Dos Mickaël

Santos

The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part II: the non-zero degree case

We consider minimizers of a Ginzburg-Landau energy with a discontinuous and rapidly oscillating pinning term, subject to a Dirichlet boundary condition of degree d > 0. The pinning term models an unbounded number of small impurities in the domain. We prove that for strongly type II superconductor with impurities, minimizers have exactly d isolated zeros (vortices). These vortices are of degree 1 and pinned by the impurities. As in the standard case studied by Bethuel, Brezis and Hélein, the macroscopic location of vortices is governed by vortex/vortex and vortex/ boundary repelling effects. In some special cases we prove that their macroscopic location tends to minimize the renormalized energy of Bethuel-Brezis-Hélein. In addition, impurities affect the microscopic location of vortices. Our technics allows us to work with impurities having different size. In this situation we prove that vortices are pinned by the largest impurities.

Introduction

In this article we let Ω ⊂ R 2 be a smooth simply connected domain and let a ε : Ω → {b, 1}, b ∈ (0, 1) be a measurable function. We associate to a ε the pinned Ginzburg-Landau energy

E ε (u) = 1 2 Ω |∇u(x)| 2 + 1 2ε 2 a ε (x) 2 -|u(x)| 2 2 dx. (1.1)
Here, u : Ω → C is in the Sobolev space H 1 (Ω, C) and ε > 0 is the inverse of the Ginzburg-Landau parameter.

Our goal is to consider a discontinuous and rapidly oscillating pinning term (the pinning term is a ε : Ω → {b, 1}). Our pinning term is periodic with respect to a δ × δ-grid with δ = δ(ε) → 0 as ε → 0 (in some cases we drop the periodic hypothesis).

We are interested in the minimization of (1.1) in H 1 (Ω, C) subject to a Dirichlet boundary condition: we fix g ∈ C ∞ (∂Ω, S 1 ) and thus the set of the test functions is

H 1 g := {u ∈ H 1 (Ω, C) | tr ∂Ω u = g}.
The situation where d = deg ∂Ω (g) = 0 was studied in detail in [START_REF] Santos | The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part I: the zero degree case[END_REF]. The non zero degree case (d = deg ∂Ω (g) > 0) is the purpose of the present article. Recall that for Γ ⊂ R 2 a Jordan curve and g ∈ H 1/2 (Γ, S 1 ), the degree (winding number) of g is defined as

deg Γ (g) := 1 2π Γ 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, τ is the direct unit tangent vector of Γ (τ = ν ⊥ where ν is the outward normal unit vector of int(Γ), the bounded open set whose boundary is Γ) and ∂ τ is the tangential derivative on Γ. This energy is a simplification of the full Ginzburg-Landau energy (see Eq. (1.2) below) whose minimizers model the state of a Type II superconductor (the parameter ε corresponds to a material parameter, this parameter is small for Type II superconductor) [START_REF] Tinkham | Introduction to Superconductivity[END_REF], [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF]. The pinning term allows to model a heterogenous superconductor (see [START_REF] Kachmar | Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint[END_REF] or Introduction of [START_REF] Santos | Défauts de vorticité dans un supraconducteur en présence d'impuretés[END_REF]).

Physical informations which can be obtained with the simplification of the full Ginzburg-Landau energy are quantization and location of zeros of minimizers. Their zeros represent the centers of small areas where the superconductivity is destroyed. These areas are called vorticity defects. Here the superconductor is a cylinder whose cross section is Ω and the vorticity defects (under some special conditions) takes the form of small wires parallel to the superconductor [START_REF] Tinkham | Introduction to Superconductivity[END_REF], [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF].

Before going further, let us summarize two previous works in related directions [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF]. In these works, the role of the pinning term is identified: its points of minimum attract the vorticity defects.

In [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], Lassoued and Mironescu considered the case where a ε ≡ a. Here, the pinning term a = b in ω 1 in Ω \ ω , 0 < b < 1, and ω is a smooth inner domain of Ω. These authors proved that the vorticity defects are quantified by deg ∂Ω (g), localized in ω and that their position is governed by a renormalized energy (in the spirit of [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF]).

In [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF], Aftalion, Sandier and Serfaty considered a smooth and ε-dependent pinning term a ε . Their study allows to consider the case where the pinning term has fast oscillations: it is a perturbation of a fixed smooth function b : Ω → [b, 1] s.t. a ε ≥ b.

In contrast with [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF] is dedicated to the study of a full Ginzburg-Landau energy GL ε with the pinning term a ε

GL ε (u, A) = 1 2 Ω |curlA -h ex | 2 + |(∇ -iA)u| 2 + 1 2ε 2 (a 2 ε -|u| 2 ) 2 . (1.2)
We denoted by A ∈ R 2 the electromagnetic vector potential of the induced field and by h ex ≫ 1 the intensity of the applied magnetic field (see [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] for more details).

They considered the following hypotheses on a ε , b:

• |∇a ε | ≤ Ch ex • there is σ ε ∈ R s.t. σ ε = o (ln | ln ε|) -1/2
and for all x ∈ Ω, we have

min B(x,σε) a ε -b = 0.
In the study of the full Ginzburg-Landau functional without pinning term GL 0 ε (GL 0 ε is obtained from (1.2) by taking a ε ≡ 1), the vorticity defects appear for large apply magnetic field. They are characterized by two facts: the presence of isolated zeros x i of a map u with a non zero degree around small circles centered in x i and the existence of a magnetic field inside the domain (curl(A) ≃ h ex inside small discs). The nature of the superconductivity makes that both facts appear together. Assume that the intensity of the applied field h ex depends on 0 < ε < 1 and that h ex /| ln ε| → Λ ∈ R * + . For the full Ginzburg-Landau energy without pinning term GL 0 ε , it is well known (see e.g. [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF]) that there is an inner domain ω Λ (non decreasing w.r.t. Λ) s.t., when ε → 0, the vorticity defects are "uniformly located" by ω Λ (in this situation the number of vortices is unbounded).

In [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF] (study of a full Ginzburg-Landau functional with a pinning term), the authors proved the existence of ω Λ , an inner set of Ω, where the penetration of the magnetic field is located. In contrast with the situation without pinning term, the presence of a ε makes that, in general, the vortices are not uniformly located in ω Λ . Although in the proofs of the main results of [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF], the minimal points of b seem play the role of a pinning site, this fact is not proved. They expect that the most favorable pinning sites should be close to the minima of b : ω Λ should be located close to the points of minimum of b.

One of our goals is to prove that the minimum points of a rapidly oscillating and discontinuous pinning term attract the vorticity defects.

Before going further, we construct our (periodic) pinning term a ε .

Construction 1. The periodic pinning term Consider

• δ = δ(ε) ∈ (0, 1), λ = λ(ε) ∈ (0, 1];

• ω ⊂ Y = (-1/2, 1/2) 2 be a smooth bounded and simply connected open set s.t. (0, 0) ∈ ω and ω ⊂ Y (here Y is the unit cell).

For k, l ∈ Z we denote

Y δ k,l := δ • Y + (δk, δl), Ω incl δ = Y δ k,l ⊂Ω Y δ k,l , ω λ = λ • ω, ω λ per = (k,l)∈Z 2 ω λ + (k, l) and ω ε = (k,l)∈Z 2 s.t. Y δ k,l ⊂Ω δ • ω λ + (δk, δl) .
For b ∈ (0, 1), we define In the rest of this article λ = λ(ε) and δ = δ(ε) are functions of ε. We assume that δ → 0 as ε → 0. In addition, we assume that either λ ≡ 1, or λ → 0 as ε → 0. The case λ → 0 is the diluted case.

a λ : R 2 → {b, 1} x → b if x ∈ ω λ
We make the (technical) assumption

lim ε | ln(λδ)| 3 | ln ε| = 0. (1.3) Remark 2.
• This is slightly more restrictive than asking that λδ ≫ ε α for all α ∈ (0, 1).

• Hypothesis (1.3) is technical, a more natural hypothesis should be λδ ≫ ε or λδ ≫ ε α for some α ∈ (0, 1).

• In [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF] and in the situation where we have a bounded number of zeros (the applied magnetic field is not too large), the smooth pinning term a 0 ε satisfies the condition |∇a 0 ε | ≤ C| ln ε|. In order to compare this assumption with (1.3), we may consider a regularization of our pinning term by a mollifier ρ t (x) = t -2 ρ(x/t). A suitable scale t to have a complete view of the variations of a ε is t = λδ. Thus, |∇(ρ λδ * a ε )| is of order 1 λδ

. Consequently, the condition (1.3) allows to consider a more rapidly oscillating than the condition in [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF]. Indeed, we have ln |∇a 0 ε | ln | ln ε| and on the other hand (1.3) is equivalent to ln |∇(ρ λδ * a ε )| | ln(λδ)| = o(| ln ε| 1/3 ).

The goal of this article is to study the minimizers of

E ε (u) = 1 2 Ω |∇u| 2 + 1 2ε 2 a 2 ε -|u| 2 2 , u ∈ H 1 g
in the asymptotic ε → 0. A standard method (initiated in [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF]) consists in decoupling E ε into a sum of two functionals. The key tool in this method is U ε the unique global minimizer of E ε in H 1 1 (see [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF]). Clearly,

U ε satisfies    -∆U ε = 1 ε 2 U ε (a 2 ε -U 2 ε ) in Ω U ε = 1 on ∂Ω . (1.4)
From the uniqueness of U ε , by construction of a test function, it is easy to get that b ≤ U ε ≤ 1. This special solution may be seen as a regularization of a ε . For example, one may easily prove that U ε is exponentially close to a ε far away from ∂ω ε (a more complete description of U ε is done Appendix D.1). Namely, we have Proposition 3. There are C, α > 0 independent of ε, R > 0 s.t.

|a ε -U ε | ≤ Ce -αR ε in V R := {x ∈ Ω | dist(x, ∂ω ε ) ≥ R}, (1.5)
|∇U ε | ≤ Ce -αR ε ε in W R := {x ∈ Ω | dist(x, ∂ω ε ), dist(x, ∂Ω) ≥ R}. (1.6)
A similar result was proved in [START_REF] Santos | Ginzburg-Landau model with small pinning domains[END_REF] (Proposition 2). The above proposition yields by the same arguments.

As in [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF], we define

F ε (v) = 1 2 Ω U 2 ε |∇v| 2 + 1 2ε 2 U 4 ε (1 -|v| 2 ) 2 .
Then we have for all v ∈ H 1 g , (see [START_REF] Lassoued | Ginzburg-landau type energy with discontinuous constraint[END_REF])

E ε (U ε v) = E ε (U ε ) + F ε (v).
Therefore, u ε is a minimizer of E ε if and only if u ε = U ε v ε where v ε is a minimizer of F ε in H 1 g . Consequently, the study of a minimizer u ε = U ε v ε of E ε in H 1 g (location of zeros and asymptotics) can be performed by combining the asymptotic of U ε with one of v ε .

Our main result is the following Theorem 1. Assume that λ, δ satisfy (1.3) and that λ → 0.

Quantization. There are ε 0 > 0, c > 0 and η 0 > 0 s.t. for 0 < ε < ε 0 :

1. v ε has exactly d zeros x ε 1 , ..., x ε d , 2. B(x ε i , cλδ) ⊂ ω ε , 3. for ρ = ρ(ε) ↓ 0 s.t. | ln ρ|/| ln ε| → 0, there is C > 0 independent of ε satisfying |v ε | ≥ 1 -C | ln ρ| | ln ε| in Ω \ ∪B(x ε i , ρ),

for ε < ε 0

• There are two repulsive effects:

|x ε i -x ε j | ≥ η 0 for i = j and dist(x ε i , ∂Ω) ≥ η 0 ; • deg ∂B(x ε i ,δ) (v ε ) = 1. Location.
• The macroscopic location of the zeros tends to minimize the renormalized energy of Bethuel-Brezis-Hélein [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF], Chapter I Eq. ( 47)):

W g : {{x 1 , ..., x d } ⊂ Ω | x i = x j for i = j} → R (defined in
lim sup W g (x ε 1 , ..., x ε d ) = min a 1 ,...,a d ∈Ω a i =a j W g (a 1 , ..., a d )
• The microscopic location of the zeros inside ω ε tends to depend only on ω and b:

since

x ε i ∈ ω ε , we have x ε i = (k ε δ, l ε δ) + λδy ε i with k ε , l ε ∈ Z and y ε i ∈ ω; -for ε n ↓ 0 s.t. y εn
i → âi , we have âi ∈ ω which minimizes a renormalized energy W1 : ω → R (given in [START_REF] Santos | Ginzburg-Landau model with small pinning domains[END_REF] Eq. ( 90)) which depends only on ω and b ∈ (0, 1).

Remark 4.

1. The renormalized energy defined in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] W

g : {{x 1 , ..., x d } ⊂ Ω | x i = x j for i = j} → R
governs the location of the zeros in the situation where a ε ≡ 1 (homogenous case): the zeros tend to minimize W g . In [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] (Chapter 1), the authors defined a renormalized energy in a more general setting

W BBH g : {(x 1 , d 1 ), ..., (x N , d N )} x i ∈ Ω, x i = x j for i = j d i ∈ Z is s.t. N i=1 d i = d → R.
Here W g (x 1 , ..., x d ) = W BBH g ({(x 1 , 1), ..., (x d , 1)}), i.e., in this article we will consider only the renormalized energy with the degrees equal 1 and thus we do not specify the degrees in its notation.

2. From smoothness of W g (see [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] and [START_REF] Comte | The behavior of a ginzburg-landau minimizer near its zeroes[END_REF]), Location part of Theorem 1 implies that up to pass to a subsequence, the zeros converge to a minimizer of W g .

3. This macroscopic location is strongly correlated with the Dirichlet boundary condition g ∈ C ∞ (∂Ω, S 1 ).

4. The result about the macroscopic position of the periodic and diluted pinning term may be sum up as: the macroscopic position of the zeros tends to be the same than in the homogenous case (a ε ≡ 1).

5.

The microscopic location of the zeros (position inside an inclusion) is independent of the boundary condition. For example, in the situation ω = B(0, r 0 ), i.e., the inclusions are discs, this location should be the center of the inclusion. This fact is not proved yet.

6. In Assertion 4. of Quantization part, deg ∂B(

x ε i ,δ) (v ε ) = deg ∂B(x ε i ,δ) (v ε /|v ε |).

Main results

We present in this section several extensions of the above result dropping either the dilution of the inclusion (λ ≡ 1 instead of λ → 0) or the periodic structure. The main results of this section are obtained under the condition: λδ satisfies (1.3).

Our sharper results are shared into four theorems:

• The first theorem (Theorem 2) gives informations on the zeros of minimizers u ε , v ε (quantization and location).

• The second theorem (Theorem 3) establishes the asymptotic behavior of v ε .

• The third theorem (Theorem 4) establishes, under the additional hypothesis λ → 0, that the microscopic position of the zeros is independent of the boundary condition g.

• The last theorem (Theorem 5) gives an expansion of F ε (v ε ).

The technics developed in this paper allows to consider either the case λ → 0 or λ ≡ 1. The results in the diluted case are more precise. One may drop the periodic structure for the pinning term and consider impurities (the connected components of ω ε = {a ε = b}) with different sizes (adding the hypothesis λ → 0).

More precisely we may consider the pinning term defined as follow:

Construction 5. The general diluted pinning term

• Fix P ∈ N * , j ∈ {1, ..., P } and 1 > ε > 0. We consider M ε j ∈ N and M ε j = ∅ if M ε j = 0 {1, ..., M ε j } if M ε j ∈ N * .
• The sets M ε j 's are s.t. (for sufficiently small ε) one may fix

y ε i,j ∈ Ω s.t. for (i, j) = (i ′ , j ′ ), i ∈ M ε j , i ′ ∈ M ε j ′ we have |y ε i,j -y ε i ′ ,j ′ | ≥ δ j + δ j ′ and dist(y ε i,j , ∂Ω) ≥ δ j . (2.1)
We denote M ε j := {y ε i,j | i ∈ M ε j }. For sake of simplicity, we assume that there is η > 0 s.t. for small ε, we have

M ε 1 ≥ d = deg ∂Ω (g) and min      min i=1,...,d dist(y ε i,1 , ∂Ω), min i,i ′ =1,...,d i =i ′ |y ε i,1 -y ε i ′ ,1 |      ≥ η. (2.2)
• We now define the domain which models the impurities: The pinning term is

ω ε = P j=1 i∈M ε j y ε i,j + δ j • ω λ , ω λ = λ • ω. a ε = b a ε = 1 ≥ 2δ ≥ δ + δ 2 ≈ λδ ≈ λδ 2 ≥ δ ≥ δ 2
a ε : R 2 → {b, 1} x → 1 if x / ∈ ω ε b if x ∈ ω ε
The values of the pinning term are represented Figure 2.

Our main results are Theorem 2. Assume that λ, δ satisfy (1.3) and if the pinning term is not periodic (represented Figure 2) then we assume also that λ → 0.

There is ε 0 > 0 s.t.:

1. for 0 < ε < ε 0 , v ε has exactly d zeros x ε 1 , ..., x ε d , 2. there are c > 0 and η 0 > 0 s.t. for ε < ε 0 , B(x ε i , cλδ) ⊂ ω ε and

min i min j =i |x ε i -x ε j |, dist(x ε i , ∂Ω) ≥ η 0 .
In particular, if the pinning term is not periodic, then the zeros are trapped by the largest inclusions (those of size λδ).

for

ρ = ρ(ε) ↓ 0 s.t. | ln ρ|/| ln ε| → 0, we have for ε < ε 0 , |v ε | ≥ 1 -C | ln ρ| | ln ε| in Ω \ ∪B(x ε i , ρ).
Here C is independent of ε.

4. for ε < ε 0 , deg ∂B(x ε i ,δ) (v ε ) = 1.
Remark 6. Hypothesis (2.2) is used to simplify the statements. Without this hypothesis, some of the results are subject to technical considerations on δ, λ, b... For example if we consider the pinning term a ε defined in Ω = B(0, 2) by

a ε : B(0, 2) → {b, 1} x → b if x ∈ B(0, λδ) ∪ B(1, λδ 2 ) 1 otherwise
, and g ∈ C ∞ (∂Ω, S 1 ) s.t. deg ∂Ω (g) = 2, then Hypothesis (2.2) is not satisfied. In this situation, we may prove that, for sufficiently small ε, v ε has exactly two zeros and if

2(1 -2b 2 )| ln λ| + (1 -3b 2 )| ln δ| → +∞ (resp.
-∞), then the zeroes are in B(0, λδ) (resp. there is one zero inside B(0, λδ) and one zero inside B(1, λδ 2 )).

Theorem 3. Assume that λ, δ satisfy (1.3) and if the pinning term is not periodic (represented Figure 2) then we assume also that λ → 0. Let ε n ↓ 0, up to a subsequence, we have the existence of a 1 , ..., a d ∈ Ω, d distinct points s.t. x εn i → a i and

|v εn | → 1 and v εn ⇀ v * in H 1 loc (Ω \ {a 1 , ..., a d }, S 1 )
where v * solves

-div(A∇v * ) = (A∇v * • ∇v * )v * in Ω \ {a 1 , ..., a d } v * = g on ∂Ω .
Here A is the homogenized matrix of

a 2 • δ Id R 2 if λ ≡ 1 and A = Id R 2 if λ → 0.
In addition, for each

M > 0, v ′ ε,i (•) = v ε x ε i + ε b • converges, up to a subsequence, in C 1 (B(0, M )) to f (|x|)
x |x| e ıθ i where f : R + → R + is the universal function defined in [START_REF] Mironescu | Les minimiseurs locaux pour l'équation de Ginzburg-Landau sont à symétrie radiale[END_REF] and θ i ∈ R.

Theorem 4. Assume, in addition to the hypotheses of Theorem 3, that λ → 0.

Let [x] = [(x 1 , x 2 )] = ([x 1 ], [x 2 ]) ∈ Z 2 be the vectorial integer part of the point x ∈ R 2 . For x ε i a zero of v ε , let
y ε i = x ε i δ -[ x ε i δ ] λ ∈ ω.
Then, as ε → 0, up to pass to a subsequence, we have

y ε i → âi ∈ ω.
Here, âi minimizes a renormalized energy W1 : ω → R (given in [START_REF] Santos | Ginzburg-Landau model with small pinning domains[END_REF] Eq. ( 90)) which depends only on ω and b. In particular, âi is independent of the boundary condition g.

Theorem 5. Assume that λ, δ satisfy (1.3) and if the pinning term is not periodic (represented Figure 2) then we assume also that λ → 0.

We have the following expansion

F ε (v ε ) = J ε,ε + db 2 (π ln b + γ) + o ε (1)
where J ε,ε is defined in (3.6) and γ > 0 is the universal constant defined in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] Lemma IX.1.

This article is divided in two parts:

• In the first one (Section 3) we consider two auxiliary minimization problems for weighted Dirichlet functionals associated to S 1 -valued maps.

• The second part (Section 4) is devoted to the proofs of Theorems 1, 2, 3, 4, 5. The main tool is an η-ellipticity result (Lemma 19). This lemma reduces (under the assumption that λ, δ satisfy (1.3)) the study of F ε to the one of the auxiliary problems considered Section 3.

Shrinking holes for weighted Dirichlet functionals

This section is devoted to the study of two minimization problems and it is divided in three subsections.

The first and the second subsections are related with minimizations of weighted Dirichlet functionals among S 1 -valued maps. In both subsections, the considered weights are the more general one:

α ∈ L ∞ (R 2 , [b 2 , 1]
). The third subsection deals the weight α = U 2 ε in the situation where U ε is the minimizer of E ε in H 1 1 with a ε represented Figure 1 (the periodic case with or without dilution) or Figure 2 (the general diluted case).

Notation 7. In Section 3 we fix :

• a smooth simply connected domain Ω ⊂ R 2 ; • a boundary condition g ∈ C ∞ (∂Ω, S 1 ) s.t. d := deg ∂Ω (g) > 0; • a smooth and bounded open subset Ω ′ ≺ R 2 s.t. Ω ⊂ Ω ′ ;
• an extension of g which is in C ∞ (Ω ′ \ Ω, S 1 ) (this extension is also denoted by g).

We will also consider (uniformly bounded) families of points/degres {(x 1 , d 1 ), ..., (x N , d N )} = {x, d} s.t.

• x i ∈ Ω, x i = x i ′ for i = i ′ ; • d i are s.t. d i ∈ N * and i d i = d (thus N ≤ d).
According to the considered problems, for 0 < ρ ≤ 8 -1 min i =i ′ |x ix i ′ | we will use the following perforated domains

• Ω ρ := Ω ρ (x) = Ω \ ∪ i B(x i , ρ) ; • Ω ′ ρ := Ω ′ ρ (x) = Ω ′ \ ∪ i B(x i , ρ).

Existence results

In this subsection we prove the existence of solutions of two minimization problems whose studies will be the purpose of the rest of Section 3 (Subsections 3.2 & 3.3).

Existence of minimal maps defined in a perforated domain

Let x = (x 1 , ..., x N ) be 1 ≤ N ≤ d distinct points of Ω and let d = (d 1 , ..., d N ) ∈ (N * ) N s.t. i d i = d. For 0 < ρ < 8 -1 min i =j |x i -x j |, we denote Ω ρ = Ω \ ∪B(x i , ρ).
We define

I ρ (x, d) = I ρ := w ∈ H 1 (Ω ρ , S 1 ) | w = g on ∂Ω and deg ∂B(x i ,ρ) (w) = d i and for 0 < ρ < 8 -1 min {min i =j |x i -x j |, min i dist(x i , ∂Ω)} J ρ (x, d) = J ρ := w ∈ H 1 (Ω ρ , S 1 ) | w = g on ∂Ω and w(x i + ρe ıθ ) = e ı(d i θ+θ i ) , θ i ∈ R .
From the compatibility condition deg ∂Ω (g

) = d = d i , we have I ρ (x, d), J ρ (x, d) = ∅ and it is clear that J ρ (x, d) ⊂ I ρ (x, d).
In Subsection 3.2, we compare the minimal energies corresponding to a weighted Dirichlet functional in the above sets. Here, we just state existence results.

Proposition 8. Let α ∈ L ∞ (Ω) be s.t. b 2 ≤ α ≤ 1.
Consider the minimization problems

I ρ,α (x, d) = inf w∈Iρ 1 2 Ωρ α|∇w| 2 and J ρ,α (x, d) = inf w∈Jρ 1 2 Ωρ α|∇w| 2 .
In both minimization problems the infima are attained. Moreover, if α ∈ W

The proof of this standard result is postponed to Appendix A.

In the special case α = U 2 ε , we denote

I ρ,ε (x, d) = inf w∈Iρ 1 2 Ωρ U 2 ε |∇w| 2 and J ρ,ε (x, d) = inf w∈Jρ 1 2 Ωρ U 2 ε |∇w| 2 .

Existence of an optimal perforated domain

For α ∈ L ∞ (R 2 , [b 2 , 1]) we define I ρ,α := inf x 1 ,...,x N ∈Ω |x i -x j |≥8ρ d 1 ,...,d N >0, d i =d inf w∈H 1 g (Ω ′ ρ ,S 1 ) deg ∂B(x i ,ρ) (w)=d i 1 2 Ω ′ ρ α|∇w| 2 (3.3)
and

J ρ,α := inf x 1 ,...,x d ∈Ω |x i -x j |≥8ρ dist(x i ,∂Ω)≥8ρ inf w∈H 1 g (Ωρ,S 1 ) w(x i +ρe ıθ )=e ı(θ+θ i ) ,θ i ∈R 1 2 Ωρ α|∇w| 2 . (3.4)
Here Ω ′ ρ = Ω ′ \ ∪B(x i , ρ). In the special case α = U 2 ε , we denote

I ρ,ε := inf x 1 ,...,x N ∈Ω |x i -x j |≥8ρ d 1 ,...,d N >0, d i =d inf w∈H 1 g (Ω ′ ρ ,S 1 ) deg ∂B(x i ,ρ) (w)=d i 1 2 Ω ′ ρ U 2 ε |∇w| 2 (3.5)
and

J ρ,ε := inf x 1 ,...,x d ∈Ω |x i -x j |≥8ρ dist(x i ,∂Ω)≥8ρ inf w∈H 1 g (Ωρ,S 1 ) w(x i +ρe ıθ )=e ı(θ+θ i ) ,θ i ∈R 1 2 Ωρ U 2 ε |∇w| 2 . (3.6)
We have the following result

Proposition 9. For α ∈ L ∞ (R 2 , [b 2 , 1]),

Dirichlet Vs Degree Conditions in a fixed perforated domain

Let η stop > 0 be s.t. η stop < 10 -5 • 9 -d 2 diam(Ω) and let N ∈ {1, ..., d}. Consider x 1 , ..., x N ∈ Ω, N distinct points of Ω satisfying the condition η stop < 10 -3 • 9 -d 2 min dist(x i , ∂Ω), and let ρ > 0 be s.t. min {η stop , min i =j |x ix j |} ≥ 8ρ. Roughly speaking η stop controls the distance between the points and ∂Ω.

The main result of this section is Proposition 10. There is C 0 > 0 depending only on g, Ω, η stop and b s.t. for α ∈ L ∞ (Ω, [b 2 , 1]) we have

I ρ,α (x, d) ≤ J ρ,α (x, d) ≤ I ρ,α (x, d) + C 0 .
Here, I ρ,α and J ρ,α are defined Proposition 8.

The rigorous proof of Proposition 10 is presented in Appendix C. Here, we simply present the main lines of the proof.

Two situations are possible:

1. N = 1 or the points x 1 , ..., x N are well separated: The domains are

• the thin domain Ω 10 -1 ηstop (x) = Ω \ ∪B(x i , 10 -1 η stop ) obtained by perforating Ω by "large", "well separated" and "far from ∂Ω" discs,

• the thick annulars B(x i , 10 -1 η stop ) \ B(x i , ρ).

The proof is made in three steps:

Step 1: Using Lemma 44, we obtain a constant C 1 (depending only on g, Ω, η stop ) s.t.

J 10 -1 ηstop,α (x, d) ≤ C 1 .
Step 2: With the help of Proposition 45, we obtain the existence of a constant C b (depending only on b) s.t. for d ∈ N, denoting A i ρ = B(x i , 10 -1 η stop ) \ B(x i , ρ), we have

inf w∈H 1 (A i ρ ,S 1 ) w(x 1 +10 -1 ηstope ıθ )=Cst 1 e ı dθ w(x 1 +ρe ıθ )=Cst 2 e ı dθ 1 2 A i ρ α|∇w| 2 ≤ inf w∈H 1 (A i ρ ,S 1 ) deg ∂B(x i ,ρ) = d 1 2 A i ρ α|∇w| 2 + C b d2 .
Step 3: By extending a minimizer of J 10 -1 ηstop,α (x, d) by the ones of

1 2 A i ρ α|∇ • | 2 with
Dirichlet conditions, we can construct a map which proves the result taking

C 0 = C 1 + d 3 C b .

Optimal perforated domains for the degree conditions

Recall that we fixed Ω ′ ⊃ Ω a smooth bounded domain s.t. dist(∂Ω ′ , Ω) > 0 and a smooth S 1 -valued extension of g to Ω ′ \ Ω (still denoted by g).

In this section, we study the minimization problem

I ρ,ε := inf x 1 ,...,x N ∈Ω |x i -x j |≥8ρ d 1 ,...,d N >0, d i =d inf w∈H 1 g (Ω ′ ρ ,S 1 ) deg ∂B(x i ,ρ) (w)=d i 1 2 Ω ′ ρ U 2 ε |∇w| 2 (3.7)
where

Ω ′ ρ = Ω ′ \ ∪B(x i , ρ) and H 1 g (Ω ′ ρ , S 1 ) = w ∈ H 1 (Ω ′ ρ , S 1 ) | w = g in Ω ′ \ Ω ∪ B(x i , ρ)
; here, we extended U ε with the value 1 outside Ω. We recall that we denoted by U ε the unique global minimizer of E ε in H 1 1 . In this subsection we assume that Hypothesis (1.3) holds (| ln(λδ)| 3 /| ln ε| → 0). This is not optimal for the statements but it makes the proofs simpler (this hypothesis may be relaxed, but it appears as a crucial and technical hypothesis for the methods developed Section 4).

A first purpose of this section is the study of the behavior of I ρ,ε when ρ = ρ(ε) → 0 as ε → 0. In view of the application we have in mind we suppose that λδ P +1 ≫ ρ(ε) ≥ ε but this is not crucial for our arguments (here P = 1 if U ε is associated associated with the periodic pinning term) .

A second objective of our study is to exhibit the behavior of almost minimal configurations {(x n 1 , ..., x n N ), (d n 1 , ..., d n N )}. For fixed ρ, ε, the existence of a minimal configuration of points x ρ,ε is the purpose of Proposition 9. In this section we consider only almost minimal configurations. Notation 11. For ε n ↓ 0, we say that {(x n 1 , ...,

x n N ), (d n 1 , ..., d n N )} is an almost minimal configuration for ρ = ρ(ε n ) ↓ 0 when x n 1 , ..., x n N ∈ Ω, |x n i -x n j | ≥ 8ρ, d n 1 , ..., d n N > 0, d n i = d and there is C > 0 (independent of n) s.t. inf w∈H 1 g (Ω ′ ρ ,S 1 ) deg ∂B(x n i ,ρ) (w)=d n i 1 2 Ω ′ ρ U 2 εn |∇w| 2 -I ρ,εn ≤ C.
Roughly speaking, we establish in this section two repelling effects for the points: point/point and point/∂Ω ; and an attractive effect for the points by the inclusions ω ε .

The case of the periodic pinning term

The main result of this section establishes that when ε n , ρ ↓ 0, an almost minimal configuration {(x Assume that λ, δ satisfy (1.3) and let a ε be the periodic the pinning term (represented Figure 1).

Let

ε n ↓ 0, ρ = ρ(ε n ) ↓ 0,
= j 0 s.t. |x n i 0 -x n j 0 | → 0. Then inf w∈H 1 g (Ω ′ ρ ,S 1 ) deg ∂B(x n i ,ρ) (w)=d n i 1 2 Ω ′ ρ U 2 εn |∇w| 2 -I ρ,εn → ∞. 2. Assume that there is i 0 ∈ {1, ..., N } s.t. dist(x n i 0 , ∂Ω) → 0. Then inf w∈H 1 g (Ω ′ ρ ,S 1 ) deg ∂B(x n i ,ρ) (w)=d n i 1 2 Ω ′ ρ U 2 εn |∇w| 2 -I ρ,εn → ∞.
3. Assume that ρ λδ → 0 and that there is

i 0 ∈ {1, ..., N } s.t. x n i 0 / ∈ ω ε or s.t. x n i 0 ∈ ω ε and dist(x n i 0 , ∂ω ε ) λδ → 0. Then inf w∈H 1 g (Ω ′ ρ ,S 1 ) deg ∂B(x n i ,ρ) (w)=d n i 1 2 Ω ′ ρ U 2 εn |∇w| 2 -I ρ,εn → ∞.
A straightforward consequence of Proposition 12 is the following Corollary 13.

Consider an almost minimal configuration {x

ρ,ε , d ρ,ε } ∈ Ω N × N * N , i.e., assume that there is w ρ,ε ∈ H 1 g (Ω ′ \ ∪B(x ρ,ε i , ρ), S 1 ) verifying deg ∂B(x ρ,ε i ,ρ) (w) = d ρ,ε i and 1 2 Ω ′ \∪B(x ρ,ε i ,ρ) U 2 ε |∇w| 2 ≤ I ρ,ε + C.
(Here, C is independent of ε.)

Then, there is some η 0 independent of ε s.t., for small ε, we have

|x ρ,ε i -x ρ,ε j |, dist(x ρ,ε i , ∂Ω) ≥ η 0 and d i = 1 for all i = j, i, j ∈ {1, ..., N }.
In particular, we have

N = d. 2. If, in addition, ρ = ρ(ε) is s.t. ρ ≥ ε and ρ λδ → 0, then there is c > 0 (independent of ε) s.t., for small ε, we have B(x ρ,ε i , cλδ) ⊂ ω ε .
Proof of Corollary 13. We prove the first part. Let C > 0. We argue by contradiction and we assume that for all n ∈ N * there are

0 < ε n ≤ ρ = ρ(ε n ) ≤ 1/n, x n = x ρ,εn , (d 1 , ..., d N )
and w n = w ρ,εn satisfying the hypotheses of Corollary 13 and s.t.

min |x n i -x n j |, dist(x n i , ∂Ω) → 0 or s.t. there is i ∈ {1, ..., N } for which we have d i = 1.
By construction we have that {x ρ,εn , d} is an almost minimal configuration for I ρ,εn with ρ = ρ(ε n ) ≥ ε n . Clearly from Proposition 12 we find a contradiction. The proof of the second part is similar.

We end this subsection by the following direct consequence of Corollary 13

Corollary 14. For sufficiently small ε, ρ, an almost minimal configuration (x 1 , ..., x d ) for J ρ,ε is an almost minimal configuration for I ρ,ε . Moreover, there is

C 0 > 0 s.t. J ρ,ε ≤ I ρ,ε + C 0 , C 0 is independent of small ε, ρ. Proof. Let C ≥ 0 and let (x 1 , ..., x d ), (x ′ 1 , ..., x ′ d ) ∈ Ω d be s.t. Ĵρ,ε (x 1 , ..., x d ) ≤ J ρ,ε + C and Îρ,ε (x ′ 1 , ..., x ′ d ) ≤ I ρ,ε + C. From Corollary 13, there is η 0 = η 0 (C) > 0 s.t. for ε ≤ ρ ≤ η 0 , min i dist(x ′ i , ∂Ω) ≥ η 0 . Using Proposition 10 we have the existence of C 0 s.t. Îρ,ε (x 1 , ..., x d ) ≤ Ĵρ,ε (x 1 , ..., x d ) ≤ J ρ,ε + C ≤ Ĵρ,ε (x ′ 1 , ..., x ′ d ) + C ≤ Îρ,ε (x ′ 1 , ..., x ′ d ) + C + C 0 ≤ I ρ,ε + 2C + C 0 .

A more precise result for the case of the periodic pinning term with dilution

In this section we focus on the periodic pinning term (represented Figure 1) with dilution: λ → 0. Notation 15. We define two kinds of configuration of distinct points of Ω:

• We say that for ε n ↓ 0 and ρ = ρ(ε n ) → 0, d distinct points of Ω, x n = (x n 1 , ..., x n d ) form a quasi-minimizer of J ρ,εn when J ρ,εn (x n ) -J ρ,εn → 0. • We say that for ε n ↓ 0 and ρ = ρ(ε n ) → 0, d distinct points of Ω, x n = (x n 1 , ..., x n d
) form a quasi-minimizer of W g , the renormalized energy of Bethuel-Brezis-Hélein (see [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF]) when W g (x n ) → min W g .

Proposition 16. [Asymptotic location of optimal perforations]

Assume that λ, δ satisfy (1.3) and that λ → 0.

Let ε n ↓ 0, ρ = ρ(ε n ) → 0, ρ ≥ ε n and x n = (x n 1 , ..., x n d ) be d distinct points of Ω. If the points x n form a quasi-minimizer of J ρ,εn , then x n = (x n 1 , ..., x n d ) form a quasi- minimizer of W g .
This proposition is proved Appendix E.

The case of a general pinning term with variable sizes of inclusions

We assume that a ε is the general pinning term represented Figure 2 with the hypothesis on the dilution: λ → 0.

Proposition 17. [The case of a non-periodic pinning term]

Assume that λ, δ satisfy (1.3) and λ → 0.

Let ρ = ρ(ε) s.t. ρ ≥ ε and ρ λδ 3/2 → 0. If {x ρ,ε , d ρ,ε } is an almost minimal configura- tion for I ρ,ε , then N = d (thus d i = 1
for all i) and there are c, η 0 > 0 (independent of ε) s.t. for sufficiently small ε:

1. |x ρ,ε i -x ρ,ε j |, dist(x ρ,ε i , ∂Ω) ≥ η 0 for all i = j, i, j ∈ {1, ..., N }.
2. B(x ρ,ε i , cλδ) ⊂ ω ε (the centers of the holes are included in the largest inclusions).

Moreover, there is

C 0 > 0 s.t. J ρ,ε ≤ I ρ,ε + C 0 , C 0 is independent of small ε, ρ. And thus an almost minimal configuration x ρ,ε for J ρ,ε is an almost minimal configuration for I ρ,ε
This proposition is proved Appendix E (Subsection E.3).

The pinned Ginzburg-Landau functional

In this section, we turn to the main purpose of this article: the study of minimizers of E ε (defined in (1.1)) in H 1 g . The pinning term is the periodic one (represented Figure 1) or the non periodic one (represented Figure 2).

Recall that we fix δ = δ(ε), δ → 0, λ = λ(ε), λ ≡ 1 or λ → 0 satisfying (1.3). If the pinning term is not periodic then we add the hypothesis λ → 0.

Sharp Upper Bound, η-ellipticity and Uniform Convergence

Sharp Upper Bound and an η-ellipticity result

We may easily prove the following upper bound.

Lemma 18. Assume that ρ λδ → 0 (or ρ λδ 3/2 → 0 if the pinning term is not periodic), then we have inf

v∈H 1 g (Ω,C) F ε (v) ≤ db 2 (π ln bρ ε + γ) + J ρ,ε + o ε (1), (4.1)
where γ > 0 is a universal constant defined in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF], Lemma IX.1.

Proof. We construct a suitable test function wε ∈ H 1 g (for sufficiently small ε). From Proposition 9, one may consider

(x ε 1 , ..., x ε d ) = x ε ∈ Ω d , a minimal configuration for J ρ,ε .
Note that since ρ λδ → 0 (or ρ λδ 3/2 → 0 if the pinning term is not periodic), from Corollaries 13 & 14 (or Proposition 17 if the pinning term is not periodic), there are η > 0 and c > 0 s.t. for small ε we have B(

x ε i , cλδ) ⊂ ω ε and min i {min i =j |x i -x j |, dist(x i , ∂Ω)} ≥ η. Let w ε be a minimal map in J ρ,ε (x ε , 1) (Proposition 8). We denote 1 := (1, ..., 1) ∈ N d Let u ε/(bρ) ∈ H 1 (B(0, 1), C) be a global minimizer of E 0 ε/(bρ) (u) = 1 2 B(0,1) |∇u| 2 + b 2 ρ 2 2ε 2 (1 -|u| 2 ) 2 , u ∈ H 1 x/|x| (B(0, 1), C).
We consider the test function

wε (x) =    w ε in Ω ρ α ε i u ε/(bρ) x -x ε i ρ in B(x ε i , ρ)
.

Here the constants [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] Lemma IX.1) and Proposition 3.

α ε i ∈ S 1 are s.t. w ε (x ε i + ρe ıθ ) = α ε i e ıθ . Estimate (4.1) is obtained by using the fact that E 0 ε (u ε ) = π| ln ε| + γ + o ε (1) as ε → 0 (see

Note that

I ρ,ε ≤ J ρ,ε ≤ πd| ln ρ| + C. (4.2)
We now turn to the η-ellipticity.

We denote by v ε a global minimizer of F ε in H 1 g . We extend |v ε | with the value 1 outside Ω.

One of the main ingredients in this work is the following result.

Lemma 19. [η-ellipticity lemma]

Let 0 < α < 1/2. Then the following results hold:

1. If for ε < ε 0 F ε (v ε , B(x, ε α ) ∩ Ω) ≤ χ 2 | ln ε| -C 1 , then we have |v ε | ≥ 1 -Cχ in B(x, ε 2α ).
Here,

χ ε ∈ (0, 1) is s.t. χ ε → 0 and ε 0 > 0, C > 0, C 1 > 0 depend only on b, α, χ, Ω, g C 1 (∂Ω) . 2. If for ε < ε 0 F ε (v ε , B(x, ε α ) ∩ Ω) ≤ C| ln ε|, then we have |v ε | ≥ µ in B(x, ε 2α ).
Here, µ ∈ (0, 1) and ε 0 , C > 0 depend only on b, α, µ, Ω,

g C 1 (∂Ω) .
This result is a direct consequence of Lemma 1 in [START_REF] Santos | Ginzburg-Landau model with small pinning domains[END_REF].

4.1.2 Uniform convergence of |v ε | outside ω ε
With the help of Lemma 19, we are in position to establish uniform convergence of |v ε | to 1 far away from ω ε .

Proposition 20. Let 10 -2 • dist(ω, ∂Y ) > µ > 0 and K µ ε = {x ∈ Ω | dist(x, ω ε ) ≥ µλδ}.
Then, for sufficiently small ε, we have

|v ε | ≥ 1 -C | ln(λδ)| | ln ε| in K µ ε .
Here C is independent of ε and µ.

Furthermore, if for some small ε, we have

|v ε (x)| < 1 -C | ln(λδ)| | ln ε| , then F ε (v ε , B(x, ε 1/4 )) ≥ 2(πd + 1) b 2 (1 -b 2 ) | ln(λδ)|.
Proof. Using Lemma 19.1 with α = 1/4 and χ = 2(πd + 1)

b 2 (1 -b 2 ) | ln(λδ)| | ln ε| , we obtain the existence of C > 0 s.t. for ε > 0 sufficiently small: if F ε (v ε , B(x, ε 1/4 )) < 2(πd + 1) b 2 (1 -b 2 ) | ln(λδ)|, then we have |v ε | ≥ 1 -Cχ in B(x, ε 1/2 ).
In order to prove Proposition 20, we argue by contradiction. There are ε n ↓ 0, µ > 0 and

x n ∈ K µ εn s.t. |v εn (x n )| < 1 -Cχ.
From (1.5), we find

|U εn -1| ≤ C 0 e -αµ 2ξ in K µ/2 εn , ξ = ε n λδ . (4.3)
Consequently, Lemma 19, the definition of C and (4.3) imply that for large n,

1 2 B(xn,ε 1/4 n ) |∇v εn | 2 + 1 2ε 2 n (1 -|v εn | 2 ) 2 ≥ 2(πd + 1) b 2 (1 -b 2 ) | ln(λδ)| + o ε (1). (4.4)
We extend v ε to Ω ′ := Ω + B(0, 1) with the help of a fixed smooth S 1 -valued map v s.t. v = g on ∂Ω. We also extend U ε and a ε with the value 1 outside Ω.

For n sufficiently large, we have [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] applied with r = 10 -2 • λδµ and for large n, implies the existence of B n = {B n j } a finite disjoint covering by balls of

1 2 Ω ′ |∇v εn | 2 + 1 2ε 2 n (1 -|v εn | 2 ) 2 ≤ C| ln ε n |. Theorem 4.1 in
x ∈ Ω ′ dist(x, ∂Ω ′ ) > ε n b and 1 -|v εn (x)| ≥ ε n b 1/8 s.t. rad (B n ) ≤ 10 -2 • λδµ satisfying 1 2 ∪B n j |∇v εn | 2 + b 2 2ε 2 n (1 -|v εn | 2 ) 2 ≥ π j d n j (| ln ε n | -| ln(λδ)|) -C = π j d n j | ln ξ| -C.
Here, rad (B n ) = j rad(B n j ), rad(B) stands for the radius of the ball B, ξ = ε n /(λδ) and the integers d n j are defined by

d n j = |deg ∂B n j (v εn )| if B n j ⊂ {x ∈ Ω ′ | dist(x, ∂Ω ′ ) > ε n b } 0 otherwise . Since B j ⊂ Ω + B 1/2 ⊂ {x ∈ Ω ′ | dist(x, ∂Ω ′ ) > ε n b }, we obtain 1 2 ∪B n j |∇v εn | 2 + b 2 2ε 2 n (1 -|v εn | 2 ) 2 ≥ πd| ln ξ| -C. (4.5)
From (4.3) and (1.3) we have

F ξ (v εn , ∪ j B j ∪ B(x n , ε 1/4 n )) ≥ b 2 (1 -b 2 ) 2 B(xn,ε 1/4 n ) |∇v εn | 2 + 1 2ε 2 n (1 -|v εn | 2 ) 2 + + b 2 2 ∪ j B j |∇v εn | 2 + b 2 2ε 2 n (1 -|v εn | 2 ) 2 + o n (1). (4.6)
By combining (4.1) (with ρ = λ 2 δ 2 ), (4.2), (4.4), (4.5) and (4.6), we find that

πdb 2 ln[(λδ)/ξ] + πd| ln[(λδ) 2 ]| ≥ F εn (v εn , Ω ′ ) -O n (1) ≥ F εn (v εn , ∪ j B j ∪ B(x n , ε 1/4 n )) -O n (1) ≥ πdb 2 | ln ξ| + 2(πd + 1)| ln(λδ)| -O n (1)
which is a contradiction. This completes the proof of Proposition 20.

Bad discs

Construction and first properties of bad discs

A fundamental tool in this article is the use of ad-hoc coverings of {|v ε | ≤ 7/8} by small discs. The best radius for a covering of {|v ε | ≤ 7/8} should be of the order ε. But the construction of such covering need some preliminary results.

Roughly speaking, the way to get a "sharp" covering is to consider a trivial covering and to "clean" it by dropping some discs with the help an "energetic test" (η-ellipticity result).

Here, we used two kinds of energetic tests: Lemma 19 and Theorem III.3 in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF]. Theorem III.3 in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] gives the most precise results (it allows to deal with discs with radius O(ε)) but it needs a bound on the potential part ε -2 Ω (1 -|v ε | 2 ) 2 which is the purpose of Proposition 30. In order to prove this bound (Proposition 30), we first use larger discs (discs with radius ρ, ε ≪ ρ ≪ λδ P +1 ). The construction of intermediate coverings is done via Lemma 19.

We first consider Notation 21. A trivial covering of Ω by discs For ε > 0, we fix a family of discs B(x i , ε 1/4 ) i∈I s.t

x i ∈ Ω, ∀ i ∈ I, 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.
Then we select discs (using Lemma 19) and we define Notation 22. The initial good/bad discs

• Let C 0 = C 0 (1/4, 7/8), ε 0 = ε 0 (1/4, 7/8) be defined by Lemma 19.2. For ε < ε 0 , we say that B(x i , ε 1/4 ) is an initial good disc if F ε (v ε , B(x i , ε 1/4 ) ∩ Ω) ≤ C 0 | ln ε| and B(x i , ε 1/4 ) is an initial bad disc if F ε (v ε , B(x i , ε 1/4 ) ∩ Ω) > C 0 | ln ε|. (4.7) • We let J = J(ε) := {i ∈ I | B(x i , ε 1/4
) is an initial bad disc}.

An easy consequence of Lemma 18 is

Lemma 23. The number of initial bad discs is bounded

There is an integer N which depends only on g and Ω s.t.

Card J ≤ N. Proof. Since each point of Ω is covered by at most C > 0 (universal constant) discs B(x i , ε 1/4 ), we have i∈J F ε (v ε , B(x i , ε 1/4 ) ∩ Ω) ≤ CF ε (v ε , Ω).
The previous assertion implies that Card J ≤ Cπd C 0 + 1.

Let ρ(ε) = ρ ↓ 0 be s.t.

ρ λδ P +1 → 0 and | ln ρ| 3 | ln ε| → 0. (4.8)
Note that from Assumption (1.3), such a ρ exists, e.g., ρ = (λδ) P +2 . (Recall that if the pinning term is periodic then P = 1)

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

Lemma 24. Separation of the initial bad discs Let ε n ↓ 0. Then (possibly after passing to a subsequence and relabeling the indices), we may choose J ′ ⊂ J and a constant κ independent of n s.t.

J ′ = {1, ..., N ′ }, N ′ = Cst, |x i -x j | ≥ 16κρ for i, j ∈ J ′ , i = j, and ∪ i∈J B(x i , ε 1/4 n ) ⊂ ∪ i∈J ′ B(x i , κρ).
Notation 25. The ρ-bad disc For i ∈ J ′ , we say that B(x i , 2κρ) is a ρ-bad disc.

Proposition 26. We have

1. ρ dist(B(x i , 2κρ), ∂Ω) → 0, 2. deg ∂B(x i ,2κρ) (v εn ) > 0, 3. F εn (v εn , B(x i , 2κρ)) ≥ πb 2 deg ∂B(x i ,2κρ) (v εn ) ln ρ ε n -O(1), 4. |v εn | ≥ 1 -C | ln ρ| | ln ε n | in Ω \ ∪ i∈J ′ B(x i , 2κρ).
Proof. We prove Assertions 1., 2. and 3.. Set

J ′ 0 := {i ∈ J ′ | deg ∂(B(x i ,2κρ)∩Ω) (v εn ) > 0}.
Since |v εn | ≥ 7 8 in Ω \ ∪ i∈J ′ B(x i , 2κρ), we have

0 < d = I∈J ′ deg ∂(B(x i ,2κρ)∩Ω) (v εn ) ≤ I∈J ′ 0 deg ∂(B(x i ,2κρ)∩Ω) (v εn ). (4.9)
Consequently J ′ 0 = ∅. Up to a subsequence, we may assume that J ′ 0 is independent of n. From Proposition 20, for all i ∈ J ′ 0 , we have dist(B(x i , ε 1/4 ), ∂Ω) δ (or δ P if the pinning term is not periodic). Consequently, for i ∈ J ′ 0 we find

dist(B(x i , 2κρ), ∂Ω) ρ → +∞ (4.10)
since ρ λδ P +1 → 0. Assertions 1., 2. and 3. will follow from the estimate

F εn (v εn , B(x i , 2κρ)) ≥ b 2 πdeg ∂B(x i ,2κρ) (v εn ) ln ρ ε n -O(1), (4.11)
valid for i ∈ J ′ 0 . Indeed, assume for the moment that (4.11) holds for i ∈ J ′ 0 . Then, by combining (4.1), (4.2), (4.7), (4.8), (4.9) and (4.11), we find that J ′ 0 = J ′ , i.e., 2. holds. Consequently, by combining Assertion 2. with (4.10), Assertion 1. yields and from Assertion 2. and (4.11), Assertion 3. holds.

We now turn to the proof of (4.11), which relies on Proposition 4.1 in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF]. We apply this proposition in the domain B = B(0, 2κ), to the function v ′ (x) = v εn [ρ(xx i )] and with the rescaled parameter ξ meso = ε ρ .

Note that, from (4.8), ε ≪ ξ meso ≪ ρ ≪ λδ P +1 and | ln ε|

∼ | ln ξ meso | ≫ | ln(λδ)|. Clearly, v ′ satisfies B |∇v ′ | 2 + 1 ξ 2 meso (1 -|v ′ | 2 ) 2 = B(x i ,2κρ) |∇v εn | 2 + 1 ε 2 (1 -|v εn | 2 ) 2 = O(| ln ε|) = O(| ln ξ meso |).
Hence, one may apply the following result of Sandier and Serfaty: there is

(B j ) j∈I , a finite covering of {x ∈ B(0, 2κ -ξ meso /b) | |v ′ (x)| ≤ 1 -(ξ meso /b) 1/8 }
with disjoint balls B j of radius r j < 10 -3 s.t.

1 2 B ∩ ∪B j |∇v ′ | 2 + b 2 ξ 2 meso (1 -|v ′ | 2 ) 2 ≥ π j d j | ln ξ meso | -O(1); here d j = |deg ∂B j (v ′ )| if B j ⊂ B(0, 2κ -ξ meso /b) 0 otherwise . Note that from construction, {|v εn | ≤ 7/8} ⊂ ∪ J B(x i , ε 1/4 n ) ⊂ ∪ J ′ B(x i , κρ). Conse- quently: if deg ∂(B j ∩B(0,2κ-ξmeso/b)) (v ′ ) = 0, then we have B j ⊂ B(0, 3 2 κ).
Therefore,

d j = deg ∂B(0,2κ) (v ′ ) = deg ∂B(x i ,2κρ) (v εn ) and 1 2 B(x i ,2κρ) |∇v εn | 2 + 1 2ε 2 (1 -|v εn | 2 ) 2 ≥ πdeg ∂B(x i ,2κρ) (v εn )| ln ξ meso | -O(1) = πdeg ∂B(x i ,2κρ) (v εn ) ln ρ ε -O(1).
Thus (4.11) holds. The last assertion is obtained using Lemmas 18 & 19. Indeed, note that the proof of (4.11) gives a more precise result

F εn (v εn , B(x i , 3 2 κρ)) ≥ b 2 πdeg ∂B(x i ,2κρ) (v εn ) ln ρ ε n -O(1). Let x ∈ Ω \ ∪ J ′ B(x i , 2κρ) then B(x, ε 1/4 n ) ∩ B(x i , 3 2 κρ) = ∅.
Consequently, using Lemma 18 and the previous lower bound, we obtain:

F εn (v εn , B(x, ε 1/4 n )) ≤ I 2κρ,εn + C 0 ≤ πd| ln ρ| + C 0 .
Therefore, from Lemma 19, there is C > 0, independent of x s.t.

|v εn (x)| ≥ 1-C | ln ρ| | ln ε n | .

Location and degree of bad discs

Let

w n = v εn |v εn | ∈ H 1 (Ω \ ∪ J ′ B(x i , 2κρ), S 1 ).
Proposition 27. The map w n is an almost minimal function for I 2κρ,εn .

Proof. Indeed, denote

K n = 1 2 Ω\∪ J ′ B(x i ,2κρ) U 2 εn |∇w n | 2 , then we have K n ≤ F εn (v εn , Ω \ ∪ J ′ B(x i , 2κρ)) + Ω\∪ J ′ B(x i ,2κρ) U 2 εn (1 -|v εn | 2 )|∇w n | 2 = F εn (v εn , Ω) -F εn (v εn , ∪ J ′ B(x i , 2κρ)) + Ω\∪ J ′ B(x i ,2κρ) U 2 εn (1 -|v εn | 2 )|∇w n | 2 ≤ (4.1), Prop 26 ≤ I 2κρ,εn + C | ln ρ| | ln ε n | Ω\∪ J ′ B(x i ,2κρ) U 2 εn |∇w n | 2 + O(1) ≤ (4.1), Prop 26 ≤ I 2κρ,εn + C | ln ρ| | ln ε n | F εn (v εn , Ω \ ∪ J ′ B(x i , 2κρ)) + O(1) ≤ (4.1), (4.2) Prop 26 ≤ I 2κρ,εn + C | ln ρ| 3 | ln ε n | + O(1) ≤ (4.8) ≤ I 2κρ,εn + O(1).
Remark 28. Note that the penultimate line in the proof of Proposition 27 is the main use of (1.3) (which is express in (4.8)).

By combining Proposition 12 with Proposition 27 in the periodic case or Proposition 17 if the pinning term is not periodic, we obtain the following

Corollary 29. The configuration {(x 1 , ..., x N ′ ), (deg ∂B(x 1 ,2κρ) (v εn ), ..., deg ∂B(x N ′ ,2κρ) (v εn ))} is an almost minimal configuration of I 2κρ,εn and consequently, N ′ = d, deg ∂B(x i ,2κρ) (v εn ) =
1 for all i and there is η 0 > 0 independent of large n s.t.

min min i =j |x i -x j |, min i dist(x i , ∂Ω) > 2η 0 , B(x i , 2η 0 λδ) ⊂ ω ε .

H 1 loc -weak convergence

In order to keep notations simple, we replace from now on, 2κρ by ρ/2. Using Corollary 29, there is {a 1 , ..., a d } ⊂ Ω s.t. possibly after passing to a subsequence, we have x n i = x i → a i . Let ρ 0 > 0 be defined as

ρ 0 = 10 -2 • min k =l {dist(a k , ∂Ω), |a k -a l |} .

The contribution of the modulus is bounded in the whole domain

We are now in position to bound the potential part of F ε (v ε ). More precisely we have Proposition 30. We have

Ω |∇|v εn || 2 + 1 ε 2 n (1 -|v εn | 2 ) 2 = O(1).
Proof. From (4.1), Proposition 26 (Assertion 1., 2. and 3.) and Proposition 27, we infer that

Ω\∪ i B(x i ,ρ/2) |∇|v εn || 2 + 1 ε 2 n (1 -|v εn | 2 ) 2 = O(1).
Consequently it suffices to obtain a similar estimate in B(x i , ρ/2). Note that B(x i , ρ) ⊂ ω ε . Thus, if we set

u ′ (x) = u εn (x i + ρx) b : B(0, 1) → C, then u ′ solves -∆u ′ = 1 [ε n /(bρ)] 2 u ′ (1 -|u ′ | 2 ) in B(0, 1).
From [START_REF] Bethuel | Improved estimates for the Ginzburg-Landau equation: the elliptic case[END_REF], we obtain

1 2 B(0,1/2) ∇|u ′ | 2 + b 2 ρ 2 2ε 2 n (1 -|u ′ | 2 ) 2 = O(1).
This estimate is the subject of Theorem 1 for the potential part and Proposition 1 in [START_REF] Bethuel | Improved estimates for the Ginzburg-Landau equation: the elliptic case[END_REF] for the gradient of the modulus (see also Corollary 1 in [START_REF] Bethuel | Improved estimates for the Ginzburg-Landau equation: the elliptic case[END_REF]).

Set

K n = 1 2 B(0,1/2) ∇|u ′ | 2 + b 2 ρ 2 2ε 2 n (1 -|u ′ | 2 ) 2 .
Using Proposition 3, we obtain

K n = O(1) = 1 2b 2 B(x i ,ρ/2) |∇|U εn v εn || 2 + b 4 2ε 2 n 1 - |U εn v εn | 2 b 2 2 = 1 2 B(x i ,ρ/2) |∇|v εn || 2 + b 2 2ε 2 n 1 -|v εn | 2 2 + o n (1).
Consequently, Proposition 30 holds.

4.3.2

We bound the energy in a fixed perforated domain Proposition 31. For 0 < η ≤ ρ 0 , there is C(η) > 0 independent of n s.t. we have

1 2 Ω\∪B(a i ,η) |∇v εn | 2 ≤ C(η). (4.12)
Proof. We argue by contradiction and we assume that there is η > 0 s.t., up to pass to a subsequence, we have

Ω\∪B(a i ,η) |∇v εn | 2 → ∞. Because Ω\∪B(a i ,η) |∇v εn | 2 = Ω\∪B(a i ,η) |v εn | 2 |∇w n | 2 + |∇(|v εn |)| 2 , from Propositions 26 & 30 we get Ω\∪B(a i ,η) |∇w n | 2 → ∞. Therefore, we have Ω\∪B(x i ,10 -1 η) |∇w n | 2 → ∞.
It is clear that we may get a map wn ∈ J 10 -1 η,εn (x εn , 1)

s.t. Ω\∪B(x i ,10 -1 η) |∇ wn | 2 ≤ C(η).
For i = 1, ..., d, using Proposition (45) (Appendix C, Section C.3, Page 44), we get the existence of a map wi,n ∈ H 1 (B(x i , 10 -1 η) \ B(x i , ρ/2), S 1 ) s.t. wi,n (x i + 10 -1 ηe ıθ ) = wn (x i + 10 -1 ηe ıθ ) and

B(x i ,10 -1 η)\B(x i ,ρ/2) U 2 εn |∇ wi,n | 2 ≤ B(x i ,10 -1 η)\B(x i ,ρ/2) U 2 εn |∇w n | 2 + O(1)
Therefore by extending wn with wi,n in B(x i , 10 -1 η)\B(x i , ρ/2) we get a map still denoted wn ∈ H 1 g (Ω \ ∪B(x i , ρ/2), S 1 ) s.t.

1 2 Ω\∪B(x i ,ρ/2) |∇w n | 2 - 1 2 Ω\∪B(x i ,ρ/2) |∇ wn | 2 → ∞
which is in contradiction with Proposition 27.

Consequently, there is v * ∈ H 1 loc (Ω \ {a 1 , ..., a d }, S 1 ) s.t., up to pass to a subsequence,

v εn ⇀ v * in H 1 loc (Ω \ {a 1 , ..., a d }).
Next section is dedicate to the limiting equation of v * .

We establish the limiting equation

In order to obtain the expression of the homogenized problem, we use the unfolding operator (see [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF], definition 2.1).

The use of the unfolding operator needs a slightly modification of the cell period. More precisely, instead of considering the δ × δ-grid whose vertices-grid are the points {δ(k, l)+(1/2, 1/2) | k, l ∈ Z}, we consider the one whose vertices-grid are {δ(k, l) | k, l ∈ Z}.

Thus instead of having cells which contain one inclusion at their center we have cells with quarters of inclusion at their vertices. (See Figure 3)

More specifically, we define, for Ω 0 ⊂ R 2 an open set, p ∈ (1, ∞) and δ > 0,

T δ : L p (Ω 0 ) → L p (Ω 0 × Ỹ ) φ → T δ (φ)(x, y) = φ δ x δ + δy for (x, y) ∈ Ωincl δ × Ỹ 0 for (x, y) ∈ Λ δ × Ỹ .
Here, Ỹ = (0, 1) 2 , [s] is the integer part of s ∈ R and

Ωincl δ := Ỹ K δ ⊂Ω 0 , K∈Z 2 Ỹ K δ =δ•(K+ Ỹ ) Ỹ K δ , Λ δ := Ω 0 \ Ωincl δ and x δ := x 1 δ , x 2
δ .

An adaptation of a result of Sauvageot ([21], Theorem 4) gives the following

Proposition 32. Let Ω 0 ⊂ R 2 be a smooth bounded open set. Let v n ∈ H 2 (Ω 0 , C) be s.t. 1. |v n | ≤ 1 and Ω 0 (1 -|v n | 2 ) 2 → 0, 2. v n ⇀ v * in H 1 (Ω 0 ) for some v * ∈ H 1 (Ω 0 , S 1 ), 3. there are H n ∈ W 1,∞ (Ω 0 , [b 2 , 1]) and δ = δ n ↓ 0 s.t. T δ (H n )(x, y) → H 0 (y) in L 2 (Ω 0 × Ỹ ), • (-δ 2 , -δ 2 ) • ( δ 2 , -δ 2 ) • (-δ 2 , δ 2 )
• (0, 0)

. -div(H n ∇v n ) = v n f n (x), f n ∈ L ∞ (Ω 0 , R). Then v * is a solution of -div(A∇v * ) = (A∇v * • ∇v * )v *
where A is the homogenized matrix of H 0 ( • δ )Id R 2 . (See Appendix F to have more details about A)

The proof of Proposition 32 is postponed to Appendix F. We apply the above proposition with Ω 0 = Ω \ ∪B(a i , η), δ = δ n ↓ 0 the sequence which defines a εn and H n = U 2 εn . By application of Proposition 3, we obtain

T δ (U 2 εn )(x, y) L 2 (Ω 0 × Ỹ ) → ã2 (y) if λ ≡ 1 1 if λ → 0 . Note that the Ỹ -periodic extension of ã in R 2 is equal to the Y -periodic extension of 1 -(1 -b 2 )1I ω λ which is a λ (defined Construction 1). We find that v * solves -div(A∇v * ) = (A∇v * • ∇v * )v * if λ ≡ 1 -∆v * = |∇v * | 2 v * if λ → 0 .
Here A is the homogenized matrix of

[a λ ( • δ )] 2 Id R 2 .
4.4 The small bad discs

Definition

With the help the bound on the potential part of the minimizers 1

ε 2 Ω (1 -|v ε | 2 ) 2 ≤ C (Proposition 30
), in the spirit of [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] (Theorem III.3), we may detect the vorticity defects (the connected components of {|v ε | ≤ 7/8}) by smaller discs (discs with radius of order ε) than the ρ-bad discs (Notation 25).

Notation 33. The small bad discs

The construction is done as follows:

• We consider a covering of Ω as in Notation 21 (page 21). We fix ρ = ρ(ε) ↓ 0 s.t. Assumption (4.8) holds. For sufficiently small ε, we denote

S ρ (ε) = {B(x, R/2) | B(x, R)
given by Notation 25, page 22}.

• Following [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] (Theorem III.3), for l ≥ 2, there are κ l , µ l > 0 (depending only on Ω, g and l)

s.t. for x ∈ Ω, if 1 ε 2 B(x,2κ l ε) (1 -|v ε | 2 ) 2 ≤ µ l then |v ε | ≥ 1 - 1 l 2 in B(x, κ l ε).
We fix l ≥ 2 and we drop the subscript l. We now consider a covering of ∪ B∈Sρ(ε) B by discs (B(x ε i , κε)) i∈I s.t

x ε i ∈ ∪ B∈Sρ(ε) B, ∀ i ∈ I, B(x ε i , κε/4) ∩ B(x ε j , κε/4) = ∅ if i = j, ∪ i∈I B(x ε i , κε) ⊃ ∪ B∈Sρ(ε) B. We say that B(x ε i , κε) is a small good disc if 1 ε 2 B(x ε i ,2κε) (1 -|v ε | 2 ) 2 < µ.
• If B(x ε i , κε) is not a small good disc, then we call it a small bad disc. We denote J ⊂ I the set of indices of small bad discs.

Following [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF], using Proposition 30, there is N l = N > 0 (depending only on Ω, g and l) s.t. Card(J) ≤ N .

Using a standard separation process (Lemma 42), for ε n ↓ 0, possibly after passing to a subsequence and relabeling the discs, there are J ′ ⊂ J and κ ′ ∈ {κ, ..., 9 N -1 κ} s.t.

{|v εn | < 1 -1/l 2 } ⊂ ∪ i∈J B(x εn i , κε n ) ⊂ ∪ i∈J ′ B(x εn i , κ ′ ε n ) and |x εn i -x εn j | ε n ≥ 8κ ′ if i, j ∈ J ′ , i = j.
By a standard iterative procedure, we may assume that the small bad discs are mutually far away in the ε-scale.

Proposition 34. Possibly after passing to a subsequence, we have, for large R and J ′′ ⊂ J ′ ,

{|v εn | < 1 -1/l 2 } ⊂ ∪ i∈J ′′ B(x εn i , Rε n ), where, for i = j, |x εn i -x εn j | ε n → ∞ as n → ∞.
Notation 35. The small and separated bad discs The discs {B(x εn i , Rε n ) | i ∈ J ′′ } obtained in Proposition 34 are the small and separated bad discs.

Each ρ-bad disc contains exactly one small bad disc

By construction, we know that the small and separated bad discs (defined Notation 35) are covered by the ρ-bad discs defined Notation 25 (page 22). We next prove that there are exactly d small bad discs and consequently, there is exactly one small bad discs per ρ-bad discs.

Proposition 36. For large n and for all i ∈ J ′′ , we have

deg ∂B(x εn i ,Rεn) (v εn ) = 1.
Proof. First we prove that, for large n and for all i, we have deg ∂B(x εn i ,Rεn) (v εn ) = 0. We argue by contradiction and we assume that, up to a subsequence, there is i s.t.

deg ∂B(x εn i ,Rεn) (v εn ) = 0. Set M n = min b min i =j |x εn i -x εn j | 8Rε n , δ -1 (4.13)
and set

u ′ n : B(0, M n ) → C x → u εn ε n b x + x εn i b .
Note that, B(x εn i , M n ε n ) ⊂ ω εn and by Proposition 34, we have

M n → ∞. It is easy to check that u ′ n solves -∆u ′ n = u ′ n (1 -|u ′ n | 2 )
. Following [START_REF] Brezis | Quantization effects for -∆u = u(1 -|u| 2 ) in R 2[END_REF], up to a subsequence,

u ′ n → u 0 in C 2 loc (R 2 ); (4.14) here u 0 : R 2 → C solves -∆u 0 = u 0 (1 -|u 0 | 2 ) in R 2 .
Then two cases occur:

R 2 (1 -|u 0 | 2 ) 2 < ∞ or R 2 (1 -|u 0 | 2 ) 2 = ∞. Assume first that R 2 (1 -|u 0 | 2 ) 2 < ∞.
From [START_REF] Brezis | Quantization effects for -∆u = u(1 -|u| 2 ) in R 2[END_REF], noting that the degree of u 0 on large circles centered in 0 is 0, we obtain that u 0 = Cst ∈ S 1 and consequently

R 2 (1-|u 0 | 2 ) 2 = 0. Since u ′ n → u 0 in L 4 (B(0, 2bR)) (R ≥ κ), we find that B(0,2bR) (1 -|u ′ n | 2 ) 2 = b 2 ε 2 n B(x εn i ,2Rεn) (1 -|u n /b| 2 ) 2 = b 2 ε 2 n B(x εn i ,2Rεn) (1 -|v εn | 2 ) 2 + o n (1) → 0.
Noting that B(x εn i , κε n ) is a small bad disc and that B(

x εn i , 2κε n ) ⊂ B(x εn i , 2Rε n ), we have a contradiction. Therefore R 2 (1 -|u 0 | 2 ) 2 = ∞. Consequently, there is M 0 > 0 s.t. B(0,bM 0 ) (1 -|u 0 | 2 ) 2 ≥ sup n 4b 2 ε 2 n Ω (1 -|v εn | 2 ) 2 .
Thus, for large n we have

B(0,bMn) (1 -|u ′ n | 2 ) 2 = b 2 ε 2 n B(x εn i ,Mnεn) (1 -|u εn /b| 2 ) 2 = b 2 ε 2 n B(x εn i ,Mnεn) (1 -|v εn | 2 ) 2 + o n (1) ≥ sup n 2b 2 ε 2 n Ω (1 -|v εn | 2 ) 2 ,
which is a contradiction with B( We define

x εn i , M n ε n ) ⊂ Ω.
ae y n := 10 -2 min i,j∈Λy, i =j |x εn i -x εn j | if Card(Λ y ) > 1 Rε n otherwise .
From Proposition 34, if Card(Λ y ) > 1 then ae n /ε n → ∞.

For simplicity, we assume that y = 0 and we let

B = B(0, 8) \ ∪ i∈Λ 0 B x εn i ρ , ae 0 n ρ .
Remark 37. Note that from Corollary 29 we have B(y, 16ρ) ⊂ ω ε . Clearly, we are in position to apply Theorem 2 in [START_REF] Han | Lower bounds for the energy of S 1 -valued maps in perforated domains[END_REF] in the perforated domain B. After scaling, we find that

1 2 B(y,8ρ)\∪B(x εn i ,ae y n ) |∇v εn | 2 ≥ π i∈Λy deg ∂B(x εn i ,Rεn) (v εn ) ln ρ ae y n -C = π ln ρ ae y n -C.
In order to prove (4.15), we observe the case where there is y s.t. Card(Λ y ) > 1. Recall that if for all y centers of ρ-bad discs we have Card(Λ y ) = 1, then (4.15) holds. Since

deg ∂B(x εn i ,Rεn) (v εn ) = 0, if Card(Λ y ) > 1, then we have i∈Λy |deg ∂B(x εn i ,Rεn) (v εn )| > 1.
We obtain easily the following lower bound for i ∈ Λ y :

1 2 B(
U 2 εn |∇v εn | 2 = πdb 2 ln ρ ε n + O n (1).
Combining the previous estimates, we obtain that

{y center of ρ-bad discs | Card(Λ y ) > 1} = ∅,
and thus deg ∂B(x εn i ,Rεn) (v εn ) = 1 for large n. Corollary 38. For large n, there is a unique zero inside each small and separated bad discs defined Notation 35 (page 28).

Proof. From Proposition 36, one may assume that v εn (x εn i ) = 0. Let i ∈ {1, ..., d}. In view of (4.14), if we denote

u ′ n : B(0, M n ) → C x → u εn ( ε n b x + x εn i ) b , (4.16)
then, up to a subsequence,

u ′ n → u 0 in C 1 (B(0, bR)). (4.17)
Here M n is defined in (4.13).

Using the main result of [START_REF] Mironescu | Les minimiseurs locaux pour l'équation de Ginzburg-Landau sont à symétrie radiale[END_REF], we have the existence of a universal function f :

R + → [0, 1] s.t. u 0 (x) = f (|x|)e ı(θ+θ i ) where x = |x|e ıθ , θ i ∈ R and f : R + → R + is increasing. (4.18)
Therefore, we may apply Theorem 2.3 in [START_REF] Bauman | On the zeros of solutions to Ginzburg-Landau type systems[END_REF] in order to obtain that, for large n, u ′ n has a unique zero in B(0, bR). Consequently, for large n, v εn has a unique zero in B(x εn i , Rε n ).

Corollary 39. One may consider that R depends only on l (R is independent of the extraction we consider), i.e, for l ≥ 2 there is R l > 0 s.t. for small ε, denoting {x ε i | i ∈ {1, ..., d}} the set of zeros of a minimizer v ε , we have

{|v ε | < 1 -1/l 2 } ⊂ ∪ i B(x ε i , R l ε).
Proof. From Corollary 38, one may assume that v εn (x εn i ) = 0. Let f : R + → R + be defined as in (4.18) and u ′ n as in (4.16). For l ≥ 2, consider R l > 0 s.t. l → R l is increasing and f (bR l ) ≥ 1 -1 2l 2 . Note that from [START_REF] Shafrir | Remarks on solutions of -∆u = (1 -|u| 2 )u in R 2[END_REF], one may consider R l ≃ √ 2l/b. By uniqueness of f , the full sequence

|u ′ n | converges to f in L ∞ [B(0, b max {R, R l })].
Consequently, for n sufficiently large, since f is not decreasing,

{|v εn | < 1 -1/l 2 } ⊂ ∪ i B(x εn i , R l ε n ).

Asymptotic expansion of F ε (v ε )

This section is essentially devoted to proof Theorem 5. The key argument in this proof is Proposition 40.

Statement of the main result and a corollary

We state a technical and fundamental result and a direct corollary.

Proposition 40. For all ε n ↓ 0, up to a subsequence, there is ρ = ρ(ε n ) s.t. ε n ≪ ρ ≪ λδ 3/2 and s.t. when n → ∞ the following holds

F εn (v εn ) ≥ J ρ,εn + db 2 (π ln bρ ε n + γ) + o n (1), (4.19)
where J ρ,ε is defined in (3.6) and γ is the universal constant defined in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF], Lemma IX.1.

Corollary 41. Let ε n ↓ 0, ρ be as in Proposition 40. Then we have

J εn,εn -J ρ,εn = πdb 2 ln ρ ε n + o n (1).
Proof of Corollary 41. Using Proposition 9, we may consider x n = (x n 1 , ..., x n d ) ∈ Ω d a minimal configuration of points for J ρ,εn , i.e. s.t.

Ĵρ,εn (x n , 1) = J ρ,εn .

Combining Corollaries 13 & 14 (or Proposition 17 if the pinning term is not periodic), we have the existence of c > 0 s.t. B(x n i , cλδ) ⊂ ω ε . Therefore, for a minimal map w n of Ĵρ,εn (x n , 1), we may easily construct a map wn ∈ H 1 (Ω \ ∪ i B(x i , ε n ), S 1 ) s.t. wn ∈ J εn (x n , 1) and

J εn,εn ≤ 1 2 Ω\∪B(x i ,εn) U 2 εn |∇ wn | 2 = 1 2 Ω\∪B(x i ,ρ) U 2 εn |∇w n | 2 + 1 2 ∪B(x i ,ρ)\B(x i ,εn) U 2 εn |∇ wn | 2 = J ρ,εn + db 2 π ln ρ ε n + o n (1). (4.20)
On the other hand, Lemma 18 combined with Proposition 40 yield

J ρ,εn + db 2 (π ln bρ ε n + γ) + o n (1) ≤ F εn (v εn ) ≤ J εn,εn + db 2 (π ln b + γ). (4.21)
We conclude with the help of (4.20) and (4.21).

Proof of Theorem 5

We are now in position to prove Theorem 5, i.e., we are going to prove that

F ε (v ε ) = J ε,ε + db 2 (π ln b + γ) + o ε (1).
Indeed, using Lemma 18, it suffices to prove that

F ε (v ε ) ≥ J ε,ε + db 2 (π ln b + γ) + o ε (1).
This estimate is equivalent to: for all ε n ↓ 0, up to subsequence, we have F εn (v εn ) ≥ J εn,εn + db 2 (π ln b + γ) + o n (1).

Let ε n ↓ 0. Then, up to a subsequence, there is ρ = ρ n given by Proposition 40 s.t.

F εn (v εn ) ≥ J ρ,εn + db 2 (π ln bρ ε n + γ) + o n (1).
We deduce from Corollary 41 that

F εn (v εn ) ≥ J εn,εn -db 2 ln ρ ε n + db 2 (π ln bρ ε n + γ) + o n (1) = J εn,εn + db 2 (π ln b + γ) + o n (1),
which ends the proof of Theorem 5.

Proof of Proposition 40

In order to construct ρ, we first define a suitable extraction. For l ∈ N \ {0, 1}, consider R l given by Corollary 39. Using Proposition 36 and Corollary 38, for sufficiently large n, v εn has exactly d zeros x n 1 = x 1 , ..., x n d = x d . Clearly, these zeros are well separated and far from ∂Ω (independently of n). Fix i ∈ {1, ..., d} and consider

u ′ n : B(0, λ 2 δ 2 /ε n ) → C x → u εn ( ε n b x + x i ) b .
For simplicity, assume x i = 0. Up to a subsequence, one has, as in (4.18),

u ′ n → u 0 in C 2 loc (R 2 , C), u 0 (x) = f (|x|)e ı(θ+θ i )
where x = |x|e ıθ , θ i ∈ R and f : R + → R + is increasing. Consequently, for l ∈ N \ {0, 1}, one may construct an extraction (n l ) l≥2 s.t., denoting

u ′ n l = u ′ l = |u ′ l |e ı(θ+φ ′ l )
and v εn l = v l , we have

{|v l | < 1 -1/l 2 } ⊂ ∪ i B(x i , R l ε n l ), (4.22)
ρ l := R l ε n l ≤ λ 2 δ 2 l , B(0,bR l ) ∇u ′ l 2 + 1 2 1 -u ′ l 2 2 - B(0,bR l ) |∇u 0 | 2 + 1 2 1 -|u 0 | 2 2 ≤ 1 l , (4.23)
and

φ ′ l -θ i C 1 (B(0,bR l )) ≤ 1 l . (4.24)
Here R l ≃ √ 2l/b and is defined in Corollary 39. Following the proof of Proposition 1, Step 2 in [START_REF] Brezis | Quantization effects for -∆u = u(1 -|u| 2 ) in R 2[END_REF], one has

B(0, λ 2 δ 2 εn l )\B(0,R l ) |∇φ ′ l | 2 ≤ C independently of l. (4.25)
In B(0, λ 2 δ 2 ) \ B(0, ε n l ), we denote v n l = v l = |v l |e ı(θ+φ l ) (e ıθ = x/|x|). By conformal invariance, (4.24) implies that

φ l -θ i L ∞ (∂B(0,ρ l )) + |φ l | H 1/2 (∂B(0,ρ l )) ≤ C l . ( 4

.26)

Denote W l = B(0, 2ρ l ) \ B(0, ρ l ) and consider ψ l i ∈ H 1/2 (∂W l , R) s.t.

ψ l i = φ l -θ i on ∂B(0, ρ l ) 0 on ∂B(0, 2ρ l ) . Using (4.26), it is clear that ψ l i L ∞ (∂W l ) +|ψ l i | H 1/2 (∂W l ) = O(1/l).
From this, it is straightforward that there exists a constant C 0 > 0 (independent of l) and

Ψ l i ∈ H 1 (W l , R) s.t.
tr ∂W l Ψ l i = ψ l i and

1 2 W l |∇Ψ l i | 2 ≤ C 0 l 2 .
Finally we define Ψ l ∈ H 1 (Ω \ ∪B(x i , ρ l ), R) by

Ψ l = Ψ l i (• -x i ) in x i + W l 0 otherwise and wl = v l |v l | e -ıΨ l ∈ J ρ l (x, 1) with x = (x 1 , ..., x d ).
Therefore, denoting

w l = v l |v l | = e ı(θ+φ l ) , U l = U εn l and Ω ρ l = Ω \ B(x i , ρ l ), we have Ĵρ l ,εn l (x, 1) ≤ 1 2 Ωρ l U 2 l |∇ wl | 2 = 1 2 Ωρ l U 2 l |∇w l | 2 + 2U 2 l ∇(θ + φ l ) • ∇Ψ l + o l (1).
From (4.25), we obtain easily that

Ωρ l ∇(θ + φ l ) • ∇Ψ l = i x i +W l ∇(θ + φ l ) • ∇Ψ l i (• -x i ) = o l (1)
and consequently Ĵρ l ,εn l (x, 1) ≤

1 2 Ωρ l U 2 l |∇w l | 2 + o l (1). (4.27)
On the other hand, from direct computations, one has

1 2 Ωρ l U 2 l |∇v l | 2 ≥ 1 2 Ωρ l U 2 l |∇w l | 2 + 1 2 Ωρ l U 2 l (|v l | 2 -1)|∇(θ + φ l )| 2 .
Using the same argument as Mironescu in [START_REF] Mironescu | Les minimiseurs locaux pour l'équation de Ginzburg-Landau sont à symétrie radiale[END_REF], one may obtain that

1 2 Ωρ l (1 -|v l | 2 ) 1/2 |∇(θ + φ l )| 2 ≤ C with C independent of l. (4.28)
From (4.28) and (4.22), we obtain

1 2 Ωρ l U 2 l |∇v l | 2 ≥ 1 2 Ωρ l U 2 l |∇w l | 2 -o l (1).
Therefore, with (4.27),

F εn l (v l , Ω ρ l ) + o l (1) ≥ 1 2 Ωρ l U 2 l |∇v l | 2 + o l (1) ≥ Ĵρ l ,εn l (x, 1). (4.29)
In order to complete the proof of (4.19), it suffices to estimate the contribution of the discs B(x i , ρ l ). One has (using (4.23))

F εn l (v l , B(x i , ρ l )) = b 2 2 B(0,ρ l ) ∇ u l b 2 + b 2 2ε 2 n l 1 - u l b 2 2 + o l (1) = b 2 2 B(0,bR l ) ∇u ′ l 2 + 1 2 1 -u ′ l 2 2 + o l (1) = b 2 2 B(0,bR l ) |∇u 0 | 2 + 1 2 1 -|u 0 | 2 2 + o l (1).
From Proposition 3.11 in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF], one has

1 2 B(0,bR l ) |∇u 0 | 2 + 1 2 1 -|u 0 | 2 2 = π ln(bR l ) + γ + o l (1), hence F εn l (v l , B(x i , ρ l )) = b 2 [π ln(bR l ) + γ] + o l (1). ( 4

Proof of Theorems 1, 2, 3 and 4

We prove Quantization part of Theorem 1 and Theorem 2:

• The existence of exactly d zeros is a direct consequence of Corollary 38.

• The facts that they are well included in ω ε , well separated and that v ε has a degree equal to 1 on small circles around the zeros are obtained by Proposition 36 and Corollary 38.

• The lower bound for |v ε | is given by Proposition 26.4.

We prove Macroscopic location part of Theorem 1:

• The macroscopic location part of Theorem 1 is a direct consequence of Theorem 5 (proved Subsection 4.5.2), Proposition 26, (4.30) and Proposition 16.

Indeed, from Theorem 5, Proposition 26.4 and (4.30), we get that the zeros form a quasiminimizer of J ρ,ε (defined Notation 15, page 17). By using Proposition 16 we deduce that they are a quasi-minimizer of the renormalized energy W g (defined Notation 15). Thus, by smoothness of W g , the zeros tend to a minimal configuration of W g .

We prove Microscopic location part of Theorem 1 & 4:

• In the case where λ → 0, the fact that we may localize the zeros inside the inclusions (microscopic location part of Theorem 1 and Theorem 4) is obtained via Theorem 4 in [START_REF] Santos | Ginzburg-Landau model with small pinning domains[END_REF].

Indeed we take f n (x) = tr ∂B((kn,ln),δ/2) v εn ((k n , l n ) + δx) with (k n , l n ) a center of a cell containing a zero of v εn . Using the main result of [START_REF] Mironescu | Explicit bounds for solutions to a Ginzburg-Landau type equation[END_REF], one may easily prove that f n satisfies the conditions (A1) and (A2) in [START_REF] Santos | Ginzburg-Landau model with small pinning domains[END_REF]. Thus we can apply Theorem 4 in [START_REF] Santos | Ginzburg-Landau model with small pinning domains[END_REF] and infer that the location of the zero inside the inclusion is governed by a renormalized energy which is independent of the boundary condition.

Theorem 3 is obtained by combining:

• The weak H 1 -convergence of v εn to v * is a direct consequence of Proposition 31. The limiting equation for v * is a direct consequence of Proposition 32 (this is explained right after Proposition 32).

• The behavior in an ε-neighborhood of the zeros of v εn is given by (4.16), (4.17) and (4.18) (noting that in (4.17) we have R = R l → +∞ as l → ∞).

A Proof of Proposition 8

We prove the existence of minimal map in I ρ and in J ρ . The main ingredient is the fact that these sets are closed under H 1 -weak convergence (see [START_REF] Lassoued | Sur quelques équations aux dérivées partielles non linéaires issues de la géométrie et de la physique[END_REF] or below). Thus, considering a minimizing sequence for 1 2 Ωρ α|∇ • | 2 in above sets, we obtained the result.

We consider

• θ i : Ω ρ → R the main argument of x -x i , i.e. e ıθ i = x-x i |x-x i | .
Note that the θ i are multivalued function with smooth gradient.

• For d i ∈ N * (given by the definition of I ρ or J ρ ) we let θ 0 = d i θ i and thus

e ıθ 0 = Π i x-x i |x-x i | d i .
From Lemma 11 in [START_REF] Brezis | Équations de Ginzburg-Landau et singularités[END_REF], there is φ 0 ∈ C ∞ (∂Ω, R) s.t. ge -ıθ 0 = e ıφ 0 . Note that w ∈ I ρ ⇐⇒ w = e ı(θ 0 +φ) with φ ∈ H 1 (Ω ρ , R) and tr ∂Ω φ = φ 0 , (A.1)

w ∈ J ρ ⇐⇒    w = e ı(θ 0 +φ) with φ ∈ H 1 (Ω ρ , R), j =i d j θ j + φ = Cst i on ∂B(x i , ρ) and tr ∂Ω φ = φ 0 . (A.2)
Clearly, from (A.1) and (A.2), I ρ and J ρ are H 1 -weakly closed.

We now prove the second part of Proposition 8. One may easily obtain that for some λ : Ω ρ → R, denoting w = e ı(θ 0 +φ) , φ ∈ H 1 (Ω ρ , R) (and thus w ∈ I ρ ), we have

-div(α∇w) = λw ⇐⇒ -div [α∇(θ 0 + φ)] = 0 and λ = α|∇w| 2 . (A.3)
This observation is a direct consequence of the following identity -div α∇e ı(θ 0 +φ) = -div [α∇(θ 0 + φ)] ıe ı(θ 0 +φ) + α|∇(θ 0 + φ)| 2 e ı(θ 0 +φ) .

Note that under these notations one has |∇w| = |∇(θ 0 + φ)|. Thus w is a minimizer in I ρ or J ρ if and only if θ 0 + φ minimizes the weighted Dirichlet functional under the condition fixed by the RHS of (A.1) or (A.2). Consequently, we find that θ 0 + φ minimizes the weighted Dirichlet functional under its Dirichlet boundary condition.

Therefore, we obtain easily that -div [α∇(θ 0 + φ)] = 0. The identity ∇(θ 0 + φ) = w × ∇w yields -div(α∇w) = λw.

Hence, the Euler-Lagrange equations in (3.1) and (3.2) are direct consequences of (A.3). The condition on the boundary of the holes for w deg ρ,α (resp. w Dir ρ,α ) follows from multiplying the equation satisfied by

θ 0 +φ deg ρ,α , w deg ρ,α = e ı(θ 0 +φ deg ρ,α ) (resp. θ 0 +φ Dir ρ,α , w Dir ρ,α = e ı(θ 0 +φ Dir ρ,α ) ) by ψ ∈ D(Ω, R) (resp. ψ ∈ D(Ω, R) s.t ψ ≡ Cst i in B(x i , ρ)).
Since α is sufficiently smooth, we can rewrite the Euler-Lagrange equation as

-∆φ = ∇α • ∇(φ + θ 0 ) α with ∇α • ∇(φ + θ 0 ) α ∈ L 2 (Ω ρ ).
So, by elliptic regularity φ deg ρ,α , φ Dir ρ,α ∈ H 2 (Ω ρ , R), and consequently w deg ρ,α , w Dir ρ,α ∈ H 2 (Ω ρ , S 1 ).

B Proof of Proposition 9

We prove the existence of a minimal configuration {x, d} = {(x 1 , ..., x N ), (d 1 , ..., d n )} for I ρ,α .

Let ({x n , d n }) n be a minimizing sequence of configuration of I ρ,α , i.e., inf

w∈H 1 (Ω n ρ ,S 1 ) s.t. w=g in Ω ′ \Ω ∪B(x n i ,ρ) deg ∂B(x n i ,ρ) (w)=d n i for all i 1 2 Ω n ρ α|∇w| 2 → I ρ,α ;
here

Ω n ρ = Ω ′ \ ∪B(x n i , ρ). Up to a subsequence, we have N n = N = Cst, d n = d = Cst and x n → x with x = (x 1 , ..., x N ) s.t. min i =j |x i -x j | ≥ 8ρ.
Consider w n ∈ I ρ (x n , d) a minimal map. Since w n is bounded independently of n in H 1 (Ω n ρ ), up to a subsequence, we have

w n ⇀ w 0 in H 1 loc (Ω 0 ρ ), Ω 0 ρ = Ω ′ \ ∪B(x i , ρ).
Clearly the following properties hold:

• w 0 ∈ H 1 loc (Ω 0 ρ , S 1 ) and w 0 = g in Ω 0 ρ \ Ω.
• For all compact K ⊂ Ω 0 ρ we have

1 2 K α|∇w 0 | 2 ≤ lim inf 1 2 K α|∇w n | 2 ≤ I ρ,α . Thus w 0 ∈ H 1 g (Ω 0 ρ , S 1 )
and

Ω 0 ρ α|∇w 0 | 2 ≤ I ρ,α .
Now, it suffices to check that deg ∂B(x i ,ρ) (w 0 ) ∈ N * for all i. Since w 0 is S 1 -valued, this fact is equivalent to deg ∂B(x i ,ρ ′ ) (w 0 ) ∈ N * for all i and for all ρ ′ ∈ (ρ, 2ρ).

In view of the facts:

• for ρ ′ ∈ (ρ, 2ρ) we have w ′ n = w n|Ω ′ \∪B(x i ,ρ ′ ) ⇀ w ′ 0 = w 0|Ω ′ \∪B(x n i ,ρ ′ )
• the set

I ′ := {w ′ ∈ H 1 (Ω ′ \ ∪B(x i , ρ ′ ), S 1 ) | deg ∂B(x i ,ρ ′ ) (w ′ ) = d i for all i ∈ {1, ..., N }} is closed under the H 1 -weak convergence (see Appendix A or [16]), since w ′ n ∈ I ′ , we obtain that w ′ 0 ∈ I ′ . Therefore {x, d} = {(x 1 , ..., x N ), (d 1 , ..., d n )} is a minimal configuration for I ρ,α .
Now we prove the existence of a minimal configuration for J ρ,α . Let (x n ) n be a minimizing sequence of configuration for J ρ,α , i.e., Ĵρ,α (x n , 1) → J ρ,α .

Up to a subsequence, one may assume that there is

x = (x 1 , ..., x d ) ∈ Ω d s.t. x n i → x i , |x i -x j | ≥ 8ρ and dist(x i , ∂Ω) ≥ 8ρ. Let η n = 8 max |x n i -x i |. There is a smooth diffeomorphism φ n : R 2 → R 2 satisfying        φ n ≡ Id R 2 in R 2 \ ∪B(x n i , ρ + η 1/2 n ) φ n [x i + (1 + η n )x] = x n i + x for x ∈ B(0, ρ) φ n -Id R 2 C 1 (R 2 ) = o n (1)
.

For example we can consider φ

n = Id R 2 + H n with      H n ≡ 0 in R 2 \ ∪B(x n i , ρ + η 1/2 n ) H n [x i + (1 + η n )x] = [1 -ψ n (|x|)] (x n i -x i -η n x) for x ∈ B(0, ρ + η 1/2 n 1 + η n ) .
Here ψ n : R + → [0, 1] is a smooth function satisfying

ψ n (r) = 0 if r ≤ ρ 1 if r ≥ ρ + η 1/2 n /2 and |ψ ′ n | = O(η -1/2 n ).
For w n ∈ J ρ (x n , 1) a minimal map, we consider

wn : Ω \ ∪ i B(x i , (1 + η n )ρ) → S 1 x → w n [φ n (x)] .
Clearly wn is well defined and we have

Ω\∪ i B(x i ,(1+ηn)ρ) α|∇ wn | 2 = Ω\∪ i B(x n i ,ρ) α|∇w n | 2 + o n (1), wn x i + (1 + η n )ρe ıθ = w n φ(x i + (1 + η n )ρe ıθ ) = w n x n i + ρe ıθ = e ı(θ+θ i ) .
We can extend wn in

∪ i B(x i , (1 + η n )ρ) \ B(x i , ρ) by wn (x i + re ıθ ) = e ı(θ+θ i ) , ρ < r < (1 + η n )ρ.
Clearly, we have wn ∈ J ρ,α (x, 1) and

1 2 Ω\∪ i B(x i ,ρ) α|∇ wn | 2 = J ρ,α + o n (1).
Thus considering w ∈ J ρ,α (x, 1) a minimizer of

1 2 Ω\∪ i B(x i ,ρ) α|∇ • | 2 , we obtain 1 2 Ω\∪ i B(x i ,ρ) α|∇w| 2 ≤ 1 2 Ω\∪ i B(x i ,ρ) α|∇ wn | 2 = J ρ,α + o n (1).
Letting n → ∞ we deduce that the configuration x = (x 1 , ..., x d ) is minimal.

C Proof of Proposition 10

As explained Section 3.2, Proposition 10 is easily established when either N = 1 or when the points are well separated. It remains to consider the case where N ≥ 2 and there are i = j s.t. |x ix j | ≤ 4η stop .

C.1 The separation process

We assume that N ≥ 2 and that the points are not well separated. Our purpose is to compare the energy of Ĵρ,α to the energy of Îρ,α . To this purpose, we decompose Ω ρ into several regions and we compare energies in each regions. These regions are constructed recursively using the following version of Theorem IV.1 in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF].

Lemma 42. Let N ≥ 2, x 1 , ..., x N ∈ R 2 and η > 0. There are κ ∈ {9 0 , ..., 9 N -1 } and {y 1 , ..., y N ′ } ⊂ {x 1 , ..., x N } s.t.

∪ N i=1 B(x i , η) ⊂ ∪ N ′ i=1 B(y i , κη)
and |y iy j | ≥ 8κη for i = j.

We let x 0 1 , ..., x 0 N denote the initial points x 1 , ..., x N and N 0 = N the initial number of points. For k ≥ 1 (here, k is an iteration in the construction of the regions), we let N k denote the number of points selected at Step k, and denote the points we select by

x k 1 , ..., x k N k . The recursive construction is made in such a way that N k > N k+1 and N k ≥ 1 for all k ≥ 1.
The process will stop at the end of Step k if and only if one of the following conditions yields Rule 1: there is a unique point in the selection (i.e. N k = 1),

Rule 2: min i =j |x k i -x k j | > 4η stop .
Step k, k ≥ 1:

Let η ′ k = 1 4 min i =j |x k-1 i -x k-1 j
|. Using Lemma 42, there are

κ k ∈ {9 1 , ..., 9 N k-1 -1 } and {x k 1 , ..., x k N k } ⊂ {x k-1 1 , ..., x k-1 N k-1 } s.t. ∪ N k-1 i=1 B(x k-1 i , η ′ k ) ⊂ ∪ N k i=1 B(x k i , κ k η ′ k ) and |x k i -x k j | ≥ 8κ k η ′ k for i = j. We denote η k = 2κ k η ′ k . We stop the construction if N k = 1 (Rule 1) or if 1 4 min |x k-1 i - x k-1 j | > η stop (Rule 2).
In Figure 4 & 5 both stop-conditions are presented.

4η stop ≥ 4η ′ 1 (a) The initial balls κ 1 η ′ 1 η 1 = 2κ 1 η ′ 1 (b) The first step: a selec- tion of two centers |x 1 1 -x 1 2 | > 4η stop (c)
The process stops at the end of the first step since there are two well separated balls.

η 1 = 2κ 1 η ′ 1 κ 1 η ′ 1 (b)
The first step: a selection of three centers

κ 2 η ′ 2 = η 2 2 η 2 (c)
The second step: it remains a unique ball (the picture is at scale 1/2)

Figure 5: The process stops when we obtain a unique ball Remark 43. i. From the definitions of η ′ k and η k , we have

N k < N k-1 and η k-1 ≤ η ′ k < η k . ii. The balls B(x k j , 2η k ) are disjoint. iii. Denoting Λ k j ⊂ {1, ..., N k-1 } the set of indices i s.t. x k-1 i ∈ B(x k j , κ k η ′ k ), then for i ∈ Λ k j we have B(x k-1 i , η ′ k ) ⊂ B(x k j , κ k η ′ k ). Furthermore, by construction, |x k-1 i -x k-1 j | ≥ 4η ′ k .

C.2 The separation process gives a natural partition of Ω

Let Ω, g, x 1 , ..., x N , d and ρ, η stop like in Section 3.2 with N ≥ 2 and s.t. the points are not well separated.

We apply the separation process. The process stops after K steps, 1 ≤ K ≤ N -1.

We denote {y 1 , ..., y N ′ } ⊂ {x 1 , ..., x N } the selection that we obtain, i.e., y j = x K j and

N ′ = N K , η = 9 N • η stop if N ′ = 1 min 9 N • η stop , 1 4 min |y i -y j | if N ′ > 1 , so η ≥ max(η K , η stop ), (C.1)
Λ j = {i ∈ {1, ..., N } | x i ∈ B(y j , η)} and η 0 = ρ.
We denote

D j,k = B(x k j , η k ) \ ∪ x k-1 i ∈B(x k j ,η k ) B(x k-1 i , η ′ k ), k ∈ {1, ..., K}, j ∈ {1, ..., N k }, (C.2) R j,k = B(x k j , η ′ k+1 ) \ B(x k j , η k ), k ∈ {0, ..., K -1}, j ∈ {1, ..., N k }, (C.3)
R j = B(y j , η) \ B(y j , η K ), j ∈ {1, ..., N ′ } (C .4)
and D = Ω \ ∪ j∈{1,...,N ′ } B(y j , η).

Note that by construction of η ′ k , η k and x k i the following properties are satisfied:

the balls B(x k-1 i , 2η ′ k ) are disjoint (C.5) and 2 • 9η ′ k ≤ η k ≤ 9 N η ′ k . (C.6)
Therefore

Ω ρ = D ∪ j,k D j,k ∪ j,k R j,k ∪ j R j with disjoint unions. (C.7)

C.3 Construction of test functions

Construction of test functions in D and D j,k

Lemma 44.

1. Let η > 0. There is C 1 (η) > 0 (depending on Ω, g and η) s.t. if x 1 , ..., x N ∈ Ω satisfy min i =j |x i -x j |, min i dist(x i , ∂Ω) > 4η and d 1 , ..., d N ∈ N * are s.t. d i = d then there is w ∈ H 1 g (Ω \ ∪B(x i , η), S 1 ) s.t. w(x) = (x-x i ) d i η d i on ∂B(x i , η) and Ω\∪B(x i ,η) |∇w| 2 ≤ C 1 (η).
Moreover C 1 can be considered decreasing with η.

D = Ω \ ∪B(y j , η) R 1 R 2 (a)
The macroscopic perforated domain and the first mesoscopic rings

η k η ′ k+1 2η ′ k R j,k D j,k R j ′ ,k-1 's B(x k-1 i ′ , η k-1 ) (b)
A mesoscopic ring and a mesoscopic perforated domain

2. Let η > 0, κ ≥ 8, d 0 , d 1 , ..., d N ∈ N * be s.t. 1≤i≤N d i = d 0 .
Then, there is C 2 (κ, d 0 ) s.t. for x 1 , ..., x N ∈ B(0, κη) satisfying min i =j |x ix j | ≥ 4η we can associate a map w ∈ H 1 (B(0, 2κη) \ ∪B(x i , η), S 1 ) s.t.

w(x) =        x d 0 (2κη) d 0 on ∂B(0, 2κη) (x -x i ) d i η d i on ∂B(x i , η)
and

B(0,2κη)\∪B(x i ,η) |∇w| 2 ≤ C 2 (κ, d 0 ).
Moreover C 2 can be considered increasing with κ, d 0 .

Proof. In order to prove 1., we consider, e.g., the test function defined in Ω η := Ω\∪B(x i , η) by

w = e ıH Π i (x -x i ) d i |x -x i | d i with H s.t.            H : Ω η → R H ≡ 0 in {dist [x, ∂Ω η ] ≥ η} -∆H = 0 in {dist [x, ∂Ω η ] < η} w ∈ H 1 g (Ω η , S 1 ) and w(x) = (x-x i ) d i η d i on ∂B(x i , η) .
Assertion 2. was essentially established in [START_REF] Han | Lower bounds for the energy of S 1 -valued maps in perforated domains[END_REF], Section 3. We adapt here the argument in [START_REF] Han | Lower bounds for the energy of S 1 -valued maps in perforated domains[END_REF]. By conformal invariance, we may assume that η = 1. We let

w(x) =                      Π i x + 2x i |x| κ -2 d i x + x i |x| κ -2 d i in B(0, 2κ) \ B(0, 3κ 2 ) Π i (x -x i ) d i |x -x i | d i in B(0, 3κ 2 ) \ ∪B(x i , 3/2) (x -x i ) d i |x -x i | d i e ı(2|x-x i |-2)ϕ i in B(x i , 3/2) \ ∪B(x i , 1)
;

here ϕ i ∈ C ∞ (B(x i , 3/2), R) is defined by e ıϕ i = Π j =i (x -x j ) d j |x -x j | d j and ϕ i (x i ) ∈ [0, 2π).
Clearly

ϕ i H 1 (B(x i ,3/2)\B(x i ,1)
) is bounded by a constant which depends only on d 0 .

By (C.1) and Lemma 44.1, one may find a map w 0 ∈ H 1 (D, S 1 ) s.t.

w 0 =      g on ∂Ω w 0 (x) = (x -y j ) dj
η dj on ∂B(y j , η) ( where dj =

x i ∈B(y j ,η)

d i )
satisfying in addition

D |∇w 0 | 2 ≤ C 1 (η) ≤ C 1 (η stop ). (C.8)
For each D j,k , combining (C.2), (C.5), (C.6) and using Lemma 44.2, there exists a map w j,k ∈ H 1 (D j,k , S 1 ) s.t.

w j,k (x) =              (x -x k j ) dj,k η dj,k k for x ∈ ∂B(x k j , η k ) (x -x k-1 i ) di,k-1 η ′ di,k-1 k for x ∈ ∂B(x k-1 i , η ′ k )
.

Here, dj,k =

x i ∈B(x k j ,η k ) d i and D j,k |∇w j,k | 2 ≤ C 2 (2κ k , d j,k ) ≤ C 2 (2 • 9 d-1 , d). (C.9)
Construction of test functions in R j 's and R j,k 's For R > r > 0 and

x 0 ∈ R 2 we denote R(x 0 , R, r) := B(x 0 , R) \ B(x 0 , r). For α ∈ L ∞ (R 2 , [b 2 , 1]), we define µ α (R(x 0 , R, r), d) = inf w∈H 1 (R(x 0 ,R,r),S 1 ) deg ∂B(x 0 ,R) (w)= d 1 2 R(x 0 ,R,r) α|∇w| 2 (C.10) and µ Dir α (R(x 0 , R, r), d) = inf w∈H 1 (R(x 0 ,R,r),S 1 ) w(x 0 +Re ıθ )=e ı dθ w(x 0 +re ıθ )e -ı dθ =Cst 1 2 R(x 0 ,R,r) α|∇w| 2 . (C.11)
In the special case α = U 2 ε , we denote d). Note that the minimization problems (C.10) and (C.11) admit solutions; this is obtained by adapting the proof of Proposition 8.

µ ε (R(x 0 , R, r), d) = µ U 2 ε (R(x 0 , R, r), d) and µ Dir ε (R(x 0 , R, r), d) = µ Dir U 2 ε (R(x 0 , R, r),
We present an adaptation of a result of Sauvageot, Theorem 2 in [START_REF] Sauvageot | Periodic Unfolding Method and Homogenization for the Ginzburg-Landau Equation[END_REF].

Proposition 45. There is

C b > 0 depending only on b ∈ (0, 1) s.t. for R > r > 0 and α ∈ L ∞ (R 2 , R) satisfying 1 ≥ α ≥ b 2 , we have µ Dir α (R(x 0 , R, r), d) ≤ µ α (R(x 0 , R, r), d) + d2 C b .
Proof. This result was obtained by Sauvageot with

α ∈ W 1,∞ (R 2 , [b 2 , 1]). We may extend this estimate to α ∈ L ∞ (R 2 , [b 2 , 1]). Indeed, let (ρ t ) 1>t>0 be a classical mollifier, namely ρ t (x) = t -2 ρ(x/t) with ρ ∈ C ∞ (R 2 , [0, 1]), Supp ρ ⊂ B(0, 1) and R 2 ρ = 1. Set α t = α * ρ t ∈ W 1,∞ (B(x 0 , R), [b 2 , 1]). We have lim t→0 µ αt (R(x 0 , R, r), d) = µ α (R(x 0 , R, r), d) (C.12)
and lim

t→0 µ Dir αt (R(x 0 , R, r), d) = µ Dir α (R(x 0 , R, r), d). (C.13)
We prove (C.12), Equality (C.13) follows with the same lines. Let w be a minimizer of µ α (R(x 0 , R, r), d). By using Dominated convergence theorem, since

α t → α in L 1 (B(x 0 , R)), we obtain that α t |∇w| 2 → α|∇w| 2 in L 1 (R(x 0 , R, r)) as t → 0. Consequently lim t→0 µ αt (R(x 0 , R, r), d) ≤ µ α (R(x 0 , R, r), d).
On the other hand, let w t be a minimizer of µ α (R(x 0 , R, r), d) and let t n ↓ 0. Up to a subsequence, [START_REF] Lassoued | Sur quelques équations aux dérivées partielles non linéaires issues de la géométrie et de la physique[END_REF]), we obtain that w 0 ∈ I. Consequently, we have lim inf

w tn ⇀ w 0 in H 1 (R(x 0 , R, r)) as n → ∞ and √ α tn ∇w tn ⇀ √ α∇w 0 in L 2 (R(x 0 , R, r)). Since the class I := {w ∈ H 1 (R(x 0 , R, r), S 1 ) | deg B(x 0 ,R) (w) = d} is closed under the H 1 -weak convergence (see Appendix A or
t→0 µ αt (R(x 0 , R, r), d) ≥ µ α (R(x 0 , R, r), d).
Thus the proof of (C.12) is complete. Therefore, without loss of generality, we may assume that α is Lipschitz.

One may easily prove that if R ≤ 4r, then µ Dir α (R(x 0 , R, r), d) ≤ 2 d2 π ln 4. Thus we assume that R > 4r. Clearly, it suffices to obtain the result for d = 1 and x 0 = 0.

Let w be a global minimizer of µ α (R(x 0 , R/2, 2r), 1). As explained Section A, denoting x/|x| = e ıθ , one may write w = e ı(θ+φ) for some φ ∈ H 2 (R(x 0 , R/2, 2r), R). Now we switch to polar coordinates. Consider

I = ρ ∈ [2r, R/2] 2π 0 α|∇(θ + φ)| 2 (ρ, θ) dθ ≤ 1 ρ 2 2π 0 α(ρ, θ) dθ .
Then I is closed (since φ ∈ H 2 ). On the other hand, I is non empty, by the mean value theorem.

Let r 1 = min I and r 2 = max I. We may assume that φ(r 2 , 0) = 0 and φ(r 1 , 0) = θ 0 . We construct a test function:

φ ′ (ρ, θ) =                    0 if 2r 2 ≤ ρ ≤ R 2r 2 -ρ r 2 φ(r 2 , θ) if r 2 ≤ ρ ≤ 2r 2 φ(ρ, θ) if r 1 ≤ ρ ≤ r 2 2ρ -r 1 r 1 φ(r 1 , θ) + 2 r 1 -ρ r 1 θ 0 if r 1 /2 ≤ ρ ≤ r 1 θ 0 if r ≤ ρ ≤ r 1 /2 .
As explained in [START_REF] Sauvageot | Periodic Unfolding Method and Homogenization for the Ginzburg-Landau Equation[END_REF], there is C depending only on b s.t.

1 2 R(0,R/2,2r) α |∇(θ + φ ′ )| 2 -|∇(θ + φ)| 2 ≤ C.
Thus the result follows.

As a direct consequence of Proposition 45 (the two first assertions of the next proposition are direct), we have We turn to the construction of test functions in R j and R j,k .

Proposition 46. Let α ∈ L ∞ (R 2 , [b 2 , 1]), R > r 1 > r > 0, d ∈ Z and x 0 ∈ R 2 , we have 1. µ α (R(x 0 , R, r), d) = d2 µ α (R(x 0 , R, r), 1), 2. b 2 π ln R r ≤ µ α (R(x 0 , R, r), 1) ≤ π ln R r , 3. µ α (R(x 0 , R, r), 1) ≤ µ α (R(x 0 , R, r 1 ), 1) + µ α (R(x 0 , r 1 ,
Using Proposition 45, there is C b depending only on b ∈ (0, 1)

s.t. for α ∈ L ∞ (Ω, [b 2 , 1]
) and for all k ∈ {1, ..., K -1}, j ∈ {1, ..., N k }, there is w α,j,k ∈ H 1 (R j,k , S 1 ) s.t.

w α,j,k (x) =              (x -x k j ) dj,k η ′ dj,k k+1 for x ∈ ∂B(x k j , η ′ k+1 ) γ α,j,k (x -x k j ) dj,k η dj,k k for x ∈ ∂B(x k j , η k ) where γ α,j,k ∈ S 1
and s.t. for all w ∈ H 1 (R j,k , S 1 ) satisfying deg ∂B(x k j ,η k ) (w) = dj,k one has

R j,k α|∇w α,j,k | 2 ≤ R j,k α|∇w| 2 + C b d2 j,k ≤ R j,k α|∇w| 2 + 2d 2 C b . (C.14)
Now we consider the rings R j . For j ∈ {1, ..., N ′ }, we denote dj =

x i ∈B(y j ,η)

d i .
Using Proposition 45, for j ∈ {1, ..., N ′ }, we obtain w α,j ∈ H 1 (R j , S 1 ) s.t.

w α,j (x) =        (x -y j ) d η d for x ∈ ∂B(y j , η) γ α,j (x -y j ) d η d K for x ∈ ∂B(y j , η K ) where γ α,j ∈ S 1
and s.t. for all w ∈ H 1 (R j , S 1 ) satisfying deg ∂B(y j ,η) (w) = dj one has From Proposition 3, we know that far away ∂ω ε , U ε is uniformly close to a ε . Here we prove that, in a neighborhood of ∂ω ε , U ε is very close to a cell regularization of a ε .

R j α|∇w α,j | 2 ≤ R j α|∇w| 2 + 2d 2 C b . (C.
Let

a λ : Y = (-1 2 , 1 2 ) × (-1 2 , 1 2 ) → {b, 1} x → b if x ∈ ω λ = λ • ω 1 otherwise . Consider V ξ the unique minimizer of E a λ ξ (V, Y ) = 1 2 Y |∇V | 2 + 1 2ξ 2 (a λ 2 -V 2 ) 2 , V ∈ H 1 1 (Y, R). (D.1)
Lemma 47. We have the existence of C, γ > 0 s.t. for ε > 0 and x ∈ Y

|U ε [y ε i,j + δ j x] -V ε/δ j (x) | ≤ Ce -γδ j ε .
Thus in the periodic case, we have

U ε which is almost a δ • (Z × Z)-periodic function in Ω incl δ in the sense that |U ε (x) -U ε [x + (δk, δl)] | ≤ Ce -γδ ε if x, x + (δk, δl) ∈ Ω incl δ and k, l ∈ Z.
Proof.

Step 1. We first prove that, for all s > 0 and for sufficiently small ε, we have

U 2 ε ≥ b 2 + 1 2 -s in Ω \ ω ε . The same argument leads to U 2 ε ≤ b 2 + 1 2 + s in ω ε and for sufficiently small ξ: V 2 ξ ≥ b 2 + 1 2 -s in Y \ ω λ and V 2 ξ ≤ b 2 + 1 2 + s in ω λ .
From Proposition 3, it suffices to prove that for

R = α -1 ln C 1 -1+b 2 2 , we have U 2 ε ≥ b 2 + 1 2 -s in {x ∈ Ω \ ω ε | dist(x, ∂ω ε ) < Rε} (for sufficiently small ε).
Here C > 1, α > 0 are given by (1.5). We fix 0 < s < 1 and we let z ε = y ε i,j + λδ j z 0 ε ∈ ∂ω ε , z 0 ε ∈ ∂ω. For x ∈ B(z ε , λδ P +1 ), we write x = z ε + εx with x ∈ B(0, λδ P +1 /ε). Here P = 1 and y ε i,j ∈ δZ × δZ if we are in the periodic situation.

We define Ũε (x) : B(0, λδ

P +1 /ε) → [b, 1] x → U ε (z ε + εx) .
It is easy to check that

-∆ Ũε = Ũε (ã 2 ε -Ũ 2 ε ) in B(0, λδ P +1 /ε) Ũε ∈ H 1 ∩ L ∞ (B(0, λδ P +1 /ε), [b, 1]) (D.2)
where

ãε =      b in ω ε -z ε ε ∩ B(0, λδ P +1 /ε) 1 in (R 2 \ ω ε ) -z ε ε ∩ B(0, λδ P +1 /ε) .
Clearly

ω ε -z ε ε ∩ B(0, λδ P +1 /ε) = λδ j ε • (ω -z 0 ε ) ∩ B(0, λδ P +1 /ε) = λδ j ε • (ω -z 0 ε ) ∩ B(0, δ P +1-j ) ,
and thus

(R 2 \ ω ε ) -z ε ε ∩ B(0, λδ P +1 /ε) = λδ j ε • (R 2 \ ω) -z 0 ε ) ∩ B(0, δ P +1-j ) .
Note that λδ P +1 /ε → ∞ and δ P +1-j → 0, thus by smoothness of ω, up to a subsequence, we have

λδ j ε • [(R 2 \ ω) -z 0 ε ] ∩ B(0, δ P +1-j ) → R θ 0 (R × R + ).
Here R θ 0 is the vectorial rotation of angular θ 0 ∈ [0, 2π).

For sake of simplicity, we assume that θ 0 = 0. From (D.2) and standard elliptic estimates, we obtain that Ũε is bounded in W 2,p (B(0, R)) for p ≥ 2, R > 0. Thus up to consider a subsequence, we obtain that Ũε

→ Ũb in C 1 loc (R 2 ) (ε → 0) where Ũb ∈ C 1 (R 2 , [b, 1]) is a solution of      -∆ Ũb = Ũb (1 -Ũ 2 b ) in R × R + -∆ Ũb = Ũb (b 2 -Ũ 2 b ) in R × R - Ũb ∈ C 1 (R 2 ) ∩ H 2 loc (R 2 ) ∩ L ∞ (R 2 ) . (D.3)
It is proved in [START_REF] Kachmar | Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint[END_REF] (Theorem 2.2), that (D.3) admits a unique positive solution. Moreover Ũb (x, y) = U b (y) ( Ũb is independent of its first variable) and U b is the unique solution of

       -U ′′ b = U b (1 -U 2 b ) in R + -U ′′ b = U b (b 2 -U 2 b ) in R - U b ∈ C 1 (R, R), U ′ b > 0, lim +∞ U b = 1, lim -∞ U b = b .
Note that since the limit is unique, the convergence is valid for the whole sequence. This solution U b may be explicitly obtained by looking for U b under the form

U b (x) =          Ae √ 2x -1 Ae √ 2x + 1 if x ≥ 0 b Be -b √ 2x -1 Be -b √ 2x + 1 if x < 0 . We get B = - 3b 2 + 1 + 2b 2(b 2 + 1) 1 -b 2 , A = B(1 + b) + 1 -b B(1 -b) + 1 + b and U b (0) = b B -1 B + 1 = 1 + b 2 + b 2(b 2 + 1) 2b + 2(b 2 + 1) = 1 -b 2 2b + 2(b 2 + 1) + b = b 2 + 1 2 Since U b (0) 2 = b 2 + 1 2 and U b is an increasing function, for x ≥ 0, U b (x) 2 ≥ b 2 + 1 2 .
From the convergence Ũε → Ũb in L ∞ (B(0, R)), we obtain that, for ε sufficiently small,

Ũ 2 ε ≥ b 2 + 1 2 -s in B(0, R) ∩ λδ j ε • [(R 2 \ ω) -z 0 ε ] . Step 2. Fix j ∈ {1, ..., P } s.t. M ε j = ∅ and fix i ∈ M ε j .
Note that if we are in the periodic case then j = 1 and we fix y k,l = (δk, δl)

∈ δZ × δZ s.t. y k,l + δ • Y ⊂ Ω. We denote ξ := δ j ε . For x ∈ Y , consider W (x) = V ξ (x) -U ε (y ε i,j + δ j x) which satisfies (using (1.5)) -ξ 2 ∆W (x) = W (x) a λ (x) 2 -[V ξ (x) 2 + U ε (y ε i,j + δ j x)V ξ (x) + U ε (y ε i,j + δ j x) 2 ] in Y 0 ≤ W ≤ Ce -γ ξ on ∂Y .
Here γ = α • dist(∂Y, ω), C and α given by (1.5). By Step 1, taking s = b 2 , for sufficiently small ε, we have for

x ∈ Y \ ω λ U 2 ε (y ε i,j + δ j x), V 2 ξ (x) ≥ max b 2 , 1 -b 2 2 ≥ 1 3 .
Thus, using the weak maximum principle, we find that W ≥ 0 in Y . Consequently, since W is subharmonic, we deduce that W ≤ Ce -γ ξ . For the general case, we divide R(x, R, r

Assume for the moment that

) into R 1 (x) ∪ R 2 (x) ∪ R 3 (x) with R 1 (x) = R (x, R, min{max(δ, r), R}) , R 2 (x) = R (x, min{max(δ, r), R}, min {R, max(λδ, r)}) , R 3 (x) = R (x, min {R, max(λδ, r)} , r) . Remark 51. 1. For k ∈ {1, 2, 3}, we have ∅ ⊆ R k ⊆ R(x, R, r).
2. It is easy to check that

• [R 1 = R(x, R, r) ⇔ r ≥ δ (Case 1.)] and [R 1 = ∅ ⇔ R ≤ δ],
•

[R 2 = R(x, R, r) ⇔ λδ ≤ r < R ≤ δ (Case 2.)] and [R 2 = ∅ ⇔ {λ = 1 or r ≥ δ or R ≤ λδ}],
•

[R 3 = R(x, R, r) ⇔ R ≤ λδ (Case 3.)] and [R 3 = ∅ ⇔ r ≥ λδ].
3. If λ ≡ 1 then Case 2. never occurs and R 2 = ∅.

Therefore we have (using Propositions 45, 46.3 and (D.5)) We now turn to the proof of (D.5) in Case 1, 2 and 3. Recall that we assumed that R > 10 2 r.

µ ε (R(x, R, r), 1) ≥ µ ε (R 1 (x), 1) + µ ε (R 2 (x), 1) + µ ε (R 3 (x), 1) (D.5) ≥ µ ε (R 1 (x per ), 1) + µ ε (R 2 (x per ), 1) + µ ε (R 3 (x per ),
We treat Case 1. (R > r ≥ δ):

µ ε (R(x, R, r), 1) ≥ (Prop. 45) ≥ µ Dir ε (R(x, R, r), 1) -C b ≥ µ Dir ε (R(x, 10R, 10 -1 r), 1) -2π ln 10 -C b {R(x per , R, 10r) ⊂ R(x, 10R, 10 -1 r)} ≥ µ ε (R(x per , R, 10r), 1) -2π ln 10 -C b ≥ (Prop. 45) ≥ µ ε (R(x per , R, r), 1) -3π ln 10 -2C b .
Thus we may take C * = 3π ln 10 + 2C b . We treat Case 2. Note that from Remark 51.3, we may assume that λ → 0. On the one hand, it is clear that

µ ε (R(x per , R, r), 1) ≤ π ln R r .
On the other hand, letting

α ε : R 2 → {b 2 , 1} x → b 2 if x ∈ ∪ M ∈Z 2 B(δM, λδ) 1 otherwise , we have from Proposition 3 that α ε ≤ U 2 ε + V ε with V ε L ∞ = o(ε 2 ). If R(x, R, r) ∩ {α ε = b 2 } = ∅, then we have µ ε (R(x, R, r), 1) ≥ π ln R r + o(ε 2 ln λ).
And thus the result holds with C * = 1 (for sufficiently small ε).

Otherwise we have R(x, R, r) ∩ {α ε = b 2 } = ∅. In this situation, because R ≤ δ, we get that R(x, R, r) ∩ {α ε = b 2 } is a union of at most four connected components. Therefore S = {ρ ∈ (r, R) | ∂B(x, ρ) ∩ {α ε = b 2 }} is a union of at most four segments whose length is lower than 8λδ. Consequently, denoting S = ∪ k i=1 [s i , t i ] (with s i < s i+1 ), we have for w * ∈ H 1 (R(x, R, r), S 1 ) which minimizes µ ε (R(x, R, r), 1)

1 2 R(x,R,r) U 2 ε |∇w * | 2 + o ε (1) ≥ 1 2 R(x,R,r) α ε |∇w * | 2 (t 0 = r & s k+1 = R) ≥ k i=0 1 2 s i+1 t i dρ ρ 2π 0 |∂ θ w * | 2 ≥ π k i=0 ln s i+1 t i = π ln R r -π k i=1 ln t i s i Since λδ ≤ s i ≤ t i ≤ s i + 8λδ, we have 1 ≤ t i s i ≤ 1 + 8λδ s i ≤ 9
(R(x, R, r), 1) ≥ πb 2 ln R r
. Therefore the estimate in the third case is proved.

Estimates for almost minimizers

In this subsection we establish a fundamental result: fix an almost minimal configuration {x, 1} for I ρ,ε (the existence of such configuration is proved Section D.3) and a map which almost minimizes 1 2

Ω ′ \∪B(x i ,ρ) U 2 ε |∇ • | 2 . Then the map almost minimizes the weighted Dirichlet functional 1 2 R(x i ,ρ ′ ,ρ) U 2 ε |∇ • | 2 , 10 -2 min i =j |x i -x j | > ρ ′ > ρ. Lemma 52. 1. Let x ∈ R 2 , 0 < r < R, α ∈ L ∞ (R 2 , [b 2 , 1]), C 0 > 0 and a map w ∈ H 1 (R(x, R, r), S 1 ) s.t. deg ∂B(x,R) (w) = 1 and 1 2 R(x,R,r) α|∇w| 2 -µ α (R(x, R, r), 1) ≤ C 0 .
Then for all r ′ , R ′ s.t. r < r ′ < R ′ < R one has

1 2 R(x,R ′ ,r ′ ) α|∇w| 2 -µ α (R(x, R ′ , r ′ ), 1) ≤ 4C b + C 0 ,
where C b depends only on b and is given by Proposition 45.

Let

x 1 , ..., x d ∈ Ω (x i = x j for i = j), d i = 1, ε < ρ < 10 -2 η, η := 10 -2 • min {|x i -x j |, dist(x i , ∂Ω)}, C 0 > 0 and w ∈ H 1 (Ω ′ ρ , S 1 ) s.t. 1 2 Ω ′ ρ U 2 ε |∇w| 2 ≤ I ρ,ε + C 0 .
Then for ρ ≤ r < R < η one has for all i

1 2 R(x i ,R,r) U 2 ε |∇w| 2 -µ ε (R(x i , R, r), 1) ≤ C 0 + C(η);
here C(η) depends only on b, g, Ω, Ω ′ and η.

3. Under the hypotheses of 2., we also have for η > ρ 0 > ρ

1 2 Ω ′ ρ 0 U 2 ε |∇w| 2 ≤ C(ρ 0 , C 0 );
here C(ρ 0 , C 0 ) depends only on b, g, Ω, Ω ′ , C 0 , ρ 0 and η.

Proof. Using the third part of Proposition 46, we have

1 2 R(x,R,r) α|∇w| 2 ≤ µ α (R(x, R, R ′ ), 1) + µ α (R(x, R ′ , r ′ ), 1) + µ α (R(x, r ′ , r), 1) + 4C b + C 0 .
We easily obtain

1 2 R(x,R,r) α|∇w| 2 ≥ µ α (R(x, R, R ′ ), 1) + 1 2 R(x,R ′ ,r ′ ) α|∇w| 2 + µ α (R(x, r ′ , r), 1)
which proves the first assertion.

The second assertion is obtained by using the same argument combined with Proposition 49.

Last assertion is a straightforward consequence of Proposition 49 and both previous assertions.

D.2.2 Lower bound on circles

In this subsection we prove an estimate for the minimization of weighted 1-dimensional Dirichlet functionals. In the following this estimate will be used to get lower bounds in rings.

Lemma 53. Let θ 0 ∈ (0, 2π) and let α ∈ L ∞ ([0, 2π], {b 2 , 1}) be s.t. H 1 ({α = b 2 }) = θ 0 . Let ϕ ∈ H 1 ([0, 2π], R) s.t. ϕ(2π) -ϕ(0) = 2π.
The following lower bound holds

1 2 2π 0 α(θ)|∂ θ ϕ(θ)| 2 dθ ≥ 2π 2 2π 0 1 α = 2π 2 2π + θ 0 (b -2 -1)
.

Here H 1 is the 1-dimensional Hausdorff measure.

Proof. The proof of this lower bound is based on the computation of the minimal energy.

It is easy to check that a minimal function ϕ min ∈ H 1 ([0, 2π], R) for 1 . Therefore

1 2 2π 0 α(θ)|∂ θ ϕ(θ)| 2 dθ ≥ 1 2 2π 0 α(θ)|∂ θ ϕ min (θ)| 2 dθ = 2π 2 2π 0 1 α = 2π 2 2π + θ 0 (b -2 -1) . U 2 εn |∇w| 2 -µ εn (R(x 0 , √ η, η), 1) → ∞. (D.
(Ω ′ \ (B(x 0 , ρ) ∪ ∪ d i=2 B(x n i , ρ)), S 1 )
• For i = 2, ..., d, we define w|R(x n i ,η 0 ,ρ) by taking a minimal map for

1 2 R(x n i ,η 0 ,ρ) U 2 εn |∇•| 2 in H 1 (R(x n i , η 0 , ρ), S 1
) with the boundary conditions w(x n i +η 0 e ıθ ) = e ıθ and w(x n i +ρe ıθ ) = Cst i e ıθ , Cst i ∈ S 1 . From Proposition 45 we have

1 2 R(x n i ,η 0 ,ρ) U 2 εn |∇ w| 2 ≤ µ εn (R(x n i , η 0 , ρ), 1) + C b .
• We divide R(x 0 , η 0 , ρ) into R(x 0 , η 0 , √ η), R(x 0 , √ η, η) and R(x 0 , η, ρ). In each of these rings we consider the minimal maps for 1 2 ring U 2 εn |∇ • | 2 with the boundary conditions w(x 0 + Re ıθ ) = e ıθ and w(x 0 + re ıθ ) = Cst i e ıθ , Cst i ∈ S 1 where ring ∈ {R(x 0 , η 0 , √ η), R(x 0 , √ η, η), R(x 0 , η, ρ)}, r < R and ring = R(x 0 , R, r).

Up to consider suitable rotations, we glue these functions to get a map w|R(x 0 ,η 0 ,ρ) ∈ H 1 (R(x 0 , η 0 , ρ), S 1 ) which is s.t. w(x 0 + η 0 e ıθ ) = e ıθ and (from Proposition 45)

1 2 ring U 2 εn |∇ w| 2 ≤ µ εn (ring, 1) + C b with ring ∈ {R(x 0 , η 0 , √ η), R(x 0 , √ η, η), R(x 0 , η, ρ)}.
• We extend w in Ω \ (B(x 0 , η 0 ) ∪ ∪ d i=2 B(x n i , η 0 )) using Lemma 44.1. Then we finally obtain w ∈ H 1 g (Ω ′ \ (B(x 0 , ρ) ∪ ∪ d i=2 B(x n i , ρ)), S 1 ).

From Lemma 48, (D.11) and by construction of w, for w n ∈ H 1 g (Ω ′ \ ∪ i B(x n i , ρ), S 1 ) we have easily that

Ω ′ \∪ i B(x n i ,ρ) U 2 εn |∇w n | 2 - Ω ′ \(B(x 0 ,ρ)∪∪ d i=2 B(x n i ,ρ)) U 2 εn |∇ w| 2 → +∞
which implies that {x n 1 , ..., x n d } can not be an almost minimal configuration of points. We now turn to the proof of (D.11). We argue by contradiction and we assume that there is w

* = w εn * ∈ H 1 g (Ω ′ \ ∪ i B(x n i , ρ), S 1 ) s.t. 1 2 R(x n 1 , √ η,η) U 2 εn |∇w * | 2 ≤ µ εn (R(x 0 , √ η, η), 1) + O(1). (D.12)
In particular (using Lemma 48) we have

1 2 R(x n 1 , √ η,η) U 2 εn |∇w * | 2 = µ εn (R(x n 1 , √ η, η), 1) + O(1).
The key ingredient to get a contradiction is the fact that the map w * is almost constant in the "half" ring R(x n 1 , √ η, η) \ Ω.

By smoothness of Ω, we may assume that the cone

K √ η,η := {x = x n 1 + ρe ıθ | θ ∈ [0, π/2], η ≤ ρ ≤ √ η} does not intersect Ω: K √ η,η ∩ Ω = ∅.
Consequently,

H 1 (C ρ ∩ ω ε ) = Ỹε∈Sj H 1 (C ρ ∩ ωε ) ≤ 8πλ Ỹε∈Sj H 1 (C ρ ∩ Ỹε \ ωε ) ≤ 8πλH 1 (C ρ ) = 16π 2 λρ.

E.2 Proof of Proposition 16

We are now in position to prove Proposition 16. The proof is done in 3 steps. Let ε n ↓ 0, ρ = ρ(ε n ) ↓ 0, ρ ≥ ε n and let x n be a quasi-minimizer for J ρ,εn (defined Notation 15).

From Corollaries 13 & 14, up to pass to a subsequence, there are η 0 > 0 and a = (a 1 , ..., a d ) ∈ Ω d s.t. x n i → a i , |a ia j |, dist(a i , ∂Ω) > 10 2 η 0 . We prove that W g (a 1 , ..., a d ) = min b 1 ,...,bn∈Ω W g (b 1 , ..., b n ). We argue by contradiction and we assume that, up to consider a smaller value for η 0 if necessary, we have the existence of

b = (b 1 , ..., b d ) ∈ Ω d s.t. |b i -b j | ≥ 10 2 η 0 , dist(b i , ∂Ω) > 10 2 η 0 and W g (b) < W g (a) -10 2 η 0 .
Step 1. We estimate the energies in perforated domains with a fixed perforation size The goal of this step is to prove the existence of small ρ 0 (independent of n) s.t. we have for c ∈ {a, b} and x ∈ Ω d satisfying max i |x ic i | ≤ ρ 0 Ĵρ 0 ,1I (x) -Ĵρ 0 ,εn (x) ≤ 2η 0 .

(E.1)

From [START_REF] Comte | The behavior of a ginzburg-landau minimizer near its zeroes[END_REF] ((15) and Lemma 2), we may fix η 0 > ρ 0 > 0 independent of n s.t. for c ∈ {a, b}, we have Ĵρ

0 ,1I (x) -Îρ 0 ,1I (x) ≤ η 0 for all x ∈ Ω d s.t. max i |x i -c i | ≤ ρ 0 , Îρ 0 ,1I (x) -πd| ln ρ 0 | -W g (x) ≤ η 0 for all x ∈ Ω d s.t. max i |x i -c i | ≤ ρ 0 and |W g (c) -W g (x)| ≤ η 0 for all x ∈ Ω d s.t. max i |x i -c i | ≤ ρ 0 . For c ∈ {a, b} and x ∈ Ω d s.t. max i |x i -c i | ≤ ρ 0 : • We let θ x = d i=1 θ x i where θ x i ∈ (-π, π], x -x i |x -x i | = e ıθx i (x = x i ) is main determina- tion of the argument of x -x i .
• We fix φ x 0 ∈ C ∞ (∂Ω, R) s.t. e ıφ x 0 = ge -ıθx . Clearly, since deg ∂Ω (ge -ıθx ) = 0, and since ge

-ıθx ∈ C ∞ (∂Ω, S 1 ), φ x 0 ∈ C ∞ (∂Ω, R) is well defined [3]. • We let φ * = φ x * , φ = φ x ∈ H 1 be the solutions of      -∆φ * = 0 in Ω \ ∪B(x i , ρ 0 ) φ * = φ 0 on ∂Ω ∂ ν φ * = -j =i ∂ ν θ x j on ∂B(x i , ρ 0 ), i = 1, ..., d and      -div(U 2 ε ∇φ) = div(U 2 ε ∇θ x ) in Ω \ ∪B(x i ,
(x) = 1 2 Ω\∪B(x i ,ρ 0 ) U 2 ε |∇w| 2 = 1 2 Ω\∪B(x i ,ρ 0 ) U 2 ε |∇(θ x + φ)| 2 .
2. ∇φ and ∇φ * are bounded independently of x and ε n in L 2 (Ω \ ∪B(x i , ρ 0 )).

3. From a Poincaré inequality we have the existence of C 0 independent of x s.t.

ψ L 2 (Ω\∪B(x i ,ρ 0 )) ≤ C 0 ∇ψ L 2 (Ω\∪B(x i ,ρ 0 )) .

Therefore, using a trace inequality in R(x i , 2ρ 0 , ρ 0 ) we obtain ψ L 2 (∂B(x i ,ρ 0 )) ≤ C ′ 0 , C ′ 0 is independent of x, n. 4. We have |∇φ * | which is bounded in L ∞ (Ω \ ∪B(x i , ρ 0 )):

|∇φ * | ≤ C 0 with C 0 independent of x.
Indeed, with standard result of elliptic interior regularity, we have φ * C 2 (∂B(x i ,8ρ 0 )) , φ * C 2 (∂B(c i ,4ρ 0 )) ≤ C ′ 0 .

Thus, from global regularity for the Laplacian, we have ∇φ * L ∞ (Ω\∪B(c i ,4ρ 0 )) , ∇φ * L ∞ (R(x i ,8ρ 0 ,ρ 0 )) ≤ C ′′ 0 .

We let Ω ρ 0 = Ω ρ 0 (x) := Ω \∪B(x i , ρ 0 ). We are now in position to prove that Cauchy-Schwarz inequality ≤ C0 ∇ψ L 2 (Ωρ 0 ) + 1 -U 2 εn L 2 (Ωρ 0 ) → 0.

Consequently we obtain

Ĵρ 0 ,1I (x) -Ĵρ 0 ,εn (x) ≤ Îρ 0 ,1I (x) -Îρ 0 ,εn (x) + η 0 ≤ η 0 + o n (1) ≤ 2η 0 which is exactly (E.1).

Thus it remains to establish that Step 2. We study the energies in R(x i , ρ 0 , max(δ, λ 2 ))

Let

• κ = max(λ, √ δ)

• x n be a quasi minimizer for J ρ,ε

• w n = e ıϕn be a minimizer of Ĵρ,εn (x n ) (ϕ n is locally defined and its gradient is globally defined in Ω \ ∪B(x i , ρ)).

We prove that there is r ∈ (κ 2 , κ) s.t. This estimate is obtained via a mean value argument. We first prove that µ εn (R(x n i , κ, κ 2 ), 1) = µ 1I (R(x n i , κ, κ 2 ), 1) + o εn (1).

Indeed we let ω ′ be a smooth open set s.t. ω ⊂ ω ′ and ω ′ ⊂ B(0, 1). We define α ′ ε = b 2 in δZ × δZ + λδ • ω ′ 1 otherwise . From Proposition 3, we have

α ′ ε ≤ U 2 ε + V ε with V ε L ∞ = O(ε 2 ).
For ρ ≥ δ and x ∈ R

|h n i × ∂ τ h n i -1| 2 = S 1 |h n i × ∂ τ h n i | 2 + 1 -2h n i × ∂ τ h n i ≤ 2/ | ln κ| → 0. Therefore h n i × ∂ τ h n i → 1 in L 2 (S 1
). Consequently, up to pass to a subsequence, we have the existence of α i ∈ S 1 s.t. α -1 i h n i e -ıθ → 1 in H 1 (S 1 ). From Propositions 12 and 13 in [START_REF] Santos | Ginzburg-Landau model with small pinning domains[END_REF] we have inf w∈H 1 (R(x n i ,ρ 0 ,r),S 1 ) w(x n i +ρ 0 e ıθ )=α i e ıθ w(x n i +re ıθ )=h n i (e ıθ ) 1 2 R(x n i ,ρ 0 ,r)

|∇w| 2 = inf w∈H 1 (R(x n i ,ρ 0 ,r),S 1 ) w(x n i +ρ 0 e ıθ )=α i e ıθ w(x n i +re ıθ )=α i e ıθ Step 3. We conclude

We are going to construct a map wn ∈ J ρ (y n ), max |y ib i | ≤ δ and s.t.

≤ (α ′ ε ≤ U 2 ε + V ε ) ≤ 1 2 ∪ i R(x n i ,ρ 0 ,r) U 2 εn |∇w n | 2 + o n (1).
Proof. In order to keep notations simple, we write, in what follows, δ rather than δ n . Since f n is real valued, we have that div(H n ∇v n ) × v n = 0. From (F.1) and (F.2), we obtain div y [T δ (H n )(x, y)T δ (∇v n )(x, y)] × T δ (v n )(x, y) = 0 in Ω 0 × Ỹ . (F.7)

Note that from the assumptions and (F.1),(F.5), passing to a subsequence, there is ŵ ∈ L 2 (Ω 0 , H 1 per ( Ỹ )) s.t.

T δ (v n )(x, y) → v * (x), T δ (∇v n )(x, y) ⇀ ∇v * (x) + ∇ y v(x, y) in L 2 (Ω 0 × Ỹ ) and T δ (H n )(x, y) → H 0 (y) in L 2 (Ω 0 × Ỹ ).

Thus we obtain the convergence:

div y [T δ (H n )(x, y)T δ (∇v n )(x, y)]×T δ (v n )(x, y) ⇀ div y [H 0 (∇v * + ∇ y v)]×v * in L 2 (Ω 0 ×H -1 ( Ỹ )).
Consequently, div y [H 0 (∇v * + ∇ y v)] × v * = 0.

Since v * is independent of y ∈ Ỹ , the previous assertion is equivalent to

-div y [H 0 ∇ y (v × v * )] = (∇ y H 0 • ∇v * ) × v * ,
which in turn is equivalent to This identity combined with (F.4) implies that

-div y [H 0 ∇ y (v × v * )] = i ∂ y i H 0 (∂ i v * × v * ).
Ω 0 × Ỹ T δ [H n (∇v n × v n ) • ∇ψ] = 0.
Therefore, using (F.3) and (F.5), we obtain:

0 = Ω 0 × Ỹ T δ [H n (∇v n × v n ) • ∇ψ] = Ω 0 × Ỹ T δ (H n )T δ (∇v n ) × T δ (v n ) • T δ (∇ψ) → n→∞ Ω 0 × Ỹ H 0 [∇v * × v * + ∇ y (v × v * )] • ∇ψ.
Finally, for all ψ ∈ D(Ω 0 ), using (F.8), we have

0 = Ω 0 × Ỹ H 0 ∇v * × v * [Id R 2 -∇ y χ] • ∇ψ = Ω 0 Ỹ H 0 [Id R 2 -∇ y χ] ∇v * × v * ∇ψ = - Ω 0 -div (A∇v * × v * ) ψ.

b if x ∈ ω ε 1 .

 1 otherwise The values of the periodic pinning term are represented Figure 1. The connected components of {a ε = b} = ω ε are called inclusions or impurities. a ε = b ∈ (0, 1) a ε = 1 δ Ω (a) The pining term is periodic on a δ × δ-grid δ ≈ λδ (b) The parameter λ controls the size of an inclusion in the cell

Figure 1 :

 1 Figure 1: The periodic pinning term

Figure 2 :

 2 Figure 2: Representation of the general diluted pinning term with P = 2

Figure 3 :

 3 Figure 3: The modification of the reference cell

4

 4

Figure 4 :

 4 Figure 4: The process stops when we obtain well separated balls

  r), 1) + 2C b where C b is given by Proposition 45 and depends only on b.

15 )C. 4 D Proof of Proposition 12 D. 1

 154121 Proof of Proposition 10Note that there are at most d 2 regions D j,k , at most d 2 rings R j,k and at most d rings R j . Consequently, denotingC 4 (η stop ) = C 1 (η stop ) + d 2 C 2 (2 • 9 d-1 , d) + 4d 4C b and using (C.7), (C.8), (C.9), (C.14), (C.15), one may construct a test function w α ∈ J ρ (up to multiply by some S 1 -Constants each function previously constructed) s.t. for all w ∈ I ρ , one has Ωρ α|∇w α | 2 ≤ Ωρ α|∇w| 2 + C 4 . (C.16) Clearly, (C.16) allows us to prove Proposition 10 with C 0 = C 4 /2. Description of the special solution U ε

  There exists C * > 0 s.t. (D.4) yields in the three previous cases with C * = C * . (D.5)

1 )

 1 -3 C * (Prop. 45) ≥ µ Dir ε (R 1 (x per ), 1) + µ Dir ε (R 2 (x per ), 1) + µ Dir ε (R 3 (x per ), 1) -3( C * + C b ) ≥ µ ε (R(x per , R, r), 1) -3( C * + C b ).The last line is obtained by constructing a test function. Therefore, it suffices to take C * := 3( C * + C b ).

2 2π0

 2 α(θ)|∂ θ • | 2 dθ under the constraint ϕ(2π)ϕ(0) = 2π exists and satisfies ∂ θ (α∂ θ ϕ min ) = 0. Thus ∂ θ ϕ min = Cst α with Cst = 2π 2π 0 α -1

• 2 ε 2 and

 22 ρ 0 ) φ = φ 0 on ∂Ω ∂ ν φ =j =i ∂ ν θ x j on ∂B(x i , ρ 0 ), i = 1, ..., d . We let ψ = φφ * be the solution of  ∇ψ) = div[(U 2 ε -1)(∇θ x -∇φ * )] in Ω \ ∪B(x i , ρ 0 ) ψ = 0 on ∂Ω ∂ ν ψ = 0 on ∂B(x i , ρ 0 ), i = 1, ..., d .Remark 57.1. From Proposition 8, the functions φ * , φ are s.t. w * = e ı(θx+φ * ) , w = e ı(θx+φ * ) ∈ I ρ 0 (x) satisfyÎρ 0 ,1I (x) = 1 2 Ω\∪B(x i ,ρ 0 ) |∇w * | 2 = 1 2 Ω\∪B(x i ,ρ 0 ) |∇(θ x + φ * )| Îρ 0 ,ε

Ωρ 0 |∇ψ| 2 →0 2 2 + 1 2

 22212 when n → ∞ uniformly on x. This estimate will easily imply (E.1). IndeedÎρ 0 ,1I (x) -Îρ 0 ,εn (x) = 1 Ωρ 0 U 2 εn |∇(θ x + φ * )| 2 -|∇(θ x + φ)| Ωρ 0 (1 -U 2 εn )|∇(θ x + φ * )| 2

Ωρ 0 |∇ψ| 2 → 2 εn - 1 ) 0 ( 1 - 1 ) 2 2 ≤ C 0 1 -U 2 εn L 2 (

 02210112222 0 when n → ∞ uniformly on x:(∇θ x -∇φ * )]ψ = Ωρ -U 2 εn )(∇θ x -∇φ * ) • ∇ψ + ∂Ωρ 0 (U 2 εn -1)∂ ν (θ xφ * )ψ.From the L 2 bound on ∇ψ and the L ∞ bounds on ∇φ * , ∇θ x we have (with C 0 independent of x)Ωρ 0 U 2 εn |∇ψ| 2 ≤ Ωρ 0 |1 -U 2 εn | 2 |∇θ x -∇φ * | 2 |∂ ν (θ xφ * )Ωρ 0 ) + 1 -U 2 εn L 2 (∂Ωρ 0 ) .From Proposition 3 and Lemma 56 we have 1 -U 2 εn L 2 (∂Ωρ 0 ) = O(λ) uniformly in x. Therefore Ω\∪B(x i ,ρ 0 ) |∇ψ| 2 → 0 when n → ∞ uniformly on x and (E.1) holds.

  n (x n i + re ıθ )| 2 dθ ≤ π + 1 | ln κ| for i = 1, ..., d. (E.2)

1 2 R

 2 (x n i ,ρ 0 ,r) |∇w| 2 + o n (1) = π ln ρ 0 r + o n (1).

  Hence, from Proposition 58 and (F.6), we obtainv × v * =i χ i (∂ i v * × v * ) = -χ • (∇v * × v * ), χ = (χ 1 , χ 2 ) . (F.8) Let ψ ∈ D(Ω 0 ) and n sufficiently large s.t. Supp(ψ) ⊂ Ωincl δ . Since -div [H n ∇v n × v n ] = 0, we have Ωincl δ H n ∇v n × v n • ∇ψ = 0.

  there are x deg ρ,α , x Dir ρ,α ∈ Ω d and d ρ,α ∈ (N * ) N (with d ρ,α = (d 1 , ..., d N ), d i = d) s.t. {x deg ρ,α , d ρ,α } minimizes I ρ,α and x Dir ρ,α minimizes J ρ,α . The proof of this result is in Appendix B.

  1 4 min i =j |x ix j | > η stop , 2. The points x 1 , ..., x N are not well separated: 1 4 min i =j |x ix j | ≤ η stop .

	If the points are well separated (or N = 1), Proposition 10 can be easily proved: it is a
	direct consequence of Proposition 45 and Lemma 44 in Appendix C. These results, whose
	statements and proofs are postponed in Appendix C, give essentially the existence of test
	functions into two kinds of domains.

  Consequently we obtain that for large n, deg ∂B(x εn i ,Rεn) (v εn ) = 0.

	Now we prove that
	deg ∂B(x εn i ,Rεn) (v εn ) = 1 for all i and large n.	(4.15)

Note that each small bad disc contains at least a zero of v εn . Consequently, for ρ satisfying (4.8), all small bad discs are included in a ρ-bad disc B(y, ρ) defined Notation 25 (page 22). (For sake of simplicity we wrote B(y, ρ) instead of B(y, 2κρ)).

If B(y, ρ) is a ρ-bad disc, we denote Λ y = {i ∈ J ′′ | x εn i ∈ B(y, ρ)}. Clearly, if Card(Λ y ) = 1, then (4.15) holds.

11 )

 11 Remark 54. Estimate (D.11) implies that {x n 1 , ..., x n d } can not be an almost minimal configuration of points. Indeed, we may construct a suitable test function w as follows: Construction 55. The test function w ∈ H 1 g

  2 , from Lemma 56, we have H 1 [{α ′ ε = b 2 } ∩ ∂B(x, ρ)] ≤ 16π 2 λρ. Therefore, using Lemma 53 we obtainµ 1I (R(x n i , κ, κ 2 ), 1) + O(λ| ln κ|) ≤ (Lemma 53) ≤ µ α ′ εn (R(x n i , κ, κ 2 ), 1) ≤ (α ′ ε ≤ U 2 ε + V ε ) ≤ µ εn (R(x n i , κ, κ 2 ), 1) + o εn (1) ≤ (U 2 ε ≤ 1) ≤ µ 1I (R(x n i , κ, κ 2 ), 1) + o εn (1).Since for s ∈ (κ 2 , κ) we have s ≥ δ, we obtain (because κ ≥ λ)µ εn (R(x n i , κ, κ 2 ), 1) = µ 1I (R(x n i , κ, κ 2 ), 1) + O(λ| ln κ|) = π| ln κ| + o εn (1). U 2 εn |∇w n | 2 = π| ln κ| + O(1). On the other hand, from a standard estimate, we have|∂ θ ϕ n (x n i + se ıθ )| 2 dθ ≥ π, ∀ s ∈ (κ 2 , κ). |∂ θ ϕ n (x n i + se ıθ )| 2 dθ.Assume that r ∈ (κ 2 , κ) s.t. (E.2) holds does not exist. Then we obtain that for s ∈ (κ 2 , κ)|∇w n | 2 ≥ | ln κ| πd + 1 | ln κ| = πd| ln κ| + | ln κ|. U 2 εn |∇w n | 2 = π| ln κ| + O(1). We are now in position to estimate the energy in R(x n i , ρ 0 , r). Let h n i :S 1 → S 1 , h n i (e ıθ ) = w n (x n i + re ıθ ). We have h n i × ∂ τ h n i (e ıθ ) = ∂ τ ϕ n (x n i + re ıθ) . Thus from (E.2): h n i × ∂ τ h n i 2 L 2 (S 1 ) ≤ 2π + 2/ | ln κ|. Consequently

	Therefore from Corollary 14 and Lemma 52.2,	1 2 R(x n i ,κ,κ 2 )
	1		2π
	2	0
	We deduce that
	πd| ln κ| + O(1) ≥	1 2 ∪R(x n i ,κ,κ 2 )	|∇w n | 2 ≥	1 2	κ κ 2	ds s i	0	2π
	i		1 2	0	2π	|∂ θ ϕ n (x n i + se ıθ )| 2 dθ > πd +	1 | ln κ|
	and consequently
	1
	2 ∪R(x n i ,κ,κ 2 )
	Clearly this lower bound contradicts	1 2 R(x n i ,κ,κ 2 )
	S 1

•

  Ω\∪B(yi ,ρ) U 2 εn |∇ wn | 2 + η 0 ≤ Ĵρ,εn (x n ). (E.3)Clearly (E.3) is in contradiction with the assumption: J ρ,εn -Ĵρ,εn (x n ) → 0. Then this contradiction will imply that a = lim x n minimizes W g . We let y n be s.t. max|y n ib i | ≤ δ and x n iy n i ∈ δZ × δZ and we define Ω \ ∪B(y n i , ρ 0 ) Cst i,n w i (xy n i + x n i ) if x ∈ R(y n i , ρ 0 , r) Cst i,n w n [xy n i + x n i ] if x ∈ R(y n i , r, ρ) Here: • w yn ρ 0 is a minimizer of Ĵρ 0 ,1I (y n ),• w i is a minimizer of infw∈H 1 (R(x n i ,ρ 0 ,r),S 1 ) w(x n i +ρ 0 e ıθ )=α i e ıθ w(x n i +re ıθ )=h n i (e ıθ ) Cst i,n ∈ S 1 is a constant s.t. wn ∈ H 1 (Ω \ ∪B(y n i , ρ), S 1 )• w n is the minimizer of Ĵρ,εn (x n ) used in Step 2.. We now compare the energies of wn and w n .U 2 εn |∇ wn | 2 ≤ πd| ln ρ 0 | + W g (y n ) + η 0 + o n (1) ≤ πd| ln ρ 0 | + W g (x n ) -10η 0 U 2 εn |∇w n | 2 -2η 0 .

			wn (x) =	    	w yn ρ 0 (x)	if x ∈ 1 2 R(x n i ,ρ 0 ,r) |∇w| 2
	Ω\∪B(y n i ,ρ)	U 2 εn |∇ wn | 2 =	Ω\∪B(y n i ,ρ 0 )	U 2 εn |∇ wn | 2 +	∪ i R(y n i ,ρ 0 ,r)	U 2 εn |∇ wn | 2 +
							+	∪ i R(y n i ,r,ρ)	U 2 εn |∇ wn | 2 .
	From Step 1. (the definition of ρ 0 and Estimate (E.1)), we have
		1
		2 Ω\∪B(y n i ,ρ 0 )
							≤	1 2 Ω\∪B(x n i ,ρ 0 )
	From Step 2., letting α ′ ε =		b 2 in δZ × δZ + λδ • ω ′ 1 otherwise	, we have
	1 2 ∪ i R(y n i ,ρ 0 ,r)	U 2 εn |∇ wn | 2	= (Step 2.) =	πd ln	ρ 0 r	+ o n (1)

≤ (Lem. 53 & 56) ≤ 1 2 ∪ i R(x n i ,ρ 0 ,r) α ′ |∇w n | 2 + o n (1)

, √ η,η)

D.2 Behavior of almost minimizers of I ρ,ε

We recall that for x 0 ∈ R 2 and R > r > 0, we denoted R(x 0 , R, r) := B(x 0 , R) \ B(x 0 , r).

D.2.1 Useful results for the periodic situation

We establish three preliminary results for the periodic situation represented Figure 1. Thus in this subsection we assume that U ε is the unique global minimizer of E ε in H 1 1 with the periodic pinning term a ε represented Figure 1.

Energetic estimates in rings and global energetic upper bounds

From Lemma 47 (U ε is close to a periodic function) we obtain Lemma 48. For all 1 ≥ R > r ≥ ε, x, x 0 ∈ R 2 s.t. B(x 0 , R) ⊂ Ω incl δ and xx 0 ∈ δ • Z 2 , we have µ ε (R(x, R, r), 1) ≥ µ ε (R(x 0 , R, r), 1)o ε [START_REF] Aftalion | Pinning Phenomena in the Ginzburg-Landau model of Superconductivity[END_REF].

Adding the condition that B(x, R) ⊂ Ω incl δ , we have

Moreover the o ε (1) may be considered independent of x, x 0 , R, r.

Lemma 48 implies easily the following estimate.

Proposition 49. Let η > 0 and η > ρ ≥ ε. Then there is C = C(Ω, Ω ′ , g, b, η) > 0 s.t. for x 0 ∈ R 2 we have I ρ,ε ≤ d µ ε (R(x 0 , η, ρ), 1) + C(η),

where C(η) is independent of x 0 and ρ.

From Lemma 47 we get the almost periodicity of µ ε (R(•, R, r), 1) w.r.t. a δ × δ-grid (expressed in Lemma 47). Therefore, the "best points" to minimize µ ε (R(•, R, r), 1) should be almost periodic.

Another important result is the next proposition. It expresses that the center of an inclusion is not too far to a good point to minimize µ ε (R(•, R, r), 1). This proposition may be seen as a first step in the proof of the pinning effect of ω ε .

Proposition 50. There is C * which depends only on ω, b and Ω s.t. for sufficiently small ε, for x ∈ Ω and x per ∈ B(x, 3δ √ 2/2) ∩ (δZ × δZ) ∩ ω ε we have for 1 > R > r > ε µ ε (R(x per , R, r), 1) ≤ µ ε (R(x, R, r), 1) + C * . (D.4)

Proof. If R ≤ 10 2 r, then the result is obvious with C * = 2π ln 10. Thus we assume that R > 10 2 r.

We share the proof in three cases:

D.3 Proof of the first part of Proposition 12

Let x n 1 , ..., x n N ∈ Ω s.t. |x n ix n j | ≥ 8ρ and d 1 , ..., d N > 0, d i = d (up to a subsequence the degrees may be considered independent of n).

Assume that there is i 0 ∈ {1, ..., N } s.t. d i 0 = 1 or there are i = j s.t.

Up to pass to a subsequence, there are a 1 , ..., a M ∈ Ω and {Λ 1 , ..., Λ M } a partition of {1, ..., N } s.t. i ∈ Λ l ⇐⇒ x n i → a l . For sake of simplicity, we drop the superscript n for the points, i.e., we write x i instead of

Note that since d i > 0, (D.6) is equivalent to

We are going to prove that (D.7) is not possible for almost minimal configurations. In order to do this, for l ∈ {1, ..., M }, we obtain a lower bound for the weighted Dirichlet functional defined around a l . Then using Proposition 49 we will conclude. For l ∈ {1, ..., M }, there are two cases:

In the first case (Card(Λ l ) > 1), we apply the separation process (defined Section C.1) in

By construction, the process stops after K steps. For k ∈ {1, ..., K} we denote:

then, in the end of the process (after K steps), we obtain a unique x K 1 = y l ∈ {x i | i ∈ Λ l } in the final selection of points and η K → 0. From (C.3) and (C.4), the following rings are mutually disjoint (denoting

In the second case (Card(Λ l ) = 1) the computations are direct

Summing the lower bounds (D.9) and (D.10) over l and applying Proposition 49, we obtain that if (D.7) occurs, then the configuration {x, d} cannot be almost minimal because η K , ρ → 0 and dl 0 > 1. Therefore (D.7) cannot occur for almost minimal configurations.

D.4 Proof of the second part of Proposition 12

We now prove the second part of Proposition 12: we establish the repelling effect of ∂Ω on the centers x i 's.

Let

From the previous subsection we may assume that there is η 0 > 0 (independent of n) s.t.

Up to pass to a subsequence, we may assume that x n i → a i ∈ Ω with a i = a j for i = j and that η = max{ dist(x n 1 , ∂Ω), ρ} → 0. For sake simplicity, we assume that for i = 2, ..., d we have a i ∈ Ω. If this condition is not satisfied, then a direct adaptation of the following argument may be done. We assume that η 0 is s.t. for i = 2, ..., d we have dist(x n i , ∂Ω) ≥ 10 2 η 0 . We fix

We are going to prove that for w ∈ H 1 g (Ω ′ \ ∪ i B(x n i , ρ), S 1 ) we have

We consider the map

For d ∈ N * (to be fixed later) we define the map

and deg ∂B(x n 1 , √ η) (w test ) = d + 1. Thus, we have

On the other hand, letting ϕ * , ϕ 0 : R(x n 1 , √ η, η) → R s.t. w * = e ıϕ * and w 0 = e ıϕ 0 , (note that ϕ * , ϕ 0 are locally defined and those gradients are globally defined and lie in

)), we have (using (D.12)),

√ η,η)∩Ω) = 0, we have (using Cauchy-Schwarz inequality)

Therefore we obtain

Clearly we obtain a contradiction taking d > (2b 2 )/(2b 2 ). Thus, by using Remark 54, the second part of Proposition 12 is proved.

D.5 Proof of the third part of Proposition 12

In this subsection, we prove the third part of Proposition 12: the attractive effect of the inclusions.

Assume that there exist

We denote x n = (x n 1 , ..., x n d ). From the first and the second assertion, there exists η 0 > 0 (independent of n) s.t.

We want to prove that there is some c > 0 s.t. for i = 1, ..., d we have (for large n) B(x n i , cλδ) ⊂ ω εn .

To this end, we argue by contradiction and we assume that either

We are going to prove that letting

Up to a subsequence, we may assume that lim n dist(x n 1 ,ωε n ) λδ exists. We divide the proof into two steps:

We now prove Step 1. Assume that

Denote w n a minimizer for I ρ,εn (x n , 1) (see Proposition 8). Using Lemma 52.2, for ρ ≤ r < R < η 0 , one has

From Lemma 52 (Assertions 2 and 3), we have

and

Thus it suffices to estimate the energies in the rings with radii κδ and ρ. We have (using (1.5))

In order to estimate µ εn (R(x n 1 , κδ, ρ), 1), we divide the argument according to the asymptotic of λ. If λ ≡ 1, then c ∈ (0, ∞) and thus dist(B(x i , cδ/3, ω εn ) ≥ cδ/3. Consequently, from Proposition 3, we have

.

Thus we obtain

Therefore if c ∈ (0, ∞], then (D.17 We now turn to Step 2. Arguing as in Step 1., it suffices to prove that µ εn (R(x n 1 , κδ, ρ), 1)µ εn (R(y n , κδ, ρ), 1), 1) → ∞ for some fixed κ. (D.18) (And

In order to prove (D.18), we divide the annular R(x n 1 , κδ, ρ) into three regions :

We are going to prove that µ εn (R(x n 1 , κλδ, r n ), 1) is too large. We consider K n the cone of vertex x n 1 and aperture π/2 which admits the line (x n 1 , Π ∂ωε n x n 1 ) for symmetry axis and s.t.

Note that since dist(x n 1 , ω εn ) λδ → 0, for large n and small κ (independently of n), by smoothness of ω, K n is well defined (see Figure 6).

We have

, then, from Lemma 53 with θ 0 = 3π/2, for w ∈

Clearly, from construction, U 2 εn ≥ α n + o(ε 2 n ), thus if w n is a minimal map for I ρ,εn (x n , 1), then we have

Now the computations are direct

Therefore, (D.18) is a direct consequence of (D. [START_REF] Lassoued | Sur quelques équations aux dérivées partielles non linéaires issues de la géométrie et de la physique[END_REF]) and (D. [START_REF] Mironescu | Les minimiseurs locaux pour l'équation de Ginzburg-Landau sont à symétrie radiale[END_REF]) since λδ r n → +∞.

E Proof of Propositions 16 and 17

We now prove the results specific to th pinning terms with dilution. We begin these proofs by their key ingredient.

E.1 An important effect of the dilution of inclusions

We first state a result which establishes that a "sufficiently large" circle has a small intersection with ω ε if λ → 0.

Lemma 56. We denote C ρ a circle with radius ρ.

1. Assume that the pinning term is periodic. Once λ ≤ 1/8π, for ρ ≥ δ/3 we have

2. Assume that the pinning term is not periodic, and recall that the inclusions with size

Proof. In order to unify the notations, we fix j = 1 if we are in the periodic case (and j ∈ {1, ..., P | M ε j ∈ N * } if we are in the non-periodic case). Assume that C ρ ∩ ω ε = ∅ and let

For Ỹε ∈ S j , we denote

in the non-periodic case .

We first treat the case where C ρ ⊂ Ỹε ∈ S j : since ρ ≥ δ j /3 and ωε ⊂ B(z j , λδ j ) (because ω ⊂ B(0, 1)), we have

Otherwise, for Ỹε ∈ S j , C ρ Ỹε and thus

and

Last estimate comes from the fact that C ρ Ỹε . Thus H 1 (C ρ ∩ Ỹε \ ωε ) is at least a radius of Ỹε (if we are in the non periodic or half of side length of Ỹε otherwise) minus the previous upper bound. Thus we obtain (for λ ≤ 1/8π)

From Lemma 47

Therefore we obtain (E.3) and consequently Proposition 16 holds.

E.3 Proof of Proposition 17

The strategy to prove Proposition 17 is the following:

Step 1. We let κ = max(λ, δ). We first characterize almost minimal configurations for I κ,ε (i.e the domain Ω is perforated by discs with radius κ).

Step 2. We make the description of almost minimal points (x ε ) ε for µ ε (R(•, κ, λδ 3/2 ), 1).

Step 3. We estimate inf x 0 ∈R 2 µ ε (R(x 0 , λδ 3/2 , ρ), 1) and we conclude.

Step 1. We study almost minimal configurations for I κ,ε , κ = max(λ, δ)

with C(η 0 ) which is independent of ε. We consider {x, d} which is almost minimal for I κ,ε . We argue as in the proof of Proposition 12 (Assertions 1 and 2, see Subsections D.3 & D.4). We use the separation process defined Subsection C.1 and the associated natural partition of

Here the key ingredients are Lemmas 53 & 56 (which replace the periodic structure of the pinning term). Combining both lemmas we get that if R > r ≥ κ, then

The rings R(x 0 , R, r) which occur in the partition of Ω κ are all s.t. Conversely, from (E.4), for η 0 > 0 and

Step 2. We study almost minimal configurations for µ ε (R(•, κ, λδ 3/2 ), 1)

For j ∈ {1, ..., P }, we denote: ω j ε := ∪ i∈M ε j {y ε i,j + λδ j ω}. And recall that the set of centers of connected components of

ε , on the one hand we may easily prove that

and on the other hand, applying Lemmas 53 & 56, we have

Therefore, from (E.5) and (E.6), we get

We are going to prove that this situation (B(x 0 ε , cλδ) ⊂ ω 1 ε ) is the only way to get the minimal energy. More precisely we prove that for a fixed constant C 0 > 0, if we have

then there is c > 0 independent of ε s.t. for sufficiently small ε we have

Up to pass to a sequence ε n ↓ 0, dropping the subscript n (we write ε instead of ε n ), we may assume that one of these cases occurs Case 0.

We want to prove that only Case 0. occurs if (E.8) holds. Case 1. From Lemmas 53 & 56, it is direct to prove that

Using (E.7) we get

Therefore, if (x ε ) ε satisfies (E.8), then Case 1. does not occur. Case 2. We let j 0 ∈ {2, ..., P } be s.t. x ε ∈ ∪ i∈M ε j 0 B(y i,j 0 , δ j 0 ). We define κ ′ := max{δ j 0 , λδ 3/2 } and we denote y 0 = y ε i,j 0 ∈ M j 0 be s.t. x ε ∈ B(y 0 , δ j 0 ). We first assume that x ε / ∈ ω ε and we let ae = max{λδ 3/2 , dist(x ε , ∂ω j 0 ε )λδ j 0 }. In order to estimate µ ε (R(x ε , κ, λδ 3/2 ), 1), we divide R(x ε , κ, λδ

From Lemmas 53 & 56 we have

Note that dist(R(x ε , κ ′ +2λδ j 0 , ae+2λδ j 0 ), ω j 0 ε ) ≥ λδ j 0 and if for some j we have R(x ε , κ ′ + 2λδ j 0 , ae + 2λδ j 0 ) ∩ ω j ε = ∅, then dist(x ε , ω j ε ) ≥ δ j (because x ε ∈ B(y 0 , δ j 0 )). Therefore, using Proposition 3 and Lemmas 53 & 56 we get

.

By definition of ae, from Proposition 3, we have

Summing these lower bounds we have

and therefore µ ε (R(x ε , κ, λδ 3/2 ), 1)inf x 0 ∈Ω µ ε (R(x 0 , κ, λδ 3/2 ), 1) → +∞ (because 0 ≤ ln(κ/λ) ≤ | ln δ| and from (E.7)). We now assume that x ε ∈ ω ε . Because j 0 ≥ 2 and x ε ∈ B(y 0 , λδ j 0 ), we have B(y 0 , 2λδ j 0 ) ∩ R(x ε , κ, λδ 3/2 ) = ∅. Therefore, from the dilution of the inclusion, if there is ωε , a connected component of ω j ε s.t. R(x ε , κ, λδ 3/2 ) ∩ ωε , then dist(x ε , ωε ) ≥ δ j /3. Consequently, from Lemmas 53 & 56, we have

From (E.7), we obtain that µ ε (R(x ε , κ, λδ 3/2 ), 1)inf x 0 ∈Ω µ ε (R(x 0 , κ, λδ 3/2 ), 1) → +∞.

We deduce that if (x ε ) ε satisfies (E.8), then Case 2. does not occur.

Case 3. We denote y 0 := y ε i,1 ∈ M ε 1 be s.t. x ε ∈ B(y 0 , δ). On the one hand, if κ ≤ 10 -2 δ, then we have µ ε (R(x ε , κ, δ), 1), µ ε (R(y 0 , κ, δ), 1) ≤ 2π ln 10.

On the other hand, if κ > 10 -2 δ, then we have R(y 0 , κ, 10δ) ⊂ R(x ε , 10κ, 10 -1 δ) and thus (using Proposition 45) we get Moreover, following the argument of Subsection D.5, we have (because

µ ε (R(x ε , δ, λδ 3/2 ), 1)µ ε (R(y 0 , δ, λδ 3/2 ), 1) → +∞.

Therefore we have the existence of H ε → +∞ as ε → 0 s.t.

Consequently µ ε (R(x ε , κ, λδ 3/2 ), 1)inf x 0 ∈Ω µ ε (R(x 0 , κ, λδ 3/2 ), 1) → +∞ and since (x ε ) ε satisfies (E.8), Case 3 does not occur.

Step 3. We study inf x 0 ∈R 2 µ ε (R(x 0 , λδ 3/2 , ρ), 1) and we conclude It is obvious that inf x 0 ∈R 2 µ ε (R(x 0 , λδ 3/2 , ρ), 1) = πb 2 ln λδ 3/2 ρ + o ε (1). Now we are in position to conclude. On the one hand, from the previous steps, for η 0 , c > 0 and a configuration of points/degrees {x ε , 1}

i , ∂Ω) → 0 or there are i = j s.t. |x ix j | → 0, then the configuration of points/degrees cannot be almost minimal for I δ,εn and thus it cannot be almost minimal for I ρ,εn . Moreover, if there is i s.t. x εn i / ∈ ω 1 εn or dist(x εn i , ∂ω 1 εn )/(λδ) → 0, then (x εn i ) n cannot be an almost minimal configuration for µ ε (R(•, κ, λδ 3/2 ), 1). And thus {x, d} cannot be an almost minimal configuration for I ρ,εn .

Therefore Assertions 1. and 2. of Proposition 17 holds. The rest of the proposition is obtained exactly as Corollary 14.

F Proof of Proposition 32

We use the unfolding operator (see [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF], definition 2.1). We define, for Ω 0 ⊂ R 2 an open set, p ∈ (1, ∞) and δ > 0:

Here, for s ∈ R, [s] is the integer part of s. We will use the following results:

T δ is linear and continuous, of norm at most 1 ([8], Proposition 2.5), (F.1)

) is such that φ δ ⇀ φ 0 in H 1 , then, up to subsequence, there exists φ ∈ L 2 (Ω 0 , H 1 per ( Ỹ )) s.t.:

T δ (φ δ ) → φ 0 and T δ (∇φ δ ) ⇀ ∇φ 0 + ∇ y φ in L 2 (Ω 0 × Ỹ ) ([8], Theorem 3.5). (F.5)

Here H 1 per ( Ỹ ) stands for the set of functions φ ∈ H 1 ( Ỹ ) s.t. the extending of φ by Ỹperiodicity is in H 1 loc (R 2 ) (see [START_REF] Cioranescu | An Introduction to Homogenization[END_REF], section 3.4). In order to define properly the homogenized matrix A we recall a classical result (see Theorem 4.27 in [START_REF] Cioranescu | An Introduction to Homogenization[END_REF]).

Proposition 58. Let H 0 ∈ L ∞ ( Ỹ , [b 2 , 1]). For all f ∈ (H 1 per ( Ỹ )) ′ s.t. f annihilates the constants there exists a unique solution h ∈ H 1 per ( Ỹ ) of div(H 0 ∇ y h) = f and M Ỹ (h) = Ỹ h = 0.

Using the previous theorem we denote χ j ∈ H 1 per ( Ỹ ) the unique solution of div(H 0 ∇ y χ j ) = ∂ y j (H 0 ) and M Ỹ (χ j ) = 0. (F.6)

With these auxiliary functions, we can give an explicit expression of A the homogenized matrix of H 0 ( • δ )Id R 2 (see Theorem 6.1 in [START_REF] Cioranescu | An Introduction to Homogenization[END_REF]):

For the convenience of the reader we restate, in larger detail, Proposition 32.

Proposition. Let Ω 0 ⊂ R 2 be a smooth bounded open set and let v n ∈ H 2 (Ω 0 , C) be s.t.

1. |v n | ≤ 1 and

Then v * is a solution of div(A∇v * ) = (A∇v * • ∇v * )v * .

Here A is the homogenized matrix of H 0 ( • δ )Id R 2 given by

Here A = Ỹ H 0 (Id R 2 -∇ y χ). Thus -div (A∇v * × v * ) = 0. Note that, since H 0 and χ are independent of x, A is a constant matrix. This fact combined with the equation -div (A∇v * × v * ) = 0 implies that v * satisfies div(A∇v * ) = (A∇v * • ∇v * )v * . (F.9) Indeed, we can always consider ϕ * which is locally defined in Ω 0 and whose gradient is globally defined and in L 2 (Ω 0 , R 2 ) s.