) = 0. Since η ε e ıψε ∈ H 1 gε (B), we have Fξ (v ε , B) ≤ Fξ (η ε e ıψε , B) ≤ 1 2 B Û 2 ε |∇ψ ε | 2 + Cξ → ε→0 1 2 B a 2 |∇ψ| 2 . ( 52
)
On the other hand, since vε ⇀ v 0 in H 1 (B), we have v 0 = e ıφ with φ ∈ H 1 ϕ (B, R) and

lim inf ε Fξ (v ε , B) ≥ lim inf ε 1 2 B Û 2 ε |∇v ε | 2 ≥ 1 2 B a 2 |∇v 0 | 2 = 1 2 B a 2 |∇φ| 2 . ( 53
) (The last inequality follows from Ûε → a in L 2 , |U ε | ≤ 1 and vε ⇀ v 0 in H 1 .)
By combining (52), (53) and the fact that ψ minimizes

B a 2 |∇ • | 2 in H 1 ϕ (B, R)
, we find that (44) holds. Furthermore, the map ψ in (52) is the same as φ in (53).

Note that since

1 2 B Û 2 ε ∇ vε |v ε | 2 -o ε (1) ≤ Fξ (v ε , B),
by comparing (52) with (53), we also have

K |∇|v ε || 2 + 1 ξ 2 (1 -|v ε | 2 ) 2 → 0. ( 54
)
In order to prove (45), it suffices to establish the convergence

φ ε → φ in L ∞ (B) with φ ε ∈ H 1 ϕε (B, R) and vε = |v ε |e ıφε , (55)
and to use the fact that |v ε | → 1 uniformly.

Proof of (55). If ∂ω ∩ B = ∅, then the argument is the same as in [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF]. Assume next that ∂ω ∩ B = ∅, and let ψ ∈ H 3/2 (B, R) be the harmonic extension of ϕ. Since ζ := φ -ψ ∈ H 1 0 (B, R) satisfies -div(a 2 ∇ζ) = div(a 2 ∇ ψ), Theorem 1 in [START_REF] Meyers | An L p -estimate for the gradient of solutions of second order elliptic divergence equations[END_REF] implies that φ ∈ W 1,p (B, R) for some p > 2.

We next prove that, for some q > 2 and B = B(y ′ , R) s.t. B(y ′ , 2 R) ⊂ B, we have φ εφ W 1,q ( B) → 0. (Once proved, this assertion will imply, via Sobolev embedding that (55) holds.)

Note that (up to a subsequence) φ ε → φ in L 2 (B, R). Thus we have

-div Û 2 ε |v ε | 2 ∇(φ ε -φ) = div ( Û 2 ε |v ε | 2 -a 2 )∇φ in B φ ε -φ L 2 (B) → 0 .
From Theorem 2 in [START_REF] Meyers | An L p -estimate for the gradient of solutions of second order elliptic divergence equations[END_REF], there is 2 < q ≤ p and C > 0 s.t.

∇(φ ε -φ) L q ( B) ≤ C R-2+2/q φ ε -φ L 2 (B) + ( Û 2 ε |v ε | 2 -a 2 )∇φ L q (B) → ε→0 0.
Consequently, φ εφ W 1,q ( B) → 0.

Step 3: We prove the third assertion Let η 0 > η > 0, with η 0 defined by (46). Denote

d k = deg ∂B(α k ,r) (v 0
). These integers do not depend on r ∈ (η, η 0 ). Moreover, we have k d k = d 0 . For r ∈ (η, η 0 ), we obtain that

2π|d k | ≤ ∂B(α k ,r) |∂ τ v 0 | ≤ √ 2πr ∂B(α k ,r) |∂ τ v 0 | 2 1/2
, and therefore

1 2 B(α k ,η 0 )\B(α k ,η) |∇v 0 | 2 ≥ πd 2 k ln η 0 η .
Consequently, we have

lim inf 1 2 ∪ k B(α k ,η 0 )\B(α k ,η) |∇v ε | 2 ≥ 1 2 ∪ k B(α k ,η 0 )\B(α k ,η) |∇v 0 | 2 ≥ π k d 2 k ln η 0 η . (56)
By combining (47) and (56), we obtain the existence of C independent of ε and η s.t.

1 2 ∪ k B(α k ,η 0 ) |∇v ε | 2 ≥ π k d 2 k ln η 0 η + πd 0 ln η ξ -C = πd 0 ln η 0 ξ + π( k d 2 k -d 0 ) ln η 0 η -C.
Therefore, d k must be either 0 or 1. Otherwise, ( 27) cannot hold for small η. Applying the strong convergence result from Step 2 with K = B(α k , η) \ B(α k , η 2 ), we have that for small ε,

d k = deg ∂B(α k ,η) vε |v ε | .
We next prove that d k = 1 for each k. By contradiction, assume that there is k 0 s.t. d k 0 = 0. We may assume that k 0 = 1. From (43), there is a (separated) µ-bad disc B(x 0 , λε 1/4 /δ) in B(α 1 , η 0 ). Thus by (41), we have Fξ (v ε , B(x 0 , λε 1/4 /δ)) > C(µ)| ln ε|.

On the other hand, since in

|v ε | ≥ 1/2 in B(α k , η 0 ) \ B(α k , η 0 /2), applying Theorem 4.1 in [27] in B(α k , η 0 ), k ∈ {2, ..., κ}, with r = 10 -4 • η 0 we find that Fξ (v ε , B(α k , η 0 )) ≥ b 2 |deg ∂B(α k ,η 0 ) (v ε )|| ln ξ| -C, k = 2, ..., κ. Since κ k=2 deg ∂B(α k ,η 0 ) (v ε ) = d 0 , the above estimates yield Fξ (v ε , B(0, ρ δ )) ≥ b 2 d 0 | ln ξ| + C(µ)| ln ε| -C
which is in contradiction with (H) and [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF]. Thus d k = 1 for k ∈ {1, ..., κ} and consequently, κ = d 0 .

We are now in position to estimate the rate of uniform convergence of

|v ε | in a compact set K ⊂ R 2 \ {α 1 , ..., α d 0 }.
Corollary 4. There is C > 0 s.t. for η 0 > η > 0 and small ε we have

|v ε | ≥ 1 -C| ln ε| -1/3 in B(0, ρ δ ) \ B(α i , η).
Proof. Due to (36), it is sufficient to establish this result in B(0, 1) \ B(α i , η). Combining Corollary 3 with (44), we obtain that

Fξ (v ε , B(0, 2) \ B(α i , η/2)) ≤ C(η).
Thus for all x ∈ B(0, ρ) s.t. B(x, ε 1/4 /δ) ⊂ B(0, 2) \ B(α i , η/2), for small ε we have

F ε (v ε , B(x, ε 1/4 )) ≤ Fξ (v ε , B(0, 2) \ B(α i , η/2)) < | ln ε| 1/3 .
From Lemma 1 (first assertion), we obtain the existence of C > 0 (independent of ε and η)

s.t. |v ε (x)| = |v ε (x)| ≥ 1 -C| ln ε| -1/3
. Finally, since for all x ∈ B(0, 1) \ B(α i , η) we have B(x, ε 1/4 /δ) ⊂ B(0, 2) \ B(α i , η/2), Corollary 4 follows.

3.6 Information about the limit v 0

Following [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] (Appendix IV, page 152) we have Proposition 9. For all 1 ≤ p < 2 and for any compact

K ⊂ R 2 , vε is bounded in W 1,p (K).
Let θ i be the main argument of x-α i |x-α i | and set θ = θ 1 + ... + θ d 0 . Note that ∇θ is smooth away from {α 1 , ..., α κ } and Π i

x-α i |x-α i | = e ıθ . Let g := tr ∂B 1 v 0 and ϕ 0 ∈ C ∞ (∂B 1 , R) be s.t. g = Π i x-α i |x-α i | e ıϕ 0 =
e ı(θ+ϕ 0 ) (see [START_REF] Brezis | Équations de Ginzburg-Landau et singularités[END_REF] for the existence of ϕ 0 ). Proposition 10. The limit v 0 satisfies -div a 2 v 0 × ∇v 0 = 0 in D ′ (R 2 ). Moreover we may write v 0 = e ı(θ+ϕ⋆) . Here ϕ ⋆ is the solution of

-div a 2 ∇(θ + ϕ ⋆ ) = 0 in B 1 ϕ ⋆ = ϕ 0 on ∂B 1 . ( 57
)
Proof. Let φ ∈ D(R 2 ), and set K = supp(φ). By Proposition 9, we have Û 2 ε vε × ∇v ε ⇀ a 2 v 0 × ∇v 0 in L p (K) for p < 2. Multiplying the equation -div Û 2 ε vε × ∇v ε = 0 by φ and integrating by parts, we obtain

0 = K -div Û 2 ε vε × ∇v ε φ = K Û 2 ε vε × ∇v ε • ∇φ → K a 2 v 0 × ∇v 0 • ∇φ = K -div a 2 v 0 × ∇v 0 φ. Consequently -div a 2 v 0 × ∇v 0 = 0 in D ′ (R 2 ).
In order to prove that -div a 2 ∇(θ + ϕ ⋆ ) = 0 in B 1 \ {α 1 , ..., α d 0 }, we follow [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], Step 12 of Theorem C.

Next, we prove that ϕ ⋆ is harmonic in a neighborhood of α k . Fix λ > 0 and x 0 ∈ ω s.t. B(x 0 , 2λ) ⊂ ω \ {α 1 , ..., α d 0 }. As we established in Proposition 8, Step 2, F ξ (v ε , B(x 0 , 2λ)) is uniformly bounded in ε. Proceeding as in Step 2, we conclude that exists λ 0 ∈ (λ, 2λ) s.t., after passing to a further subsequence, we have

∂B(x 0 ,λ 0 ) |∇v ε | 2 + 1 2ξ 2 (1 -|v ε | 2 ) 2 ≤ C (58)
with C and λ 0 independent of ε. Now, if ûε minimizes

Êξ (û) = 1 2 B(0, ρ δ ) |∇û| 2 + 1 2ξ 2 (a 2 -|û| 2 ) 2
subject to û(x) = f ε (δx) on ∂B(0, ρ δ ), then ûε minimizes Êξ (û, B(x 0 , λ 0 )) with respect to its own boundary conditions. In other words, ŵε := ûε b minimizes the classical energy

1 2 B(x 0 ,λ 0 ) |∇ ŵε | 2 + b 2 2ξ 2 (1 -| ŵε | 2 ) 2
among w ∈ H 1 (B(x 0 , λ 0 )) such that w = h ε := ûε b on ∂B(x 0 , λ 0 ). It follows from (58) and Proposition 2 that h ε also satisfies

∂B(x 0 ,λ 0 ) |∂ τ h ε | 2 + 1 2ξ 2 (1 -|h ε | 2 ) 2 ≤ C + 1. (59)
Note that by Proposition 2 we have

ŵε L ∞ (B(x 0 ,λ 0 )) ≤ 1 + ce -c 0 ξ . (60)
Using (60) and the uniform bound from Corollary 4, we may repeat the arguments of Theorem 2 in [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF] and conclude that, up to a subsequence, there exists an S 1 -valued map w 0 s.t. for every compact K ⊂ (ω \ {α 1 , ...

α d 0 }) we have ŵε → w 0 in C ∞ (K) (61) and b 2 (1 -| ŵε | 2 ) ξ 2 → |∇w 0 | 2 in C ∞ (K). ( 62
) Fix now r < min min |α k -α j | 8 , dist(α k , ∂ω) 8
and denote ω r := {x ∈ ω, dist(x, ∂ω) > r}. It follows from (61) that ŵε → q 0 := tr ∂ωr w 0 in C ∞ (∂ω r ). In view of Proposition 9, we have w 0 ∈ W 1,p (ω r ), p < 2. By Remark I.1 in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF], this implies that

w 0 = w exp i k c k ln |x -α k | + iχ .
Here:

• w is the canonical harmonic map (see [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF], Sec. I.3.) having singularities {α k , k = 1, ..., d 0 } and equal to q 0 on ∂ω r ;

• the c k 's are real coefficients;

• χ is the solution of ∆χ = 0 in ω r χ(x) + k c k ln |xα k | = 0 on ∂ω r .

Repeating the argument of [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF], Theorem VII.1, Step 2 (the key ingredients of this proof are (61), (62) and Corollary 3), we find that c k ≡ 0, k = 1, ..., d 0 , and, consequently, w 0 ≡ w in ω r . Finally, by [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF], Corollary I.2., we know that the canonical harmonic map w is of the form w = e i(θ+ϕ⋆) with ϕ ⋆ harmonic in ω r .

Uniqueness of zeros

Proposition 11. For ε sufficiently small, the minimizer vε has exactly d 0 zeros.

Proof. It suffices to prove that for small ε there is a unique zero of vε in B(α k , r), k = 1, ..., d 0 , with r defined in the proof of Proposition 10. Since ŵε = vε Ûε b , from Proposition 2 and Proposition 10 we see that

w 0 = v 0 = e i(θ k +H k ) in B(α k , r), where θ k is the phase of x-α k |x-α k | and H k = ϕ ⋆ + ψ k is harmonic in B(α k , r
). Using (61) and ( 62) and arguing as in the alternative proof of Theorem VII.4 in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] (page 74) we obtain that ∇H k (α k ) = 0.

Finally, we are now in position to obtain, as in Theorem IX.1 [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] (using the main result of [START_REF] Bauman | On the zeros of solutions to Ginzburg-Landau type systems[END_REF]), that there is a unique zero of ŵε (and, therefore, of vε ) in B(α k , r).

Summary

We have thus proved Theorem 3. Let ε n ↓ 0 and vεn be a minimizer of [START_REF] Larbalestier | High-T c superconducting material for electric power applications[END_REF] in [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF] for ε = ε n . Then there exist d 0 distinct points α 1 , ..., α d 0 ∈ ω and a function

v 0 ∈ H 1 loc (R 2 \ {α 1 , ..., α d 0 }, S 1 ) ∩ W 1,p loc (R 2 , S 1 ) (p < 2) s.t., up to a subsequence 1. vεn → v 0 in H 1 loc (R 2 \ {α 1 , ..., α d 0 }) and C 0 loc (R 2 \ {α 1 , ..., α d 0 }), 2. vεn ⇀ v 0 in W 1,p loc (R 2 ) (p < 2), 3. for K ⋐ R 2 \{α 1 , ..., α d 0 }, |v εn | ≥ 1-| ln ε n | -1/3 in K and K |∇|v εn || 2 + 1 ξ 2 (1 -|v εn | 2 ) 2 → 0, 4. for K ⋐ R 2 \ ω, vεn → v 0 in C ∞ (K) and 1 -|v εn | ≤ C K ξ 2 ,
5. vεn has exactly d 0 zeros x n 1 , ..., x n d 0 and

x n i → α i , 6. v 0 satisfies -div a 2 v 0 × ∇v 0 = 0 in D ′ (R 2 ).
Let us summarize the proof of Theorem 3:

• Statement 1. is established in Proposition 8,
• Statement 2. follows from Propositions 9 and 10,

• Statement 3. is a consequence of Corollary 4 and (54),

• Statement 4. is Corollary 2,

• Statement 5. is proved in Proposition 11,

• Statement 6. is established in Proposition 10.

The proof of Theorem 3 is complete.

Renormalized energy for the model problem

In this section, we establish the expansion for F ε (v ε , B(0, ρ)) = Fξ (v ε , B(0, ρ δ )); specifically, we derive the expression for

lim ε Fξ (v ε , B(0, ρ δ )) -πd 2 0 | ln δ| -πd 0 b 2 | ln ξ| . (63)
(We are going to prove that this limit exists).

In order to find an expression for (63), our strategy is the following:

• (Section 4.1) We first study the minimization of the Dirichlet functional among S 1 -valued maps in annulars B(0, ρ/δ) \ B(0, 1) with the Dirichlet boundary conditions: f δ (δ•) on ∂B(0, ρ/δ) and g δ on ∂B(0, 1). Here f δ = fε |fε| where f ε is given in the model problem and g δ , g 0 ∈ C ∞ (∂B 1 , S 1 ) are s.t. g δ → g 0 in C 1 . We get that the Dirichlet energy has the form πd 2 0 ln(ρ/δ) + W0 (f 0 ) + W1 (g 0 ) + o δ (1).

• (Sections 4.2, 4.3 and 4.4) In B(0, 1), we study the weighted Ginzburg-Landau functional with the Dirichlet boundary condition g δ on ∂B(0, 1). Making use of the previous bullet point, one may obtain the matching upper and lower bounds and use them to derive the third term of renormalized energy, which depends on the limiting locations of the zeros β = (β 1 , ..., β d 0 ) ∈ ω d 0 and on g 0 . We establish that inf

H 1 g δ Fξ (•, B(0, 1)) = πd 0 b 2 ln b ξ + d 0 b 2 γ + W2 (β, g 0 ) + o ε (1).
• (Section 4.5) Finally, we make a fundamental observation: the limiting function g 0 = lim tr ∂B 1 vε and the points α obtained form Theorem 3 form a minimal configuration for W 1 (g) + W 2 (β, g). Thus, introducing W (β) = inf g∈C ∞ (∂B 1 ,S 1 ) with deg ∂B 1 (g)=d 0 W1 (g) + W2 (β, g) we conclude that α minimizes W .

In this section we prove the following theorem.

Theorem 4. The following energy expansion holds when

ε → 0 Fξ (v ε , B(0, ρ δ )) = πd 0 b 2 ln b ξ + πd 2 0 ln ρ δ + W0 (f 0 ) + W (α) + d 0 b 2 γ + o ε (1). ( 64
)
Here the points α = (α 1 , ..., α d 0 ) are obtained from Theorem 3, γ > 0 is an absolute constant and W0 (f 0 ), W (α) are renormalized energies:

• W0 is independent of the points α 1 , ..., α d 0 and given by (72),

• W is given by (90), it is independent of f 0 and the limiting points (α 1 , .., α d 0 ) minimize W .

Remark 2. The renormalized energy in the expansion (64) decouples into the part that depends only on the external boundary conditions W0 (f 0 ) and the part that depends only on the location of the vortices W (α). Since α minimizes W , the external boundary data has no effect on the location of vortices inside the inclusion. This is a drastic difference with the results of [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] and [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], where the Dirichlet boundary data on the external boundary influences the location of the vortices.

Minimization among S 1 -valued maps away from the inclusion

Denote B ρ := B(0, ρ). Let (f δ ) 0<δ<1 ⊂ C ∞ (∂B ρ , S 1 ), f 0 ∈ C ∞ (∂B ρ , S 1 ) be s.t. f δ → δ→0 f 0 in C 1 (∂B ρ ) deg ∂Bρ (f δ ) = d 0 ,
and

(g δ ) 0<δ<1 ⊂ C ∞ (∂B 1 , S 1 ), g 0 ∈ C ∞ (∂B 1 , S 1 ) be s.t. g δ → δ→0 g 0 in C 1 (∂B 1 ) deg ∂B 1 (g δ ) = d 0 .
For δ ∈ (0, 1), we denote A δ = B ρ/δ \ B 1 and

W δ = {u ∈ H 1 (A δ , S 1 ) | tr ∂B ρ/δ u(•) = f δ (δ•) and tr ∂B 1 u = g δ }, Y δ = {u ∈ H 1 (A δ , S 1 ) | tr ∂B ρ/δ u(•) = f 0 (δ•) and tr ∂B 1 u = g 0 }.
Consider the following minimization problems:

I δ (f δ , g δ ) = I δ = inf u∈W δ 1 2 A δ |∇u| 2 . (P δ ) J δ (f 0 , g 0 ) = J δ = inf u∈Y δ 1 2 A δ |∇u| 2 . (Q δ )
Proposition 12. For small ε, I δ is close to J δ , namely

I δ = J δ + o δ (1). ( 65
)
Proof. In this subsection θ stands for the main argument of z i.e. z |z| = e ıθ . For δ ≥ 0, let φ δ ∈ C ∞ (∂B 1 , R) be s.t. g δ = e ı(d 0 θ+φ δ ) and ζ δ ∈ C ∞ (∂B ρ , R) be s.t. f δ = e ı(d 0 θ+ζ δ ) . We may assume that φ δ → φ 0 in C 1 (∂B 1 ) and

ζ δ → ζ 0 in C 1 (∂B ρ ). Note that u ∈ W δ ⇐⇒ u = e ı(ϕ+d 0 θ) with ϕ ∈ w δ . ( 66
)
Here

w δ := {ϕ ∈ H 1 (A δ , R) | tr ∂B ρ δ ϕ(•) = ζ δ (δ•) and tr ∂B 1 ϕ = φ δ }.
Since ∆θ = 0 in A δ and ∂ ν θ = 0 on ∂A δ , for u ∈ W δ we have

A δ |∇u| 2 = A δ |∇(ϕ + d 0 θ)| 2 = d 2 0 A δ |∇θ| 2 + A δ |∇ϕ| 2 .
Consequently, the problem (P δ ) has a unique solution u δ = e ı(d 0 θ+ϕ δ ) , with ϕ δ being the unique solution of

     -∆ϕ δ = 0 in A δ ϕ δ (•) = ζ δ (δ•) on ∂B ρ δ ϕ δ = φ δ on ∂B 1 .
With the same argument, the problem (Q δ ) admits a unique solution v δ = e ı(d 0 θ+ψ δ ) with ψ δ being the unique solution of

     -∆ψ δ = 0 in A δ ψ δ (•) = ζ 0 (δ•) on ∂B ρ δ ψ δ = φ 0 on ∂B 1 . Denote η δ = ϕ δ -ψ δ . Then η δ is the unique solution of      ∆η δ = 0 in A δ η δ = ζδ -ζ0 on ∂B ρ δ η δ = φ δ -φ 0 on ∂B 1 . (Here ζ(x) := ζ(δx)).
One may prove that ψ δ L 2 (A δ ) is bounded and more precisely we have the following result. We have

Proposition 13. 1 2 A δ |∇ψ δ | 2 → |φ 0 | 2 H 1/2 (S 1 ) + |ζ 0 | 2 H 1/2 (∂Bρ) , as δ → 0. ( 67
) Proof. Let (a n ) n∈Z , (b n ) n∈Z ⊂ C be s.t. φ 0 (e ıθ ) =
|φ 0 | 2 H 1/2 (S 1 ) = Z |n||a n | 2 and |ζ 0 | 2 H 1/2 (∂Bρ) = | ζ0 | 2 H 1/2 (∂B ρ δ ) = Z |n||b n | 2 .
From [START_REF] Berlyand | Ginzburg-Landau minimizers in perforated domains with prescribed degrees[END_REF] (Appendix D.), denoting R(δ) = ρ δ , we have

1 2π A δ |∇ψ δ | 2 = |b 0 -a 0 | 2 ln R(δ) + n =0 |n| R(δ) 2|n| -1 (|a n | 2 + |b n | 2 )(R(δ) 2|n| + 1) -2(a n b n + a n b n )R(δ) |n| = |φ 0 | 2 H 1/2 (∂B 1 ) + |ψ| 2 H 1/2 (∂Bρ) + |b 0 -a 0 | 2 ln R(δ) + n =0 2 R(δ) 2|n| -1 (|a n | 2 + |b n | 2 ) -(a n b n + a n b n )R(δ) |n| = |φ 0 | 2 H 1/2 (∂B 1 ) + |ζ 0 | 2 H 1/2 (∂Bρ) + o δ (1).
Consequently, as δ → 0, we obtain (67).

Following the same lines as Proposition 13 we obtain

∇ϕ δ L 2 (A δ ) ≤ C with C independent of δ, (68)
and

∇η δ L 2 (A δ ) → 0. ( 69
)
It follows from ( 68) and (69) that

I δ = d 2 0 2 A δ |∇θ| 2 + 1 2 A δ |∇ϕ δ | 2 = d 2 0 2 A δ |∇θ| 2 + 1 2 A δ |∇ψ δ | 2 + A δ ∇ψ δ • ∇η δ + 1 2 A δ |∇η δ | 2 = J δ + o δ (1). ( 70
)
From ( 70) and (67), we deduce that

I δ = J δ + o δ (1) = πd 2 0 ln ρ δ + W0 (f 0 ) + W1 (g 0 ) + o δ (1) (71) with W0 (f 0 ) = |ζ 0 | 2 H 1/2 (∂Bρ) and W1 (g 0 ) = |φ 0 | 2 H 1/2 (∂B 1 ) . ( 72
)
One of the main ingredients in the study of the renormalized energy is that the Dirichlet condition f min (x) = γ 0

x d 0 |x| d 0 , γ 0 ∈ S 1 minimizes W 0 . More precisely, for all f 0 ∈ C 1 (∂B 1 , S 1 ) s.t. deg ∂B 1 (f 0 ) = d 0 , we have W 0 (f min ) = 0 ≤ W 0 (f 0 ). ( 73
)
4.2 Energy estimates for S 1 -valued maps around the inclusion

Let g 0 ∈ C ∞ (∂B 1 , S 1 ) be s.t. deg ∂B 1 (g 0 ) = d 0 > 0, β 1 , ..., β d 0 are d 0 distinct points of ω, η 0 := 1 4 min i dist(β i , ∂ω), min j =i |β i -β j | .
For r ∈ (0, η 0 ), we define

Ω r := B 1 \ ∪ k B(β k , r), E r := {u ∈ H 1 (Ω r , S 1 ) | tr ∂B 1 u = g 0 and deg ∂B(β i ,r) (u) = 1}
and

F r := u ∈ H 1 (Ω r , S 1 ) | tr ∂B 1 u = g 0 and there are γ i ∈ S 1 s.t. tr ∂B(β i ,r) u(x) = γ i x -β i |x -β i | .
Consider two minimization problems

K(r, g 0 , β) = K(r) = inf u∈Er 1 2 Ωr a 2 |∇u| 2 (R r ) and L(r, g 0 , β) = L(r) = inf u∈Fr 1 2 Ωr a 2 |∇u| 2 , β = {β 1 , ..., β d 0 }. (S r )
We denote θ = θ 1 + ... + θ d 0 where θ i is the main argument of x -

β i |x -β i | , i.e., x -β i |x -β i | = e ıθ i .
Let ψ 0 be the unique (up to an additive constant in 2πZ) solution of

-div a 2 (∇ψ 0 + ∇θ) = 0 in B 1 e ı(θ+ψ 0 ) = g 0 on ∂B 1 . ( 74
)
Lemma 4. ( [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], Appendix A.)

K(r) = 1 2 Ωr a 2 |∇θ + ∇ψ 0 | 2 + O(r| ln r|), L(r) = 1 2 Ωr a 2 |∇θ + ∇ψ 0 | 2 + O(r| ln r|), with 1 2 Ωr a 2 |∇θ + ∇ψ 0 | 2 = πd 0 b 2 | ln r| + W2 (β, g 0 ) + O(r 2 ). ( 75
)
In (75), W2 (β, g 0 ), whose explicit expression is given in [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], formula (106), depends only on β and g 0 .

Upper bound for the energy

Lemma 5. Fix ρ > 0 and let f ε ∈ C ∞ (∂B ρ ), f 0 ∈ C ∞ (∂B ρ , S 1 ) be s.t. f ε → f 0 in C 1 (∂B ρ ). Let β = (β 1 , ..., β d 0 ) ∈ ω d 0 be s.t. β i = β j for i = j.
Then, for each g 0 ∈ C ∞ (∂B 1 , S 1 ), the following upper bound holds:

inf H 1 fε (Bρ) F ε ≤ πd 0 b 2 ln b ξ + πd 2 0 ln ρ δ + W0 (f 0 ) + W1 (g 0 ) + W2 (β, g 0 ) + d 0 b 2 γ + o ε (1). ( 76
)
Here W0 , W1 are defined by (72) and W2 by (75).

Proof. We construct a test function w ε ∈ H 1 fε (B ρ/δ , C) which gives (76). Fix 0 < r < η 0 . Let u δ be the minimizer of (P δ ) with g δ ≡ g 0 and f δ = f ε |f ε | and u r be the minimizer of (S r ).

Note that

f δ → f 0 = lim ε f ε in C 1 (∂B ρ ).
For each i = 1, ..., d 0 let u ξ,r i be the global minimizer of the classic Ginzburg-Landau energy in B(β i , r) with the parameter ξ/b and the boundary condition

u ξ,r i (x) = h r i (x) := γ i x-β i r on ∂B(β i , r), γ i ∈ S 1 is defined through u r . Denote I(ξ/b, r) := inf H 1 h r i (B(β i ,r)) 1 2 B(β i ,r) |∇u| 2 + b 2 2ξ 2 (1 -|u| 2 ) 2 (77) = 1 2 B(β i ,r) |∇u ξ,r i | 2 + b 2 2ξ 2 (1 -|u ξ,r i | 2 ) 2 .
Lemma IX.1 in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] implies that

I(ξ/b, r) = π ln br ξ + γ + o ξ (1). ( 78
)
We next extend the u i 's to B ρ/δ . For this purpose, we consider

ζ ∈ C ∞ (R, [0, 1]) s.t. ζ = 0 in R - and ζ = 1 in [1, ∞) and set χ ε (se ıθ ) = ζ s - ρ δ + 1 |f ε |(ρe ıθ ) -1 + 1.
In view of ( 14), we have

χ ε -1 L 2 (B ρ/δ ) ≤ Cε. Consider the following test function ũ =      χ ε u δ in B ρ/δ \ B 1 u r in B 1 \ ∪B(β i , r) u ξ,r i in B(β i , r) . ( 79
)
Clearly, inf

H 1 fε F ε ≤ Fξ (w ε ) ≤ πd 2 0 ln ρ δ + W1 (g 0 ) + W2 (β, g 0 ) + W0 (f 0 ) + πd 0 b 2 ln b ξ + d 0 b 2 γ + o ε (1) + h(r)
with h(r) = o r (1). Thus, letting r → 0 as ε → 0 we obtain the desired upper bound.

Lower bound

We prove that the upper bound ( 76) is sharp by constructing the matching lower bound.

Lemma 6. Let ε n ↓ 0, vεn be a minimizer of [START_REF] Larbalestier | High-T c superconducting material for electric power applications[END_REF] in [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF] for ε = ε n and α = (α 1 , ..., α d 0 ) ∈ ω d 0 be given by Theorem 3. Denote g 0 := lim tr ∂B 1 vε ∈ C ∞ (∂B 1 , S 1 ). Then the following lower bound holds:

F (v ε , B ρ δ ) ≥ πd 0 b 2 ln b ξ + πd 2 0 ln ρ δ + W0 (f 0 ) + W1 (g 0 ) + W2 (α, g 0 ) + d 0 b 2 γ + o ε (1). ( 80
)
Proof. As in the proof of Lemma 5, we split B ρ δ into three parts:

B ρ δ \ B 1 , B 1 \ ∪B(α i , r
) and ∪B(α i , r) with small 0 < r < η 0 .

In B ρ δ \ ∪B(α i , r) one may write vε = |v ε |w ε . Using Corollary 4 and (44) we have

Fξ (v ε , B ρ δ \ B 1 ) = 1 2 B ρ δ \B 1 Û 2 ε |v ε | 2 |∇w ε | 2 + Û 2 ε |∇|v ε || 2 + Û 4 ε 2ξ 2 (1 -|v ε | 2 ) 2 = 1 2 B ρ δ \B 1 Û 2 ε |∇w ε | 2 + Û 2 ε |∇|v ε || 2 + Û 4 ε 2ξ 2 (1 -|v ε | 2 ) 2 + o ε (1) ≥ 1 2 B ρ δ \B 1 Û 2 ε |∇w ε | 2 + o ε (1). (81)
We take

g δ = tr ∂B 1 vε |tr ∂B 1 vε | and f δ = tr ∂Bρ v ε |tr ∂Bρ v ε |
. Note that with this choice of f δ , g δ one may apply the results of Sections 4.1 and 4.2. From (81) we obtain the lower bound in B ρ δ \ B 1 :

Fξ (v ε , B ρ δ \ B 1 ) ≥ J δ + o ε (1) (82)
with J δ the energy associate to the minimization problem (Q δ ) (see page 20). Let v 0 be defined by (44). Since we have v ε → v 0 in H 1 (B 1 \ ∪B(α i , r)) and Ûε → a in L 2 (B 1 \ ∪B(α i , r)), from Proposition 10 and Lemma 4 we obtain

Fξ (v ε , B 1 \ ∪B(α i , r)) ≥ 1 2 B 1 \∪B(α i ,r) a 2 |∇v 0 | 2 + o ε (1) ≥ 1 2 B 1 \∪B(α i ,r) a 2 |∇θ + ∇ψ 0 | 2 + o ε (1) = K(r) + O(r| ln r|) + o ε (1), (83)
where K(r) is defined by (R r ) (see page 22). In order to complete the proof of the lemma, we need to obtain a sharp lower bound in each ball B(α i , r). Actually we will prove that

Fξ (v ε , B(α i , r)) ≥ b 2 I(ξ/b, r) + o r (1) + o ε (1), (84)
with I(ξ/b, r) being defined in (77). The estimate (84) is equivalent to

Fξ (v ε , B(α i , r)) ≥ b 2 I(ξ/b, r + r 2 ) + o r (1) + o ε (1). (85)
Indeed by (78) we have I(ξ, r + r 2 ) -I(ξ, r) = o r (1). We now make use of the construction by Lefter and Rȃdulescu in [START_REF] Lefter | On the Ginzburg-Landau energy with weight[END_REF] and [START_REF] Lefter | Minimization problems and corresponding renormalized energies[END_REF]. From Proposition 10, we know that v 0 = e ı(θ i +ϕ⋆+ψ i ) with ϕ ⋆ , ψ i harmonic, and therefore smooth in B(α i , η) (η > r small). Set σ i = ϕ ⋆ + ψ i . Without loss of generality, we can assume that α i = 0 and σ i (0) = 0. Consequently, |σ i (x)| ≤ C|x| with C independent of η and |x| ≤ η. Let vε = λ ε e ı(θ i +σ i ε ) where λ ε := |v ε |.

From Proposition 8 and (54), we obtain that

σ i ε → σ i in H 1 (B r+r 2 \ B r ), (86)
λ ε → 1 in H 1 (B r+r 2 \ B r ) and 1 ξ 2 B r+r 2 \Br (1 -λ ε ) 2 → 0. (87)
Let

β ε (se ıθ i ) =    vε (se ıθ i ) if s ∈ [0, r) 1 -λ ε r 2 (s -r) + λ ε exp ı θ i + σ i ε -s + r 2 + r r 2 if s ∈ [r, r + r 2 ] .
Clearly,

β ε ∈ H 1 x/|x| (B r+r 2 ). Consequently, b 2 I(ξ/b, r + r 2 ) ≤ Fξ (v ε , B r ) + Fξ (β ε , B r+r 2 \ B r ) + o ε (1).
From (87), we easily obtain that

B r+r 2 \Br |∇|β ε || 2 + 1 ξ 2 (1 -|β ε |) 2 = o ε (1).
It remains to estimate

B r+r 2 \Br ∇ θ i + σ i ε -s + r 2 + r r 2 2 .
From (86)

B r+r 2 \Br ∇ θ i + σ i ε -s + r 2 + r r 2 2 = B r+r 2 \Br ∇ θ i + σ i -s + r 2 + r r 2 2 + o ε (1). Since |σ i (se ıθ )| ≤ Cs, |∂ s σ i | ≤ C and |∂ θ i σ i | ≤ Cs we have ∇ θ i + σ i -s + r 2 + r r 2 2 = ∂ s σ i -s + r 2 + r r 2 - σ i r 2 2 + 1 r 2 1 + ∂ θ i σ i -s + r 2 + r r 2 2 ≤ C (1 + r -2 ) + r -2 = O(r -2 ). Since |B r+r 2 \ B r | = O(r 3 ) we find that B r+r 2 \Br ∇ θ i + σ i ε -s + r 2 + r r 2 2 = O(r). It follows that Fξ (β ε , B r+r 2 \ B r ) = O(r) + o ε (1)
. Consequently, (85) holds and thus we obtain (84). Combining (82), ( 83) and (84), together with (71) and (75), we obtain

Fξ (v ε , B ρ δ ) ≥ I δ + K(r) + b 2 I(ξ/b, r) + o ε (1) + o r (1) = πd 2 0 ln ρ δ + πd 0 b 2 ln b ξ + W0 (f 0 ) + W (α, g 0 ) + + d 0 b 2 γ + o ε (1) + o r (1). ( 88
)
The conclusion of the Lemma follows by letting r → 0 as ε → 0.

4.5

The function g 0 and the points {α 1 , ..., α d 0 } minimize the renormalized energy

In the previous section, we obtained an expansion for the energy Fξ (v ε , B ρ δ ) of the model problem. To summarize, using (76), (80) and Theorem 3 we get that there are g 0 = lim tr ∂B 1 vε and α = (α 1 , ..., α d 0 ) ∈ ω d 0 s.t.

Fξ (v ε , B ρ δ ) = πd 0 b 2 ln b ξ + πd 2 0 ln ρ δ + W (α, g 0 ) + W0 (f 0 ) + d 0 b 2 γ + o ε (1), (89)
with W (α, g 0 ) = W1 (g 0 ) + W2 (α, g 0 ).

The goal of this section is to underline an important property of the points α, namely, that they minimize the quantity inf g 0 ∈C ∞ (∂B 1 ,S 1 ) W (•, g 0 ).

We have the following

Proposition 14. Let β = (β 1 , ..., β d 0 ) ∈ ω d 0 be a d 0 -tuple of distinct points and let g 0 ∈ C ∞ (∂B 1 , S 1 ) be s.t. deg ∂B 1 (g 0 ) = d 0 . Then W (α, g 0 ) ≤ W (β, g 0 ).
Proof. Let (β, g 0 ) be as in Proposition 14. Using the test function given by (79), we obtain that for all ε > 0 and r > 0 (small) there is

w ε ∈ H 1 fε (B ρ δ , C) s.t. Fξ (w ε ) = πd 0 b 2 ln b ξ + πd 2 0 ln ρ δ + W (β, g 0 ) + W0 (f 0 ) + d 0 b 2 γ + h 1 ε + h 2 r here h 1 ε = o ε (1) and h 2 r = O(r).
On the other hand, taking into account the minimality of vε and (89) we have W (β, g 0 ) ≥ W (α, g 0 ) + o ε (1) + h 2 r . The previous estimate implies (letting ε → 0 and r → 0) that W (β, g 0 ) ≥ W (α, g 0 ) which completes the proof. Thus, for β = (β 1 , ..., β d 0 ) ∈ ω d 0 we define

W (β) = inf g∈C ∞ (∂B 1 ,S 1 ) deg ∂B 1 (g)=d 0 W (β, g) = inf g∈C ∞ (∂B 1 ,S 1 ) deg ∂B 1 (g)=d 0 W1 (g) + W2 (β, g) (90)
with W1 and W2 given by ( 72) and (75) respectively. It follows that for α given by Theorem 3 and

g 0 = tr ∂B 1 v 0 : W (α) = W (α, g 0 ) ≤ W (β) for all β = (β 1 , ..., β d 0 ) ∈ ω d 0 .
5 Proofs of Theorems 1 and 2

In this section v ε is a minimizer of F ε in H 1 g (Ω, C). We split the proofs of Theorem 1 and 2 in three steps:

• (Section 5.1) Using estimates on |v ε |, we first localize the vorticity to the neighborhoods of selected inclusions. Then we find two separate energy expansions in two sub-domains of Ω: away from the selected inclusions and around them.

• (Section 5.2) We study the asymptotic behavior of v ε . We prove that, for small ε, v ε has exactly d zeros of degree 1.

• (Section 5.3) We give an expansion of F ε (v ε ) up to o ε (1) terms and relate the choice of the inclusions with vortices to the renormalized energy of Bethuel, Brezis and Hélein.

Locating bad inclusions

The following result gives a uniform bound on the modulus of minimizers away from the inclusions.

Lemma 7. There exists C > 0 s.t. for small ε we have

1. |v ε | ≥ 1 -C| ln ε| -1/3 in Ω \ ∪ M i=1 B(a i , δ), 2
. there are at most d points a i 1 , ..., a

i d ′ (1 ≤ d ′ = d ′ ε ≤ d) s.t. {|v ε | < 1 -C| ln ε| -1/3 } ⊂ ∪ d ′ k=1 B(a i k , δ).
Proof. Using Lemma 1 with χ = | ln ε| -1/3 , we obtain that there exist

C, C 1 > 0 s.t. for ε > 0 small, if F ε (v ε , B(x, ε 1/4 )) < | ln ε| 1 3 -C 1 then |v ε | ≥ 1 -Cχ in B(x, ε 1/2 ).
We prove 1. by contradiction. Assume that, up to a subsequence, there is

x ε ∈ Ω \ ∪ M i=1 B(a i , δ), s.t. |v ε (x ε )| < 1 -C| ln ε| -1/3
with C given by Lemma 1. From Lemma 1 and Proposition 2

1 2 B(xε,ε 1/4 ) |∇v ε | 2 + 1 2ε 2 (1 -|v ε | 2 ) 2 ≥ | ln ε| 1/3 -O(1). (91)
Fix a bounded, simply connected domain Ω ′ such that Ω ⊂ Ω ′ , and extend v ε by a fixed smooth

S 1 -valued map v in Ω ′ \ Ω, s.t. v = g on ∂Ω.
In view of [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] for Case I or [START_REF] Brezis | Équations de Ginzburg-Landau et singularités[END_REF] for Case II, there exists C > 0 s.t. for small ε

1 2 Ω ′ |∇v ε | 2 + 1 2ε 2 (1 -|v ε | 2 ) 2 ≤ C| ln ε|.
Therefore, the map v ε in Ω ′ satisfies the condition of Theorem 4.1 [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF]. This theorem guarantees that

• there exists B ε = {B ε j }, a finite disjoint covering of the set

{x ∈ Ω ′ | dist(x, ∂Ω ′ ) > ε/b and |v ε (x)| < 1 -(ε/b) 1/8 },
• such that rad(B ε ) := j rad(B ε j ) ≤ 10 -2 • dist(ω, ∂B(0, 1)) • δ,

• and, denoting

d j = |deg ∂B j (v ε )| if B ε j ⊂ {dist(x, ∂Ω ′ ) > ε/b} and d j = 0 otherwise, we have 1 2 ∪B ε j |∇v ε | 2 + b 2 2ε 2 (1 -|v ε | 2 ) 2 ≥ π j d j ln δ ε -C = π j d j | ln ξ| -C, (92)
with C independent of ε.

Note that since |v

ε | ≡ 1 in Ω ′ \ Ω, if d j = 0 then B ε j ⊂ {dist(x, ∂Ω ′ ) > ε/b}. Consequently, we have d j = |deg ∂B ε j (v ε )|. Assertion 1.
follows as in the proof of Proposition 5 (use (91), (92) instead of ( 20) and ( 22)). The proof of Assertion 2. of Lemma 7 goes along the same lines.

We next obtain the following lower bounds for the energy. Lemma 8. For k ∈ {1, ..., d ′ }, we denote

d k = d ε k = deg ∂B(a i k ,δ) (v ε ).
There exist C, η 0 > 0 s.t. for small ε and ρ ∈ [2δ, η 0 ] we have

1 2 Ω\∪ d k=1 B(a i k ,ρ) |∇v ε | 2 ≥ π d k=1 d 2 k | ln ρ| -C (93) and F ε (v ε , B(a i k , 2δ)) ≥ π|d k |b 2 | ln ξ| -C. ( 94
)
Proof. Let η 0 = 10 -2 min i {dist(a i , ∂Ω), min j =i |a ia j |} and 0 < ρ < η 0 . We prove (93). By Lemma 7,

|v ε | ≥ 1/2 in Ω \ ∪ d k=1 B(a i k , ρ), therefore, w ε = vε |vε| is well-defined in this domain. From direct computations in B(a i k , η 0 ) \ B(a i k , ρ) we have 1 2 Ω\∪ d k=1 B(a i k ,ρ) |∇w ε | 2 ≥ π d i=1 d 2 i ln η 0 ρ . ( 95
)
We claim that the bound (93) holds with C = | ln η 0 | + 1. Argue by contradiction: assume that up to a subsequence we have:

1 2 Ω\∪ d k=1 B(a i k ,ρ) |∇v ε | 2 ≤ π d i=1 d 2 i ln η 0 ρ -1. ( 96
)
On the other hand, we have

|∇v ε | 2 = |v ε | 2 |∇w ε | 2 + |∇|v ε || 2
and therefore

Ω\∪ d k=1 B(a i k ,ρ) |∇v ε | 2 ≥ Ω\∪ d k=1 B(a i k ,ρ) |∇w ε | 2 -(1 -|v ε | 2 )|∇w ε | 2 . ( 97
)
Using the fact that |v ε | ≥

Ω\∪ d k=1 B(a i k ,ρ) (1 -|v ε | 2 )|∇w ε | 2 ≤ C| ln ε| -1 3 Ω |∇v ε | 2 ≤ C | ln ρ| | ln ε| 1 3 → 0. ( 98
)
By combining (95), ( 97) and (98), we see that (96) cannot hold for small ε; this implies (93). We now prove (94). Performing the rescaling x =

x-a i k δ , we obtain

F ε (v, B(a i k , 2δ)) = Fξ (v, B(0, 2)) = 1 2 B(0,2) Û 2 ε |∇v| 2 + 1 2ξ 2 Û 4 ε (1 -|v| 2 ) 2 dx,
where, as in the model problem we set v(x) = v(δx) and ξ = ε δ .

By Theorem 4.1 [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF], for r = 10 -2 there are C > 0 and a finite covering by disjoint balls B 1 , ..., B N (with the sum of radii at most r) Let h ε := tr ∂K v ε . Since ∂K ⊂ Ω \ {a 1 , ..., a M }, we have h ε → h 0 in C ∞ (∂K) (possibly after passing to a subsequence). Since deg(h ε , ∂K) = 0 we have deg(h 0 , ∂K) = 0 and consequently there is some ϕ 0 ∈ C ∞ (∂K, R) s.t. h 0 = e iϕ 0 .

of {x ∈ B(0, 2 -ξ/b) | 1 -|v ε (x)| ≥ (ξ/b) 1/8 } and 1 2 ∪ j B j |∇v ε | 2 + b 2 2ξ 2 (1 -|v ε | 2 ) 2 ≥ πD k | ln ξ| -C, (99)
D k = j |m j | and m j = deg ∂B j (v ε ) if dist(B j , ∂B(0, 2
Let ṽ be a minimizer of K |∇v| 2 in the class H 1 h 0 (K, S 1 ). Clearly,

K |∇ṽ| 2 ≤ K |∇v * | 2 .
On the other hand, since U ε ≤ 1, we may construct (in the spirit of [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF]) a test function and find that (see formula (93) in [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF])

F ε (v ε , K) ≤ 1 2 K |∇ψ ε | 2 + Cε, (102)
where ψ ε is the solution of ∆ψ ε = 0 in K ψ ε = ϕ ε on ∂K .

Here, ϕ ε is defined by

e iϕε = h ε |h ε | on ∂K.
As ε → 0, we have

ψ ε → ψ 0 strongly in H 1 (K), where ∆ψ 0 = 0 in K ψ 0 = ϕ 0 on ∂K . (103)
From the fact that

v ε ⇀ v * in L 2 (K), U ε → 1 in L 2 (K) and |U ε | ≤ 1 we have U 2 ε ∇v ε ⇀ v * in L 2 (K). Consequently, we obtain 1 2 K |∇v * | 2 ≤ lim inf ε→0 1 2 K U 2 ε |∇v ε | 2 ≤ lim inf ε→0 F ε (v ε , K). (104)
Combining ( 102), ( 103) and (104) we deduce that

K |∇v * | 2 ≤ K |∇ψ 0 | 2 = K |∇ṽ| 2 .
It follows that v * minimizes the Dirichlet functional in

H 1 h 0 (K, S 1 ) := {v ∈ H 1 (K, S 1 ), v = h 0 on ∂K}.
We find that hence ṽ = v * in K. By a classic result of Morrey [START_REF] Morrey | Multiple integrals in the calculus of variations[END_REF] (see also [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF]), v * satisfies (5). Moreover, as follows from weak lower semicontinuity of Dirichlet integral, (102), ( 103) and (104)

1 2 K |∇v * | 2 ≤ lim inf ε→0 1 2 K |∇v ε | 2 ≤ lim sup ε→0 F ε (v ε , K) ≤ 1 2 K |∇v * | 2 .
Therefore,

v ε converges to v * strongly in H 1 (K). (105)
From ( 101) and (105) we obtain that v ε → v * in H 1 loc (Ω \ {a 1 , ..., a d ′ }). The convergence up to ∂Ω will be established in the next section.

In order to prove Assertion 3. of Theorem 1 and Assertion 2. of Theorem 2, note that, for small ρ > 0, the estimate (101) implies that f ε := tr ∂B(a i k ,ρ) v ε satisfies the conditions ( 13) and ( 14) of Theorem 3. This gives us 3. of Theorem 1 and 2. of Theorem 2.

Assertion 3. of Theorem 2 is is a consequence of Corollary 6.

The macroscopic position of vortices minimizes the Bethuel-Brezis-Hélein renormalized energy

Let us recall briefly the concept of the renormalized energy W

g ((b 1 , d 1 ), ..., (b k , d k )) with      g ∈ C ∞ (∂Ω, S 1 ) s.t. deg ∂Ω (g) = d b 1 , ..., b k ∈ Ω, b i = b j for i = j d i ∈ Z and i d i = d .
For small ρ > 0, consider Ω ρ = Ω \ ∪ i B(b i , ρ) and the minimization problem

I ρ ((b 1 , ..., b k ), (d 1 , ..., d k )) = inf w∈H 1 (Ωρ,S 1 ) s.t. w=g on ∂Ω w(b i +ρe ıθ )=α i e ıd i θ , α i ∈S 1 1 2 Ωρ |∇w| 2 .
Such problem is studied in detail in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] (Chapter 1). In particular Bethuel, Brezis and Hélein proved that for small ρ, we have

I ρ ((b 1 , ..., b k ), (d 1 , ..., d k )) = πd| ln ρ| + W g ((b 1 , d 1 ), ..., (b k , d k )) + o ρ (1).
This equality plays an important role in the study done in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF]. In the minimization problem of the classical Ginzburg-Landau functional

1 2 Ω |∇u| 2 + 1 2ε 2 (1 -|u| 2 ) 2 , u ∈ H 1 g ,
the vortices (with their degrees) of a minimizer tend to form (up to a subsequence) a minimal configuration for W g . We prove in this section that the (macroscopic) location of the vorticity of minimizers of We present here the argument for Case I (Theorem 1). The argument in Case II is analogous. The proof of Assertion 4. relies on two lemmas, providing sharp upper and lower bounds.

Lemma 9. There exists ρ 0 > 0 s.t., for every ρ < ρ 0 and every ε > 0, we have

F ε (v ε ) ≤ πd| ln ρ| + dJ(ε, ρ) + W g ((a i 1 , 1), ..., (a i d , 1)) + o ρ (1), (106)
where J(ε, ρ) = inf u∈H 1 gρ (Bρ(0)) F ε (u) with g ρ = z ρ on ∂B(0, ρ).

Proof. The proof, via construction of a test function, is the same as proof of Lemma VIII.1 in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF].

Lemma 10. Let ρ > 0, ρ < ρ 0 . Then for small ε we have

F ε (v ε ) ≥ πd| ln ρ| + dJ(ε, ρ) + W g ((a i 1 , 1), ..., (a i d , 1)) + o ρ (1). ( 107
)
Proof. Split the domain Ω into two sub-domains: Ω \ ∪ i B(a k i , ρ) and ∪ i B(a k i , ρ). We start with the lower bound in the first sub-domain. By the previous estimate, v ε weakly converges to

v * in H 1 (Ω \ ∪ i B(a k i , ρ)). This implies that lim inf 1 2 Ω\∪ k B(a i k ,ρ) U 2 ε |∇v ε | 2 ≥ 1 2 Ω\∪ k B(a i k ,ρ) |∇v * | 2 .
Here, we used the fact that, since

U ε → 1 in L 2 (Ω), |U ε | ≤ 1 and ∇v ε ⇀ ∇v * in L 2 (Ω \∪ k B(a i k , ρ)), we have U ε ∇v ε ⇀ ∇v * in L 2 (Ω \ ∪ k B(a i k , ρ)).
Thus we deduce that, for small ε,

1 2 Ω\∪ k B(a i k ,ρ) U 2 ε |∇v ε | 2 ≥ 1 2 Ω\∪ k B(a i k ,ρ) |∇v * | 2 -ρ 2 . ( 108
)
On the other hand, as proved in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF],

1 2 Ω\∪ k B(a i k ,ρ) |∇v * | 2 ≥ πd ln 1 ρ + W g ((a i 1 , 1), ..., (a i d , 1)) + o ρ (1). (109)
Thus, combining (108), (109) and using Proposition 2, for ε sufficiently small, we have

F ε (v ε , Ω \ ∪ k B(a i k , ρ)) ≥ πd ln 1 ρ + W g ((a i 1 , 1), ..., (a i d , 1)) + o ρ (1). ( 110
)
By Theorem 4 and Corollary 5 we have the following energy expansion:

F ε (v ε , B(a i k , ρ)) = π ln ρ + πb 2 | ln ε| + π(1 -b 2 )| ln δ| + W (α) + W0 (f 0 ) + b 2 γ + o ε (1). (111)
Similarly, applying Theorem 4 to J(ε, ρ) we obtain

J(ε, ρ) = π ln ρ + πb 2 | ln ε| + π(1 -b 2 )| ln δ| + W (α) + W0 (z/|z|) + b 2 γ + o ε (1). (112)
Here, the local renormalized energy W (α) is given by (90) and is the same in ( 111) and (112). From (73), W0 (f 0 ) ≥ 0 while W0 ( z |z| ) = 0. Consequently, we have

F ε (v ε , B(a i k , ρ)) -J(ε, ρ) ≥ o ε (1). Hence ∀ρ > 0 there exists ε ρ > 0 s.t. for ε < ε ρ we have F ε (v ε , B(a i k , ρ)) ≥ J(ε, ρ) -ρ 2 and thus F ε (v ε , ∪ k B(a i k , ρ)) ≥ dJ(ε, ρ) -dρ 2 , (113)
which gives the lower bound in the second sub-domain. From (110) and (113) the bound (107) follows.

Combining Lemma 9 and Lemma 10, we see that the points {a i k , 1 ≤ k ≤ d} minimize W g among a 1 , ..., a M . The expansion (6) follows from (106), (107) and (112).

We next turn to convergence of v ε up to the boundary. It suffices to prove the H 1 -convergence of v ε in Ω ρ = Ω \ ∪ m B(a im , ρ) (for small ρ > 0). We argue by contradiction and we assume that there are some ρ 1 > 0 and η > 0 s.t.

lim inf 1 2 Ωρ 1 |∇v ε | 2 ≥ 1 2 Ωρ 1 |∇v * | 2 + η. (114)
Note that for all ρ ≤ ρ 1 , (114) still holds in Ω ρ . If, in the proof of Lemma 10, we replace (108) by (114) (with ρ 1 replaced by ρ), then we obtain for small ρ a contradiction with Lemma 9. The proof of Theorem 1 is complete. The last assertion of Theorem 2 is obtained along the same lines.

A Proof of Proposition 2

Let x 0 ∈ V R be s.t. B R = B(x 0 , R) ⊂ Ω \ ω δ and assume that x 0 = 0. We follow the proof of Lemma 2 in [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF].

In B R , η = 1 -U ε satisfies -ε 2 ∆η + tη = -η(η 2 -3η + 2 -t) in B R η ≤ 1 on ∂B R ,
here, t will be chosed later. Since η ∈ (0, 1b), if we take t = b(1 + b), then we have

-ε 2 ∆η + tη ≤ 0 in B R .
On the other hand, the function w

(x) = e γ(|x| 2 -R 2 ) satisfies -ε 2 ∆w + tw = -4ε 2 γ(1 + γ|x| 2 ) + t w in B R w = 1 on ∂B R .
A simple computation gives that

-ε 2 ∆w + tw ≥ 0 in B R γ > 0 ⇔ 0 < γ ≤ -ε + √ ε 2 + tR 2 2R 2 ε . Take γ = -ε + √ ε 2 + tR 2 2R 2 ε > 0. Setting v = η -w, we have -ε 2 ∆v + tv ≤ 0 in B R v ≤ 0 on ∂B R .
By the maximum principle, we have v ≤ 0 in B R . Therefore,

η(0) ≤ exp - -ε + √ ε 2 + tR 2 2ε ≤ Ce - √ tR 4ε .
Consequently, [START_REF] Berlyand | Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain[END_REF]

holds in {x ∈ Ω | dist(x, ∂Ω) ≥ R, dist(x, ω δ ) > R}. The estimate close to the ∂Ω is a direct consequence of 0 ≤ U ε ≤ 1, (9) holds in {x ∈ Ω | dist(x, ∂Ω) ≥ R, dist(x, ω δ ) > R} and the equation -∆U ε = 1 ε 2 U ε (1 -|U ε | 2 ) in {x ∈ Ω | dist(x, ω δ ) > R}.
Using a similar argument, we establish (9) in the case V R ∩ ω δ . The proof of (9) is complete.

In order to prove [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF]

, note that in W R := {x ∈ Ω | dist(x, ∂ω δ ) ≥ R, dist(x, ∂Ω) ≥ R} the function η = a ε -U ε satisfies ∆η = Uε ε 2 (a 2 ε -U 2 ε
). Thus, applying Lemma A.1 [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF] to η in conjunction with [START_REF] Berlyand | Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain[END_REF] and the fact that R ≥ ε, we obtain

|∇η| ≤ C 1 e -cR ε ε in W R .
Thus [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF] holds far away from ∂Ω and the inclusions. We next prove that the bound (10) holds near ∂Ω. Indeed, fix a smooth compact K ⊂ Ω s.t. for small δ we have ω δ ⊂ K. Clearly, by (9), 0 ≤ η K := tr ∂K η ≤ Ce Note that η 2 L ∞ ≤ Ce -cR ε and thus η 1 L ∞ ≤ Ce -cR ε . Lemma A.2 in [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF] implies the existence of a constant C Ω\K > 0 s.t.

|∇η 1 | ≤ C Ω\K e -cR ε ε in Ω \ K.
In order to estimate ∇η 2 near ∂Ω, we express η 2 in terms of Green's function G(x, y) in Ω \ K: function, i.e.

B Proof of Proposition 3

This appendix is devoted to the proof of Proposition 3. We prove the first assertion: when M ≥ d we have Fix first d distinct points-centers of inclusions a 1 , ..., a d . Let ρ 0 := 10 -2 •min(dist(a i , ∂Ω), min i =j |a ia j | > 0). Consider ṽ to be a smooth fixed function in Ω \ ∪ d i=1 B(a i , ρ 0 ), such that |ṽ| = 1 in Ω \ ∪ d i=1 B(a i , ρ 0 ) and ṽ = g on ∂Ω ṽ(x) = x-a i |x-a i | on ∂B(a i , ρ 0 ) .

Such a function clearly exists since the compatibility condition deg ∂Ω (g) = d i=1 deg ∂B(a i ,ρ 0 ) (ṽ) is satisfied. Let c 0 = 10 -2 • dist(0, ∂ω). For every 1 ≤ i ≤ M , consider a disc B(a i , c 0 δ) ⊂ ω i δ . By the choice of c 0 , we have dist(∂ω δ , B(a i , c 0 δ)) ≥ c 0 δ. Therefore, using Proposition 2

U 2 ε -b 2 ≤ Ce -cδ ε in B(a i , c 0 δ). (116)
Consider the test function v ε 0 defined as

v ε 0 (x) =          ṽ(x) for x ∈ Ω \ ∪ i B(a i , ρ 0 ) x -a i |x -a i |
for x ∈ B(a i , ρ 0 ) \ B(a i , ε) xa i ε for x ∈ B(a i , ε)

.

Using ( 116) and (H) we have (For example, the map considered in Remark I.5 in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] has these properties).

The necessary test function that satisfies the bound ( 12) is obtained by rescaling the v ε i 's (in order to have maps defined in balls of size δ) and gluing the rescaled maps with ṽε 0 .

C Proof of the η-ellipticity Lemma

The main argument in the proof of the η-ellipticity result is the following convexity lemma which is a generalization of Lemma 8 in [START_REF] Berlyand | Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain[END_REF]. The proof of Lemma 11 is given in [START_REF] Santos | The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part I: the zero degree case[END_REF].

Lemma 11. [Convexity Lemma]

Let C be a chord in the closed unit disc, C different from a diameter. Let S be the smallest of two regions enclosed by the chord and the boundary of the disc.

Let O be a Lipschitz, bounded, connected domain and let g ∈ C(∂O, S).

Assume that v minimizes Ginzburg-Landau type energy We prove the first part of the lemma 1. Let x ∈ Ω be s.t. dist(x, ∂Ω) ≥ ε 1/4 . We have

F (v) =
F ε (v ε , B(x, ε 1/4 )) ≥ b 2 B(x,ε 1/4 )\B(x,ε 1/2 ) |∇v ε | 2 + 1 ε 2 (1 -|v ε | 2 ) 2 = b 2 ε 1/4 ε 1/2 1 r • r ∂B(x,r) |∇v ε | 2 + 1 ε 2 (1 -|v ε | 2 ) 2 .
By Mean Value theorem, exists r ∈ (ε 1/2 , ε 1/4 ) s.t r ∂B(x,r)

|∇v ε | 2 + 1 ε 2 (1 -|v ε | 2 ) 2 ≤ 2 b F ε (v ε , B(x, ε 1/4
)) Clearly, the previous assertion gives contradiction. From (117) and (118), there is C = C(χ, b) > 0 and ε 0 = ε 0 (χ) > 0 s.t. for ε < ε 0 ,

v ε : ∂B r → {z ∈ B 1 | ℜz > 1 -Cχ}.
Using Convexity Lemma (Lemma 11), we find that |v ε | ≥ 1 -Cχ in B(x, r) ⊃ B(x, ε 1/2 ). If dist(x, ∂Ω) < ε 1/4 , we denote S r = Ω ∩ ∂B(x, r), r ∈ (ε 1/2 , ε 1/4 ). Clearly, we have

2 b F ε (v ε , B(x, ε 1/4 ) \ B(x, ε 1/2 )) ≥ ε 1/4 ε 1/2 1 r • r dr Sr |∇v ε | 2 + 1 ε 2 (1 -|v ε | 2 ) 2 .
Using mean value argument and the facts that g ε → g 0 in C 1 (∂Ω, S 1 ) and that 0 ≤ 1 -|g ε | ≤ ε, there are r ∈ (ε 1/2 , ε 1/4 ) and

C 1 = C 1 ( g 0 C 1 , Ω) s.t r ∂(B(x,r)∩Ω) |∂ τ v ε | 2 + 1 ε 2 (1 -|v ε | 2 ) 2 ≤ 2 b F ε (v ε , B(x, ε 1/4 )) + C 1 1 4 | ln ε| .
Using the same argument as before (taking O = Ω ∩ B(x, r)) we obtain the desired result.

3 A model problem: one inclusion 3 . 1

 31 Uniform convergence of |v ε | to 1 away from inclusions . . . . . . . . . . . . . . . . . 3.2 Distribution of Energy in B(0, ρ δ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Convergence in C ∞ (K) for a compact K s.t. K ∩ ω = ∅ . . . . . . . . . . . . . . . . 3.4 The bad discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Convergence in H 1 loc (R 2 \ {α 1 , ..., α κ }) . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Information about the limit v 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Uniqueness of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

n∈Z a n e ınθ and ζ 0

 0 (ρe ıθ ) = n∈Z b n e ınθ .

  )) ≥ ξ/b 0 otherwise . Since, by Lemma 7, |v ε | ≥ 1/2 in B(0, 2) \ B(0, 1), D k ≥ d k , and (94) follows from (99) and the estimate U ε ≥ b.

  F ε is related to the minimization problem of W g ((b 1 , ..., b k ), (d 1 , ..., d k )) with b 1 , ..., b k ∈ {a 1 , ..., a M }.

ε 2 U∆η 2

 22 (1 + U )η in Ω \ K η = 0 on ∂Ω η = η K on ∂K . Let η = η 1 + η 2 be s.t. η 1 solves ∆η 1 = 1 ε 2 U (1 + U )η in Ω \ K η 1 = 0 on ∂Ω ∪ ∂K and η = 0 in Ω \ K η 2 = 0 on ∂Ω η 2 = η K on ∂K .

η 2

 2 (x) = -∂K η K (y) ∂G ∂ν (x, y)dS(y).(115)It follows from (115) and (9) that |∇η 2 | ≤ C 0 e -cR ε away from ∂K. The estimate (10) is proved.

F

  ε (v, Ω) ≤ πdb 2 | ln ξ| + πd| ln δ| + O(1).

inf v∈H 1 g

 1 (Ω) F ε (v, Ω) ≤ F ε (v ε 0 ) ≤ πdb 2 | ln ε| + πd(1b 2 )| ln δ| + C = πdb 2 | ln ξ| + πd| ln δ| + C.Now we prove the second assertion: when M < d we have infv∈H 1 g (Ω) F ε (v, Ω) ≤ πdb 2 | ln ξ| + π i d 2 i | ln δ| + C. Let d 1 , ..., d M ∈ N be s.t. d i = d. Set c 0 = 10 -2d • dist(0, ∂ω). For i ∈ {1, ..., M } s.t. d i > 0, fix α 1,i , ..., α d i ,i ∈ B(0, 10 d c 0 ) ⊂ ω s.t. min min j =k |α j,iα k,i |, dist(α j,i , ∂ω) > 4c 0 . Consider an ε-dependent map ṽε 0 ∈ H 1 (Ω \ ∪ d i >0 B(a i , 10 d c 0 δ), S 1 ) s.t. a i ) d i |xa i | d i on ∂B(a i , 10 d c 0 δ)and satisfyingΩ\∪ d i >0 B(a i ,10 d c 0 δ) |∇ṽ ε 0 | 2 ≤ π d 2 i | ln δ| + Cwith C depending only on Ω, ω and g.(Such maps do exist, e.g., consider the map introduced in[START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF], Remark I.5.)For i ∈ {1, ..., M } s.t. d i > 0, we consider a map v ε i ∈ H 1 (B(0, 10 d c 0 ) \ ∪ d i j=1 B(α j,i , ξ), S 1 ) s.t. • v ε i (x) = x d i /|x| d i on ∂B(0, 10 d c 0 ), • v ε i (x) = (xα j,i )/|xα j,i | on ∂B(α j,i,ξ), • B(0,10 d c 0 )\∪ d i j=1 B(α j,i ,ξ) |∇v ε i | 2 ≤ πd i | ln ξ| + C with C depending only on ω.

O

  α(x)|∇v| 2 + β(x)(1 -|v| 2 ) 2 dx in H 1 g (O), with α, β ∈ L ∞ (O, R) satisfying essinf α > 0, essinf β > 0. Then v(O) ⊂ S.

≤ 2 b 4 |

 24 F εn (v εn , B(x, ε n 1/4 )) 1 ln ε n | ≤ 8χ 2 b .

  Properties of U ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Identifying bad discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	1 Introduction and main results
	2 Main tools
	2.1

  1 2 in Ω \ ∪ d k=1 B(a i k , ρ) we see that |∇w ε | ≤ 2|∇v ε |. Therefore, by (96), (H) and Lemma 7 we estimate the last term in (97):

  There isC 2 = C 2 (χ, b) > 0 s.t if F ε (v ε , B(x, ε 1/4 )) ≤ χ 2 | ln ε|, we have Var(v ε , ∂B(x, r)) ≤ C 2 χ, where Var (v ε , ∂B(x, r)) := It follows that |v ε | 2 ≥ 1 -3C 2 χ on ∂B(x, r).(118)Indeed, arguing by contradiction, assume that there is ε n ↓ 0 and y n ∈ ∂B(x, r) s.t. |v εn (y n )| 2 < 1 -3C 2 χ. Using (117) we obtain that |v εn | 2 ≤ 1 -C 2 χ on ∂B(x, r)

			1 4 | ln ε|	.
			∂B(x,r)	|∂ τ v ε |.	(117)
	which implies that
	2πC 2 2 χ 2 ε 2( 1 2 -1) n	≤	2πC 2 2 r 2 χ 2 ε 2 n
		≤	r ε 2 n ∂B(x,r)

(1 -|v εn | 2 ) 2

Acknowledgement. The authors thank Prof. Leonid Berlyand and Prof. Petru Mironescu for useful and stimulating discussions. The work of M. Dos Santos and O. Misiats was partially supported by NSF grant DMS-0708324. Part of this work was done when M. Dos Santos was visiting Penn State University. He is grateful to the Math Department and to Prof. Leonid Berlyand for the hospitality and support of his visit.

Proof of Corollaries 5 and 6. By combining (93) and (94) we obtain the lower bound for F ε in Ω:

The conclusions of the above corollaries are obtained by solving the discrete minimization problem of the RHS of (100).

As a direct consequence of Proposition 3 and Lemma 8, we have Corollary 7. There is C > 0 independent of ε s.t. for 1 > ρ > 2δ we have

Existence of the limiting solution

We now return to the proof of Theorems 1 and 2.

Recall that {i ε 1 , ..., i ε d ′ } is a set of distinct elements of {1, ..., M }. We choose ε small enough so that i j 's are independent of ε, thus we may simply denote this set by {i 1 , ..., i d ′ }. In Case I, we have d ′ = d and we may assume that {i 1 , ..., i d ′ } = {1, ..., d}. In Case II, we have d ′ = M . Lemma 7 and Corollary 7 imply that for an appropriate extraction ε = ε n ↓ 0 and for a compact K ⊂ Ω \ {a i 1 , ..., a i d ′ }, there is C K > 0 s.t. for small ε we have

Therefore, when ε → 0, up to a subsequence, there exists

We now fix such sequence and a compact

for small ε. By exactly the same argument as in Proposition 7 we deduce that v ε is bounded in C k (K) for all k ≥ 0 and 1

Consequently, up to subsequence we have for a compact set

Now, assume that K is s.t. K ⊂ Ω \ {a i 1 , ..., a i d ′ } but K ∩ ω δ = ∅ (then we are in Case I). Without loss of generality, assume K = B(a k 0 , R), where a k 0 ∈ {a d+1 , ..., a M } and R > 0 is sufficiently small in order to have K ∩ {a 1 , ..., a M } = {a k 0 }.

We prove the second part of the lemma. Let µ ∈ (0, 1) and x ∈ {dist(x, ∂Ω) ≥ ε 1/4 }. Using mean value argument, there is r ∈ (ε 1/2 , ε 1/4 ) s.t r ∂B(x,r)

There exists

and using the conditions on g ε , by mean value argument there is r ∈ (ε

Using the same argument as before, the statement of the lemma follows.