Etienne Sandier

Sylvia Serfaty

From the Ginzburg-Landau Model to Vortex Lattice Problems

Keywords: Ginzburg-Landau, vortices, Abrikosov lattice, triangular lattice, renormalized energy, Gamma-convergence. MSC classification: 35B25, 82D55, 35Q99, 35J20, 52C17

We introduce a "Coulombian renormalized energy" W which is a logarithmic type of interaction between points in the plane, computed by a "renormalization." We prove various of its properties, such as the existence of minimizers, and show in particular, using results from number theory, that among lattice configurations the triangular lattice is the unique minimizer. Its minimization in general remains open.

Our motivation is the study of minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields H c1 and H c2 . In that regime, minimizing configurations exhibit densely packed triangular vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, W as a Γ-limit of the Ginzburg-Landau energy. More precisely we show that the vortices of minimizers of Ginzburg-Landau, blown-up at a suitable scale, converge to minimizers of W , thus providing a first rigorous hint at the Abrikosov lattice. This is a next order effect compared to the mean-field type results we previously established.

The derivation of W uses energy methods: the framework of Γ-convergence, and an abstract scheme for obtaining lower bounds for "2-scale energies" via the ergodic theorem that we introduce.

Introduction

In this paper, we are interested in deriving a "Coulombian renormalized energy" from the Ginzburg-Landau model of superconductivity. We will start by defining and presenting the renormalized energy in Section 1.1, then state some results about it in Section 1.2. In Section 1.3, we then present an abstract method for lower bounds for two-scale energies using ergodic theory. In Sections 1.5-1.10 we turn to the Ginzburg-Landau model, and give our main results about it as well as ingredients for the proof.

The Coulombian renormalized energy W

The interaction energy W that we wish to define just below is a natural energy for the Coulombian interaction of charged particles in the plane screened by a uniform background: it could be called a "screened Coulombian renormalized energy". It can be seen in our context as the analogue for an infinite number of points in R 2 of the renormalized energy W introduced in Bethuel-Brezis-Hélein [BBH] for a finite number of points in a bounded domain, or of the Kirchhof-Onsager function. We believe that this energy is quite ubiquitous in all problems that have an underlying Coulomb interaction: it already arises in the study of weighted Fekete sets and of the statistical mechanics of Coulomb gases and random matrices [START_REF] Sandier | 2D Coulomb Gases and the Renormalized Energy[END_REF], as well as a limit in some parameter regime for the Ohta-Kawasaki model [START_REF] Goldman | The Γ-limit of the two dimensional Ohta-Kawasaki energy. Part II: Derivation of the renormalized energy[END_REF]. In [SS7] we introduce a one-dimensional analogue (a renormalized logarithmic interaction for points on the line) which we also connect to one-dimensional Fekete sets as well as "log gases" and random matrices.

We will discuss more at the end of this subsection and in the next, but let us first give the precise definition.

In all the paper, B R denotes the ball centered at 0 and of radius R, and | • | denotes the area of a set.

Definition 1.1. Let m be a positive number. Let j be a vector field in R 2 . We say j belongs to the admissible class A m if

(1.1) curl j = ν -m, div j = 0,
where ν has the form

ν = 2π p∈Λ δ p for some discrete set Λ ⊂ R 2 ,
and

(1.2) ν(B R ) |B R | is bounded by a constant independent of R > 1.
For any family of sets {U R } R>0 in R 2 we use the notation χ U R for positive cutoff functions satisfying, for some constant C independent of R,

(1.3) |∇χ U R | ≤ C, Supp(χ U R ) ⊂ U R , χ U R (x) = 1 if d(x, U R c ) ≥ 1.
We will always implicitly assume that {U R } R>0 is an increasing family of bounded open sets, and we will use the following set of additional assumptions:

• (1.4) {U R } is a Vitali family and lim R→+∞ |(λ + U R ) △ U R | |U R | = 0.
for any λ ∈ R 2 . Here, a Vitali family (see [Ri]) means that the intersection of the closures is {0}, that R → |U R | is left continuous, and that |U R -U R | ≤ C|U R | for some constant C > 0 independent of R.

• There exists θ < 2 such that for any R > 0

(1.5) U R + B(0, 1) ⊂ U R+C , U R+1 ⊂ U R + B(0, C), |U R+1 \ U R | = O(R θ ).
Definition 1.2. The Coulombian renormalized energy W is defined, for j ∈ A m , by

(1.6) W (j) = lim sup R→∞ W (j, χ B R ) |B R | ,
where for any function χ we denote

(1.7) W (j, χ) = lim η→0   1 2 R 2 \∪ p∈Λ B(p,η) χ|j| 2 + π log η p∈Λ χ(p)   .
We similarly define the renormalized energy relative to the family {U R } R>0 by

(1.8) W U (j) = lim sup R→∞ W (j, χ U R ) |U R | .
Let us make several remarks about the definition.

1. We will see in Theorem 1 that the value of W does not depend on {χ B R } R as long as it satisfies (1.3). The corresponding statement holds for W U under the assumptions (1.4)-(1.5).

2. Since in the neighborhood of p ∈ Λ we have curl j = 2πδ p -1, div j = 0, we have near p the decomposition j(x) = ∇ ⊥ log |x -p| + f (x) where f is smooth, and it easily follows that the limit (1.7) exists. It also follows that j belongs to L p loc for any p < 2. 3. From (1.1) we have j = -∇ ⊥ H for some H, and then

-∆H = 2π p∈Λ δ p -m.
Then the energy in (1.7) can be seen as the (renormalized) interaction energy between the "charged particles" at p ∈ Λ and between them and a constant background -m. We prefer to take j = -∇ ⊥ H as the unknown, though, because it is related to the superconducting current j ε .

4. We will see in Theorem 1 that the minimizers and the value of the minimum of W U are independent of U , provided (1.4) and (1.5) hold. However there are examples of admissible j's (nonminimizers) for which W U (j) depends on the family of shapes {U R } R>0 which is used.

5. Because the number of points is infinite, the interaction over large balls needs to be normalized by the volume and thus W does not feel compact perturbations of the configuration of points. Even though the interactions are long-range, this is not difficult to justify rigorously.

6. The cut-off function χ R cannot simply be replaced by the characteristic function of B R because for every p ∈ Λ lim

R→|p| R<|p| W (j, 1 B R ) = +∞, lim R→|p| R>|p| W (j, 1 B R ) = -∞.
7. It is easy to check that if j belongs to A m then j ′ = 1 √ m j(•/ √ m) belongs to A 1 and

(1.9) W (j) = m W (j ′ ) -1 4 log m .

so we may reduce to the study of W over A 1 .

When the set of points Λ is periodic with respect to some lattice Z u + Z v then it can be viewed as a set of n points a 1 , • • • , a n over the torus T ( u, v) = R 2 /(Z u + Z v). There also exists a unique periodic (with same period) j {a i } with mean zero and satisfying (1.1) for some m which from (1.1) and the periodicity of j {a i } must be equal to 2πn divided by the surface of the periodicity cell. Moreover j {a i } minimizes W among ( u, v)-periodic solutions of (1.1) (see Proposition 3.1). The computation of W in this setting where both Λ and j are periodic is quite simpler (the need for the limit R → ∞ and the cutoff function disappear). By the scaling formula (1.9), we may reduce to working in A n in a situation where the volume of the torus is 2π. Then we will see in Section 3.1 the following Lemma 1.3. With the above notation, we have

(1.10) W (j {a i } ) = 1 2 i =j G(a i -a j ) + nc ( u, v)
where c ( u, v) is a constant depending only on ( u, v) and G is the Green function of the torus with respect to its volume form, i.e. the solution to

-∆G(x) = 2πδ 0 -1 in T ( u, v) .
Moreover, j {a i } is the minimizer of W (j) among all T ( u, v) -periodic j's satisfying (1.1).

Remark 1.4. The Green function of the torus admits an explicit Fourier series expansion, through this we can obtain a more explicit formula for the right-hand side of (1.10):

(1.11)

W (j {a i } ) = 1 2 i =j p∈(Z u+Z v) * \{0} e 2iπp•(a i -a j ) 4π 2 |p| 2 + n 2 lim x→0   p∈(Z u+Z v) * \{0} e 2iπp•x 4π 2 |p| 2 + log |x|  
where * refers to the dual of a lattice.

The function i =j G(a i -a j ) is the sum of pairwise Coulombian interactions between particles on a torus. It arises for example in number theory (Arakelov theory), see [START_REF] Lang | Introduction to Arakelov theory[END_REF] p. 150, where a result attributed to Elkies is stated: i =j G(a i -a j ) ≥ -n 4 log n + O(n) (on any Riemann surface of genus ≥ 1). Note that we can retrieve this estimate in the case of the torus by using the fact that min A 1 W is finite and formula (1.12) with m = n.

So conversely, another way of looking at our energy W is that it provides a way of computing an analogue of i =j G(a i -a j ) in an infinite-size domain.

Results and conjecture on the renormalized energy

The following theorem summarizes the basic results about the minimization of W . Note that by the scaling relation (1.9) we may reduce to the case of A 1 , and we have (1.12) min

Am W = m min A 1 W - 1 4 log m .
Theorem 1. Let W be as in Definition 1.2.

1. Let {U R } R>0 be a family of sets satisfying (1.4)-(1.5), then for any j ∈ A 1 , the value of W U (j) is independent of the choice of χ U R in its definition as long as it satisfies (1.3).

2. W is Borel measurable on L p loc (R 2 , R 2 ), p < 2.

3. min A 1 W U is achieved and finite, and it is independent of the choice of U , as long as {U R } satisfies (1.4)-(1.5).

4. There exists a minimizing sequence {j n } n∈N for min A 1 W consisting of vector-fields which are periodic (with respect to a square lattice of sidelength √ 2πn).

The question of identifying the minimum and minimizers of W seems very difficult. In fact it is natural to expect that the triangular lattice (of appropriate volume) minimizes W over any A m , we will come back to this below. We show here a weaker but nontrivial result: the triangular lattice is the unique minimizer among lattice configurations.

When the set of points Λ itself is a lattice, i.e. of the form Z u ⊕ Z v, denoting by j Λ the j which is as in Lemma 1.3, that lemma shows that W (j Λ ) is equal to c ( u, v) and only depends on the lattice Λ. We will denote it in this case by W (Λ). For the sake of generality, we state the result for any volume normalization:

Theorem 2. Let L = {Λ | Λ is a lattice and j Λ ∈ A m }. Then the minimum of Λ → W (Λ) over L is achieved uniquely, modulo rotation, by the triangular lattice

Λ m = 4π m √ 3 (1, 0)Z ⊕ 1 2 , √ 3 2 Z .
The normalizing factor ensures that the periodicity cell has area 2π/m, or equivalently that Λ m ∈ A m .

Remark 1.5. The value of W for the triangular lattice with m = 1 estimated numerically from formula (3.6) or (3.10) is ≃ -0.2011. For the square lattice it is ≃ -0.1958.

Theorem 2 is proven by expressing j Λ using Fourier series. Then minimizing W over L is a limit case of minimizing the Epstein ζ function Λ → p∈Λ\{0} 1 |p| 2+x over L when x → 0. This question was answered by Cassels,Rankin,Ennola,Diananda,[START_REF] Cassels | On a problem of Rankin about the Epstein zeta-function[END_REF][START_REF] Rankin | A minimum problem for the Epstein zeta function[END_REF][START_REF] Ennola | A remark about the Epstein zeta function[END_REF][START_REF] Ennola | On a problem about the Epstein zeta-function[END_REF][START_REF] Diananda | Notes on two lemmas concerning the Epstein zeta-function[END_REF]. In a later self-contained paper [Mont], Montgomery shows that the ζ-function is actually the Mellin transform of another classical function from number theory: the Theta function

(1.13) θ Λ (α) = p∈Λ e -πα|p| 2 .
He then deduces the minimality of the ζ function at the triangular lattice from the corresponding result for the θ function (we will give more details in Section 3).

The main open question is naturally to show that the triangular lattice is a minimizer among all configurations:

Conjecture 1. The lattice Λ m being defined as in Theorem 2, we have W (Λ m ) = min Am W.

Note that a minimizer of W cannot be unique since compact perturbations do not affect the value of W , as seen in the fifth remark in Section 1.1. However, it could be that the triangular lattice is the only minimizer which is also a local minimizer in the following sense: if Λ ′ is any set of points differing from Λ by a finite number of points, and j Λ ′ a corresponding perturbation of j Λ , then

lim R→∞ W (j Λ ′ , χ B R ) -W (j Λ , χ B R ) ≥ 0.
(Note that here we do not normalize by |B R |.)

A first motivation for this conjecture comes from the physics: in the experiments on superconductors, triangular lattices of vortices, called Abrikosov lattices, are observed, as predicted by the physicist Abrikosov from Ginzburg-Landau theory. But we shall prove here, cf. Theorem 4, that vortices of minimizers of the Ginzburg-Landau energy functional (or rather, their associated "currents") converge in some asymptotic limit to minimizers of W , so if one believes experiments show ground states, then it can be expected that the triangular lattice corresponds to a minimizer of W . Another motivation for this conjecture is that, returning to the expression (1.6)-(1.7), W can be seen as a renormalized way of computing 2π p δ p -1 H -1 , thus minimizing W over A 1 is heuristically like trying to minimize 2π p δ p -1 H -1 over points in the plane, or trying to allocate points in the plane in the most uniform manner. By analogy with packing problems and other crystallisation problems, it seems natural, although far out of reach, that this could be accomplished by the triangular lattice. Positive answers are found in [Rad] for packing problems, [Th] for some very short-range pairwise interaction potential, and one also finds the same conjecture and some supporting arguments in [CK] Section 9, for a certain (but different) class of interaction potentials. But we note however again that here the interaction between the points is logarithmic hence long range, in contrast with these known results. Finally, in dimension 1, the situation is much easier, since we can prove in [SS7] that the minimum of the one-dimensional analogue of W is indeed achieved by the perfect lattice Z (suitably rescaled).

We have seen in Theorem 1 that W has a minimizer which is a limit of periodic configurations with large period. This will be used crucially for the energy upper bound on Ginzburg-Landau in Section 7. It also connects the question of minimizing W in all generality to the simpler one of minimizing it in the periodic setting. By the formula (1.10) the problem in the periodic setting reduces to minimizing i =j G(a i -a j ) or (1.11) over the torus. The points that achieve such minima (on all types of surfaces) are called Fekete points (or weighted Fekete sets) and are important in potential theory, random matrices, approximation, see [ST]. Their average distribution is well-known (see [ST]), in our setting it is uniform, but their precise location is more delicate to study (in [START_REF] Sandier | 2D Coulomb Gases and the Renormalized Energy[END_REF] we make progress in that direction and connect them to W ). As already mentioned in the last remark of the previous subsection, estimates on the minimum value of W in this setting are also used in number theory.

The energy W also bears some ressemblance with a nonlocal interaction energy related to diblock copolymers, sometimes called "Ohta-Kawasaki model" and studied in particular in a recent paper of Alberti, Choksi and Otto [ACO], see also [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF][START_REF] Goldman | The Γ-limit of the two dimensional Ohta-Kawasaki energy. Part I: Droplet density[END_REF] (and previous references therein): there, one also has a logarithmic interaction, but the Dirac masses are replaced by nonsingular charges, and so no renormalization is needed. More precisely the interaction energy is u -m H -1 where m is a fixed constant in [-1, 1] and u takes values in {-1, 1}, whose BV norm is also penalized. There, triangular lattice configurations are also observed, it can even be shown [START_REF] Goldman | The Γ-limit of the two dimensional Ohta-Kawasaki energy. Part II: Derivation of the renormalized energy[END_REF] that the energy W can be derived as a limit in the regime where m → 1 (in this regime, the problem becomes singular again). Also, the analogue result to our Theorem 2 is proven for that model in [CO] using also modular functions. In [ACO] it is proven that for all m ∈ [-1, 1] the energy for minimizers is uniformly distributed. Our study of W in Section 4 is similar in spirit. Note that results of equidistribution of energy analogous to [ACO] could also most likely be proven with our method.

Lower bounds for two-scale energies via the ergodic theorem

In this subsection we present an abstract framework for proving lower bounds on energies which contain two scales (one much smaller than the other). This framework will then be crucially used in this paper, both for proving the results of W in Theorem 1 and for obtaining the lower bounds for Ginzburg-Landau in Theorem 4 and 5. We believe it is of independent interest as well.

The question is to deduce from a Γ-convergence (in the sense of De Giorgi) result at a certain scale a statement at a larger scale. The framework can thus be seen as a type of Γconvergence result for 2-scale energies. The lower bound is expressed in terms of a probability measure, which can be seen as a Young measure on profiles (i.e. limits of the configuration functions viewed in the small scale). Following the suggestion of Varadhan, this is achieved by using Wiener's multiparameter ergodic theorem, as stated in Becker [Be]. Alberti and Müller introduced in [AM] a different framework for a somewhat similar goal, with a similar notion of Young measure, that they called "Young measures on micropatterns". In contrast with Young measures, these measures (just like ours) are not a probability measure on values taken by the functions, but rather probability measures on the whole limiting profile. The spirit of both frameworks is the same, however Alberti and Müller's method did not use the ergodic theorem. It was also a bit more general since it dealt with problems that are not homogeneous at the larger scale but admit slowly varying parameters (however we generalize to such dependence in forthcoming work), and it adressed the Γ-limsup aspect as well. Our method is more rudimentary, but maybe also more flexible. In forthcoming work, we refine it and include a version with dependence on the "slow (larger scale) variable" in [START_REF] Sandier | 2D Coulomb Gases and the Renormalized Energy[END_REF], as well as an application to random homogenization in [BSS].

Let X denote a Polish metric space (for reference see [Du]). When we speak of measurable functions on X we will always mean Borel-measurable. We assume that there exists an nparameter group of transformations θ λ acting continuously on X. More precisely we require that -For all u ∈ X and λ, µ ∈ R n , θ λ (θ µ u) = θ λ+µ u, θ 0 u = u.

-The map (λ, u) → θ λ u is measurable on R n × X.

-The map (λ, u) → θ λ u is continuous with respect to each variable (hence measurable with respect to both).

Typically we think of X as a space of functions defined on R n and θ as the action of translations, i.e. θ λ u(x) = u(x + λ).

We also consider a family {ω ε } ε of domains of R n such that for any R > 0 and letting ω ε,R = {x ∈ ω ε | dist (x, ∂ω ε ) > R}, we have

(1.14) |ω ε | ∼ |ω ε,R | as ε → 0.
In particular the diameter of ω ε tends to +∞ as ε → 0. Finally we let {f ε } ε and f be measurable nonnegative functions on X, and assume that for any family {u ε } ε such that (1.15) ∀R > 0, lim sup

ε→0 B R f ε (θ λ u ε ) dλ < +∞
the following holds:

1. (Coercivity) {u ε } ε admits a convergent subsequence.

2. (Γ-liminf) If {u ε } ε is a convergent subsequence (not relabeled) and u is its limit, then

(1.16) lim inf ε→0 f ε (u ε ) ≥ f (u).
The abstract result is Theorem 3. Let {θ λ } λ , {ω ε } ε and {f ε } ε , f be as above. Let {U R } R>0 be a family satisfying (1.4). Let

(1.17)

F ε (u) = - ωε f ε (θ λ u) dλ.
Assume that {F ε (u ε )} ε is bounded. Let P ε be the image of the normalized Lebesgue measure on ω ε under the map λ → θ λ u ε . Then P ε converges along a subsequence to a Borel probability measure P on X invariant under the action θ and such that

(1.18) lim inf ε→0 F ε (u ε ) ≥ f (u) dP (u).
Moreover, for any family {U R } R>0 satisfying (1.4), we have

(1.19) f (u) dP (u) = f * (u) dP (u),
with f * given by

(1.20) f * (u) := lim R→+∞ - U R f (θ λ u) dλ.
In particular, the right-hand side of (1.19) is independent of the choice of {U R } R>0 .

Remark 1.6. The result (1.19) is simply the ergodic theorem in multiparameter form. Part of the result is that the limit in (1.19) exists for P -almost every u.

The probability measure P is the "Young measure" on limiting profiles we were referring to before, indeed it encodes the limit of all translates of u ε . Note that P ε → P implies that P almost every u is of the form lim ε→0 θ λε u ε . The limit defining f * can be viewed as a "cell problem", using the terminology of homogenization, providing the limiting small scale functional.

To illustrate this result, we shall give two examples, in order of increasing generality, both models for what we will use here.

Example 1. Consider X = M(R n ) the set of positive bounded measures on R n , and θ λ the action of translations of R n . Let χ be a given nonnegative smooth function with support in the unit ball of R n , and define f ε (µ) = f (µ) = R n χ dµ. Then one can check that F ε , as defined in (1.17), is

F ε (µ) = 1 |ω ε | R n χ * 1 ωε dµ.
The result of the theorem (choosing balls for example) is the the assertion that

lim inf ε→0 F ε (µ ε ) ≥ f (µ) dP (µ) = E P lim R→+∞ χ * 1 B R dµ .
The probability P gives a measure over all limiting profiles of µ ε (depending on the centering point), and the result says that the quantity we are computing, here the average of µ ε over the large sets ω ε , can be bounded below by an average, this time over P , of a similar quantity for the limiting profiles. This implies in particular that there is a µ 0 in the support of P , hence of the form

lim ε→0 µ ε (λ ε + •) such that lim inf ε→0 1 |ω ε | R n χ * 1 ωε dµ ε ≥ lim R→+∞ χ * 1 B R dµ 0 .
In other words we can find a good centering sequence λ ε such that the average of µ ε over the large sets ω ε can be bounded from below by the average over large balls of the limit after centering, µ 0 . The averages over ω ε or B R , and the average with respect to P are not of the same nature (in standard ergodic settings, the first ones are usually time averages, while the latter is a space average).

Example 2. Let us assume we want to bound from below an energy which is the average over large (as ε → 0) domains ω ε of some nonnegative energy density e ε (u), defined on a space of functions X (functions over R n ), -ωε e ε (u(x)) dx, and we know the Γ-liminf behavior of e ε (u) on small (i.e. here, bounded) scales, say we know how to prove that lim inf ε→0 B R e ε (u) dx ≥

B R e(u) dx. We cannot always directly apply such a knowledge to obtain a lower bound on the average over large domains, this may be due to a loss of boundary information, or due to the difficulty to reverse limits R → ∞ and ε → 0. (These are two obstructions that we encounter specifically both for Ginzburg-Landau and W , as illustrated by the 6th remark in Section 1.1) What we can do is let χ be a smooth cutoff function as above, and define the functions f ε by

f ε (u) = e ε (u(x)) χ(x) dx,
that is f ε can be seen as the small scale local functional. Since we know the Γ-liminf behavior of the energy density e ε on small scales, let us assume we can prove that (1.16) holds for some function(al) f , a function on u (we may expect that f will also be of the form e(u(x)) χ(x) dx.) Defining F ε as in (1.17) and using Fubini's theorem, we see that

F ε (u) = - ωε R n e ε (u(x + y))χ(x) dx dy = 1 |ω ε | z∈ωε-x x e ε (u(z))χ(x) dx dz.
Since χ is supported in B 1 , χ = 1, and ω ε satisfies (1.14), we check that

F ε (u) ∼ ε→0 - ωε e ε (u(y) dy,
hence F ε is asymptotically equal to the average we wanted to bound from below. We may thus apply the theorem and it yields a lower bound for the desired quantity:

lim inf ε→0 F ε (u ε ) ≥ f * (u) dP (u) with (1.21) f * (u) = lim R→∞ - B R f (u(x + •)) dx
The "cell-function" f * is simply an average over large balls of the local Γ-liminf f . If typically f is of the form e(u(x)) χ(x) dx then we can compute by Fubini that

f * (u) = lim R→∞ 1 |B R | e(u(x))(χ * 1 B R )(x) dx.

The Ginzburg-Landau model

Our original motivation in this article is to analyze the behaviour of minimizers of the Ginzburg-Landau energy, given by

(1.22) G ε (u, A) = 1 2 Ω |∇ A u| 2 + |curl A -h ex | 2 + (1 -|u| 2 ) 2 2ε 2 .
This is a celebrated model in physics, introduced by Ginzburg and Landau in the 1950's as a model for superconductivity. Here Ω is a two dimensional bounded and simply connected domain, u is a complex-valued function, "order parameter" in physics, describing the local state of the superconductor, A : Ω → R 2 is the vector potential of the magnetic field h = curl A = ∇ × A, and ∇ A denotes the operator ∇ -iA. Finally the parameter h ex denotes the intensity of the applied magnetic field and ε is a constant corresponding to a characteristic lengthscale (of the material). It is the inverse of κ, the Ginzburg-Landau parameter in physics. We are interested in the ε → 0 asymptotics. The quantity |u| 2 measures the local density of superconducting electron pairs (|u| ≤ 1). The material is in the superconducting phase wherever |u| ≃ 1 and in the normal phase where |u| ≃ 0. We focus our attention on the zeroes of u with nonzero topological degree (recall that u is complex-valued), also known as the vortices of u. Here the typical lengthscale of the set where |u| is small, hence of the vortex "cores", is ε.

The Euler-Lagrange equations associated to this energy with natural boundary conditions are the Ginzburg-Landau equations

(1.23)          -(∇ A ) 2 u = u ε 2 (1 -|u| 2 ) in Ω -∇ ⊥ h = (iu, ∇ A u) in Ω h = h ex on ∂Ω ν • ∇ A u = 0 on ∂Ω.
where ∇ ⊥ denotes the operator (-∂ 2 , ∂ 1 ), ν the outer unit normal to ∂Ω and (•, •) the canonical scalar product in C obtained by identifying C with R 2 .

This model is also famous as the simplest "Abelian gauge theory". Indeed it admits the U(1) gauge-invariance : the energy (1.22), the equation (1.23), and all the physical quantities are invariant under the transformation

u → ue iΦ A → A + ∇Φ.
For a more detailed mathematical presentation of the functional, one may refer to [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] and the references therein.

Physical behaviour: critical fields and vortex lattices

Here and in all the paper a ≪ b means lim a/b = 0.

For ε small, the minimizers of (1.22) depend on the intensity h ex of the applied field as follows.

-If h ex is below a critical value called the first critical field and denoted by H c 1 , the superconductor is in the so-called Meissner state characterized by the expulsion of the magnetic field and the fact that |u| ≃ 1 everywhere. Vortices are absent in this phase.

-If h ex is above H c 1 , which is equivalent to λ Ω |log ε| as ε → 0, then energy minimizers have one vortex, then two,... The number of vortices increases with h ex , so as to become equal to leading order to h ex /2π for fields much larger than |log ε| (see [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF]). In this case, according to the picture we owe to A. Abrikosov and dating back to the late 1950's, vortices repell each other and organize themselves in triangular lattices named Abrikosov lattices. The theoretical predictions of Abrikosov have received ample experimental confirmation and there are numerous and striking observations of the lattices.

-If h ex increases beyond a second critical value H c 2 = 1/ε 2 , another phase transition occurs where superconductivity disappears from the material, except for a boundary layer. Even this boundary layer completely disappears when h ex is above a third critical field H c 3 .

Since the works of Ginzburg, Landau and Abrikosov, this model has been largely studied in the physics literature. We refer to the classic monographs and textbooks by De Gennes [DeG], Saint-James Sarma -Thomas [SST], Tinkham [Ti].

The above picture describes the phenomenology of the model for small values of ε (or high values of κ) in a casual way, but by now many rigorous mathematical results support this picture. Except for the third critical field however, whose existence was proven in a strong sense by Fournais and Helffer [FH1,[START_REF] Fournais | Spectral Methods in Surface Superconductivity[END_REF], mathematical results really prove the existence of intervals where the transition between the different types of behaviour occur, rather than critical values, with estimates on these intervals as ε → 0.

Our goal is the study and description of the vortices in the whole range H c 1 < h ex ≪ H c 2 , where minimizers of the Ginzburg-Landau energy have a large number of vortices, expected to form Abrikosov lattices.

Connection to earlier mathematical works

There is an abundant mathematical literature related to the study of (1.22) and to the justification of the physics picture above. A relatively extensive bibliography is given in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF], Chap. 14.

The techniques most relevant to us for the description of vortices originate in the pioneering book of Bethuel-Brezis-Hélein [BBH] and have been further expanded by several authors including in particular Jerrard, Soner [Je, JS], ourselves, etc... For a more detailed description of these techniques we refer to [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF]. In that book we describe how these techniques allow to derive the asymptotic values of the critical field H c 1 as ε → 0 as well as the mean-field description of minimizers and their vortices in the regimes h ex ≪ H c 2 .

In order to describe the vortices of minimizers, one introduces the vorticity associated to a configuration (u, A), defined by

(1.24) µ(u, A) = curl j(u, A) + curl A, where j(u, A) := (iu, ∇ A u)
is the superconducting current. Here (a, b) denotes the Euclidean scalar product in C identified with R 2 , so j(u, A) may also be written as i 2 u∇ A u -ū∇ A u , or as ρ 2 (∇ϕ -A) if u = ρe iϕ , at least where ρ = 0.

This vorticity is the appropriate quantity to consider in this context, rather than the quantity curl (iu, ∇ A u) which could come to mind first. It may be seen as a gauge-invariant version of curl (iu, ∇u) which is also (twice) the Jacobian determinant of u. Indeed if A = 0, then µ(u, A) = 2∂ x u × ∂ y u. One can prove (this is the so-called Jacobian estimate, see [START_REF] Jerrard | The Jacobian and the Ginzburg-Landau energy[END_REF][START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF]) that assuming a suitable bound on G ε (u, A), the vorticity µ(u, A) is well approximated, in some weak sense, as ε → 0 by a measure of the form 2π i d i δ a i . As points of concentration of the vorticity, the points {a i } i are naturally called vortices of u and d i , which is an integer, is called the degree of a i . The vorticity µ(u, A) may either describe individual vortices or, after normalization by the number of vortices, their density.

In previous work (summarized in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF], Chap. 7) we showed that

(1.25) H c 1 ∼ λ Ω |log ε|,
for a constant λ Ω > 1 2 depending only on Ω (and such that λ Ω → 1 2 as Ω → R 2 ). More precisely, letting h ex = λ|log ε|, we established by a Γ-convergence approach that minimizers

(u ε , A ε ) of G ε satisfy (1.26) curl A ε h ex converges to h λ , µ(u ε , A ε ) h ex converges to µ λ , as ε → 0,
where µ λ = -∆h λ + h λ , and h λ is the solution of the following minimization problem

(1.27) min h-1∈H 1 0 (Ω) 1 2λ Ω | -∆h + h| + 1 2 Ω |∇h| 2 + |h -1| 2 .
This problem is in turn equivalent to an obstacle problem, and as a consequence there exists a subdomain ω λ such that

(1.28)

µ λ = m λ 1 ω λ , where m λ = 1 - 1 2λ ,
and where 1 A denotes the characteristic function of A. In other words the optimal limiting vortex density µ λ is uniform and equal to m λ over a subregion of Ω, completely determined by λ i.e. by the applied field h ex . The constant λ Ω introduced in (1.25) is characterized by the fact that

ω λ = ∅ if and only if λ < λ Ω . Since λ Ω > 1 2 , if ω λ = ∅ we have (1.29) 1 ≥ m λ ≥ 1 - 1 2λ Ω > 0.
Since the number of vortices is proportional to h ex = λ|log ε|, it tends to +∞ as ε → 0. It is also established in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] that as ε → 0, the minimal energy has the following expansion,

(1.30) min G ε = 1 2 h ex log 1 ε √ h ex Ω µ λ + h ex 2 2 Ω |∇h λ | 2 + |h λ -1| 2 + o h ex log 1 ε √ h ex .
When the applied field is much larger than |log ε|, but much less than 1/ε 2 , (1.26) and (1.30) still hold, replacing λ by +∞. In this case h λ = 1 and ω λ = Ω.

There are other cases where the distribution of vortices for minimizers of the Ginzburg-Landau functional is understood. First in a periodic setting: Minimizing the Ginzburg-Landau energy among configurations (u, A) which are periodic (modulo gauge equivalence) with respect to a certain lattice independent of ε, one obtains (see [Ay, AyS]) as above a limiting vortex density µ λ = -∆h λ + h λ where h λ minimizes the energy in (1.27) among periodic functions. The minimizer in this case is clearly a constant, which is easily found to be max 1 -1 2λ , 0 . Second, in the regime of applied fields h ex where the number of vortices tends to +∞, but is negligible compared to |log ε|, (this corresponds to applied fields such that log |log ε| ≪ h ex -λ Ω |log ε| ≪ |log ε| as ε → 0), it is shown in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] that for simply connected domains satisfying a certain generic property (see (1.31) below) -including convex domains -vortices concentrate around a single point (a finite number of points for general simply connected domains). Then, blowing up at the suitable scale and normalizing the vorticity, one obtains in the limit a probability measure µ which describes the distribution of vortices around the point, and µ is the unique minimizer among probability measures in R 2 of

I(µ) = -π log |x -y| dµ(x) dµ(y) + π Q(x) dµ(x).
Note that here Q is a positive definite quadratic form which depends on the domain Ω. It is the Hessian of a certain function at the point of concentration of the vortices. Note that the precise regime of applied field modifies the number of vortices, i.e. the normalizing factor of the vorticity, and the scaling, but does not influence the limit distribution µ, which is a characteristic of Q.

In our previous work, the treatment for the "intermediate" regime log |log ε| ≪ h exλ Ω |log ε| ≪ |log ε| and for the regime h ex = λ|log ε|, λ > λ Ω were different, the former one being more delicate. Here we provide (and this is part of the technical difficulties) a unified approach for both, and treat all regimes h ex ≪ H c 2 where the number of vortices is blowing up.

Main result on Ginzburg-Landau

The mean field description above tells us that the vortices tend to be distributed uniformly in ω λ but is insensitive to the pattern formed by vortices. This pattern is in fact, as we shall see, selected by the minimization of the next term in the asymptotic expansion of the energy as ε → 0. The proof of this is achieved in this paper by a splitting of the energy that separates the leading order term found in (1.30) from a remainder term, and then by studying the remainder term after blow up at the scale of the expected intervortex distance, which from the above considerations is of the order of 1/ √ h ex ; this remainder term is then shown to Γ-converge to W (as introduced in Section 1.1), hence allows to distinguish among vortex configurations.

As before, we use the current j(u, A) = (iu, ∇ A u) (cf. (1.24)) to describe the vortex locations, we study through the abstract framework of Section 1.3 the probability measure carried by all possible limiting profiles of blow ups of j at the scale 1/ √ h ex centered at all possible blow-up points in ω λ , and show that this probability is concentrated on minimizers of W .

Before stating the simplest form of our main result, let us explain the main assumptions we need to make. First, we make the simplifying assumption that the domain Ω is convex. This is only used at one point in the proof (the upper bound construction) and avoids the possibility that ω λ may have cusps. We believe our results still hold without this restriction. Then letting h 0 be the solution to -∆h 0 + h 0 = 0 with h 0 = 1 on ∂Ω -which is consistent with the definition of h λ in (1.27) -it is known (see Caffarelli-Friedman [CF], the result in dimensions greater than two can be found in [KL]) that in the case where Ω is convex, h 0 is strictly convex hence achieves its minimum at a unique point x 0 ∈ Ω. Moreover this minimum is nondegenerate:

(1.31) min Ω h 0 is achieved at a unique point x 0 and Q := D 2 h 0 (x 0 ) is positive definite.
For a family (u ε , A ε ), we denote

(1.32) ε,x (•) := 1 √ h ex j(u ε , A ε ) x + • √ h ex
their blown-up current, where in the right-hand side j(u ε , A ε ) is implicitly extended by 0 outside the domain Ω.

We have Theorem 4. Assume that Ω is convex, so that (1.31) is satisfied, and that

h ex = λ|log ε| with λ > λ Ω .
Let (u ε , A ε ) be a minimizer of G ε , and let ε,x be as in (1.32) for x ∈ ω λ . Then, given 1 < p < 2, there exists a probability measure P on L p loc (R 2 , R 2 ) such that the following hold:

1. Up to extraction, for any bounded continuous function Φ on L p loc (R 2 , R 2 ), we have

(1.33) lim ε→0 1 |ω λ | ω λ Φ( ε,x ) dx = Φ(j) dP (j).
2. P -almost every j minimizes W over A m λ and

(1.34) G ε (u ε , A ε ) = G N 0 ε + h ex |ω λ |(min W + cst) + o(h ex ) as ε → 0, where G N 0 ε ∈ R is explicited in (1.39) below.
We may informally describe G N 0 ε : It is the minimum value for an obstacle problem similar to (1.27). But whereas (1.27) is derived assuming the vortex energy is π|log ε|, to obtain G N 0 ε we must use instead the more precise value π| log ε ′ |, where

ε ′ = ε h ex . As ε → 0 we have |log ε| ≈ | log ε ′ |, but using | log ε ′ | induces a correction which is not o(h ex ), hence is important to us.
As we have seen in Section 1.2, W allows to distinguish between configurations of points since it distinguishes between lattices, and it is expected to favor the triangular lattice. The result of Theorem 4 can be informally understood as follows: if one chooses a point at random in ω λ and blows up at the scale 1/ √ h ex , then in the limit ε → 0, almost surely (with respect to the blow up center), one sees a minimizer of W . This derivation of this limiting energy W is, to our knowledge, the first rigorous justification of the Abrikosov triangular lattice in this regime: at least the triangular lattice is the best among lattice configurations, and it is conjectured to be a global minimizer (see Section 1.2).

In Theorem 5 below, we will give a more precise and a full Γ-convergence version of Theorem 4, valid for the other regimes of applied field where vortex lattices are expected to arise. The latter result encompasses the regimes of Theorems 1.3, 1.4 and 1.5 in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] and allows to reprove these results.

Returning to the reference to the Theta function (1.13) in Section 1.2, it is striking to observe that the problem of minimizing the Theta function also arises in the context of Ginzburg-Landau, but in a very different regime: when h ex ∼ H c 2 or more precisely when h ex = b ε 2 with b ր 1. As seen in [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF]AS] this is a regime which is essentially linear (contrarily to the one we study here) and the energy minimization can be reduced to the minimization of a function on a finite dimensional space (the "lowest Landau level"-this is essentially the result of Abrikosov's original calculation). This function can be viewed as the linear analogue of W and reduces, in the case where the points are on a lattice, to the θ function θ Λ (and so again the optimal lattice is the triangular one). In that sense the limiting lattice energies for Ginzburg-Landau in the regime H c 1 ≪ h ex ≪ H c 2 and in the regime h ex ∼ H c 2 can be viewed as Mellin transforms of each other.

We now go into more detail on the method of the proof of Theorem 4. It follows from a result of Γ-convergence, i.e. by showing a general lower bound for the energy and a matching upper bound via an explicit construction. Thus the minimality of (u ε , A ε ) and the Euler-Lagrange equation it solves is not used per se. The proof of the lower bound involves three ingredients: an energy splitting, a blow-up, and the abstract method of Theorem 3.

The energy splitting

The first ingredient of the proof, detailed in Section 5, is a new algebraic splitting of the energy, which allows to isolate the constant leading order part from the next-order.

First we define a mean field h 0,ε similar to h ex h λ , except that when computing it we take more precisely into account the cost of a vortex which is π| log ε ′ |, where we recall ε ′ = ε √ h ex . Accordingly, assuming h ex < 1/ε 2 , we let h 0,ε be the minimizer of (1.35) min

h-hex∈H 1 0 (Ω) 1 2 | log ε ′ | Ω | -∆h + h| + 1 2 Ω |∇h| 2 + |h -h ex | 2 .
Thus h 0,ε /h ex solves (1.27), with λ replaced by λ ε = hex | log ε ′ | . This is equivalent (see [Br, BS]) to saying that h 0,ε /h ex minimizes the H 1 norm subject to the constraints h 0,ε /h ex = 1 on ∂Ω and

(1.36) h 0,ε h ex ≥ m 0,ε := 1 - | log ε ′ | 2h ex in Ω.
In other words h 0,ε is the solution to an obstacle problem with constant obstacle. Letting then µ 0,ε = -∆h 0,ε + h 0,ε , we have

(1.37) µ 0,ε = m 0,ε h ex 1 ω 0,ε , where ω 0,ε = {x | h 0,ε (x) = h ex m 0,ε } .
We define

(1.38) N 0 = 1 2π Ω µ 0,ε .
The splitting function could be taken to be h 0,ε in most regimes of applied field. Then one should define G N 0 ε in Theorem 4 as

(1.39) G N 0 ε := πN 0 | log ε ′ | + 1 2 h 0,ε -h ex 2 H 1 (Ω) ,
where

• H 1 denotes the Sobolev space norm • 2 L 2 + ∇ • 2 L 2 1 2
. However, when h ex -λ Ω |log ε| ≪ |log ε| -i.e. when the number of vortices is small compared to h ex though divergent as ε → 0 -then (1.35), which is a refinement of (1.27), must itself be refined to take into account the constraint that the vorticity is quantized. For the other regimes, the error made by ignoring this constraint is negligible in our analysis.

More precisely, given N such that 0 ≤ N ≤ 1 2π h ex |Ω|, we consider h ε,N the minimizer of (1.40) min

h-hex∈H 1 0 (Ω) R Ω |∆h+h|=2πN 1 2 Ω |∇h| 2 + |h -h ex | 2 ,
and we define

(1.41) G N ε := πN | log ε ′ | + 1 2 h ε,N -h ex 2 H 1 (Ω) .
Since h ex will be a given function of ε, we denote the dependence of h ε,N as of ε instead of h ex . G N ε is minimal at N = N 0 and we will see that h ε,N 0 = h 0,ε . The refinement with respect to (1.35) consists in taking N to be an integer, when N 0 is not necessarily one. More precisely, N will be taken to be either N - 0 , the largest integer ≤ N 0 , or N + 0 , the smallest integer ≥ N 0 . With that choice, we will sometimes call h ε,N the "splitting function".

The leading order term in the energy is not exactly G N 0 ε but rather min N ∈{N - 0 ,N + 0 } G N ε . We may immediately check however that for N ∈ {N - 0 , N + 0 }, as ε → 0,

G N ε ∼ G N 0 ε ∼ 1 2 h ex | log ε ′ | Ω µ λ + h ex 2 2 Ω |∇h λ | 2 + |h λ -1| 2
when h ex ∼ λ|log ε| in the notation of Section 1.6, hence it recovers the leading order term of the minimal energy (1.30). The difference between min N ∈{N -

0 ,N + 0 } G N ε and G N 0 ε is o(h ex )
hence it is negliglible for the precision of o(N ) we want to achieve as soon as λ > λ Ω , but not always when λ = λ Ω .

h ε,N is the solution to an obstacle problem and we have h ε,N ≥ m ε,N h ex and -∆h ε,N + h ε,N = h ex m ε,N 1 ω ε,N where ω ε,N = {h ε,N = m ε,N h ex } is called the coincidence set, for some constant m ε,N such that m ε,N → m λ as ε → 0. Note that ω ε,N will depend on the choice of N = N - 0 or N + 0 but sometimes we will forget it and simply write ω ε . We let

µ ε,N = -∆h ε,N + h ε,N = h ex m ε,N 1 ω ε,N .
It is a perturbation of µ λ and ω ε,N a perturbation of ω λ , defined in Section 1.6. We have the relation

(1.42) |ω ε,N |m ε,N h ex = Ω µ ε,N = 2πN.
Then we (temporarily) introduce A 0,ε = ∇ ⊥ h ε,N . Letting (u, A) be a configuration of finite energy, we will write

A 1,ε = A -A 0,ε = A -∇ ⊥ h ε,N ,
where A 0,ε = ∇ ⊥ h ε,N is understood as the leading order term, and A 1,ε as a remainder term. The energy-splitting is the observation of the following identity (valid even if N is not an integer):

(1.43)

G ε (u, A) = G N ε + Ω (h ε,N -h ex )µ(u, A 1,ε ) + 1 2 Ω |∇ A 1,ε u| 2 + |curl A 1,ε -µ ε,N | 2 + (1 -|u| 2 ) 2 2ε 2 -c ε,N Ω µ ε,N - 1 2 Ω (1 -|u| 2 )|∇h ε,N | 2 ,
where we recall (1.24), and c ε,N is a constant explicited in (5.3). The last term in (1.43) can be shown to be negligible if G ε (u, A) is not too large. Thus the study of the energy near its minimum reduces to that of the remainder

(1.44) F ε (u, A 1,ε ) := 1 2 Ω |∇ A 1,ε u| 2 + |curl A 1,ε -µ ε,N | 2 + (1 -|u| 2 ) 2 2ε 2 + Ω (h ε,N -h ex )µ(u, A 1,ε ) -c ε,N Ω µ ε,N .
It turns out that when we make the right specific choice N = N - 0 or N = N + 0 (depending on (u ε , A ε )), this expression simplifies and one has

(1.45) F ε (u, A 1,ε ) ≥ 1 2 Ω |∇ A 1,ε u| 2 + |curl A 1,ε -µ ε,N | 2 + (1 -|u| 2 ) 2 2ε 2 - Ω ζ ε µ(u, A 1,ε ) + o(1)
where ζ ε is a positive function, equal to its maximum 1 2 | log ε ′ | on ω ε,N = Supp(µ ε,N ). It is this remainder F ε (with this choice of N ) whose Γ-convergence we study. We will see through the upper bound construction that min F ε ≤ Ch ex |ω ε,N | (this is equivalent to min F ε ≤ CN or ≤ CN 0 from (1.42)), and we will work in that class F ε (u, A 1,ε ) ≤ CN 0 , thus reducing to a relatively narrow class of "almost minimizers" i.e. configurations whose leading order energy is the minimal one G N 0 ε . Note that once we know that this remainder F ε is of lower order, this shows that the leading order component in

A = ∇ ⊥ h ε,N +A 1,ε is ∇ ⊥ h ε,N (in other words A 1,ε ≪ A 0,ε
) so also at leading order µ(u ε , A ε ) ∼ µ ε,N ∼ µ λ and this allows to recover in essence the results of [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] Theorems 1.3, 1.4, 1.5 for all configurations in this almost minimizing class.

The next step is to make the change of scales x ′ = √ h ex x in order to study (1.44). Under this rescaling, the inter-vortex distance becomes of order 1 (recall that the average vortex density is precisely m ε,N h ex 1 ωε with m ε,N → m λ and (1.28)). After this change of scales the right-hand side of (1.45) becomes, in terms of u ′ (x ′ ) = u(x) and A ′ (x ′ ) = A(x)/ √ h ex , and

ζ ′ ε (x ′ ) = ζ(x), (1.46) F ′ ε (u ′ , A ′ ) = 1 2 Ω ′ ε |∇ A ′ u ′ | 2 + h ex |curl A ′ -m ε 1 ω ′ ε | 2 + (1 -|u ′ | 2 ) 2 2ε ′ 2 - Ω ′ ε ζ ′ ε (x ′ )µ(u ′ , A ′ ) + o(1)
where ω ′ ε is the rescaled domain √ h ex ω ε,N and we recall ε ′ = ε √ h ex . Combining all these elements, the conclusion of the splitting procedure, found in Section 5 is Proposition 1.7. For any (u, A), there exists

N ∈ {N + 0 , N - 0 } such that G ε (u, A) ≥ G N ε + F ′ ε (u ′ , A ′ ) + o(1)
where

F ′ ε is as in (1.46), u ′ (x ′ ) = u(x ′ / √ h ex ), A ′ (x ′ ) = A(x ′ / √ h ex ) and ζ ′ is a positive function, equal to its maximum 1 2 | log ε ′ | on ω ′ ε .

Full version of the main result

We may now give the more complete version of Theorem 4 and the stronger statement of Γ-convergence of 1 N (G ε (u, A) -G N ε ). In all the paper, the weak convergence of probabilities will mean convergence against bounded continuous test-functions, see [Bi]. We will say that a probability measure is concentrated on a set if that set has probability 1.

We consider configurations (u ε , A ε ) and assume that |u ε | ≤ 1 everywhere together with the second Ginzburg-Landau equation, i.e. we assume

(1.47) |u| ≤ 1 in Ω, -∇ ⊥ curl A = (iu, ∇ A u) in Ω
is satisfied. This is obviously true for minimizers and critical points of (1.22). Moreover given (u, A), replacing u by u/|u| wherever |u| ≥ 1 and replacing A by the minimizer of A → G ε (u, A), with u remaining fixed, decreases the energy without displacing the vortices, and the modified (u, A) verify (1.47). Since the change can only decrease the energy, our results allow to bound from below G ε (u ε , A ε ) for the original arbitrary (u, A). Thus, assuming (1.47) is no loss of generality. Note that the only consequence of (1.47) that we will really use is that div j(u, A) = 0. Note that when h ex -λ Ω |log ε| = O(log |log ε|) we already know from [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF], Chap. 12, that the number of vortices remains bounded as ε → 0 and we already characterized their limiting location.

G ε (u ε , A ε ) ≤ min N ∈{N - 0 ,N + 0 } G N ε + CN 0
for some C independent of ε. Then there is a choice of N ∈ {N - 0 , N + 0 } such that, letting P ε be the probability measure on L p loc (R 2 , R 2 ) defined as the push-forward of the normalized uniform measure on ω ε,N by the map x → ε,x (cf. (1.32)), as ε → 0 the measures {P ε } ε converge up to extraction to a probability measure P on L p loc (R 2 , R 2 ) which is invariant under the action of translations, concentrated on A m λ (m λ as in (1.28)) and

(1.49) G ε (u ε , A ε ) ≥ min N ∈{N - 0 ,N + 0 } G N ε + N 2π m λ W U (j) dP (j) + γ + o(1) ,
where γ is defined in (1.52), and W U is computed according to (1.8) relatively to any family of sets satisfying (1.4)-(1.5).

In the case h ex ≤ 1 ε β for some β > 0 small enough, then

(1.50) µ(u ε , A ε ) -µ ε,N W -1,p (Ω) ≤ C p √ N .
2. For any probability P on L p loc (R 2 , R 2 ) which is invariant under the action of translations and concentrated on A m λ and for every N ∈ {N - 0 , N + 0 }, there exists (u ε , A ε ) such that, letting P ε be the push-forward of the normalized Lebesgue measure on ω ε,N by the map

x → 1 √ hex j(u ε , A ε ) x + • √ hex , we have as ε → 0, P ε → P and (1.51) G ε (u ε , A ε ) ≤ G N ε + N 2π m λ W K (j) dP (j) + γ + o(1) ,
where W K is the renormalized energy relative to the family of squares {K R } R , where

K R = [-R/2, R/2] 2 , as defined in (1.8).
3. If we assume that (u ε , A ε ) minimizes G ε then it satisfies all the assumptions of item 1), P -almost every j minimizes W over A m λ , and there is equality in (1.49).

Remark 1.8.

-The constant γ in (1.49) and (1.51) was introduced in [BBH] and may be defined by

(1.52) γ = lim R→∞ 1 2 B R |∇u 0 | 2 + (1 -|u 0 | 2 ) 2 2 -π log R ,
where u 0 (r, θ) = f (r)e iθ is the unique (up to translation and rotation) radially symmetric degree-one vortex (see [BBH, Mi]).

-There exists {(u ε , A ε )} ε satisfying (1.51) for any N such that 1 ≪ N ≤ hex|Ω| 2π , see Theorem 7 in Section 7.

-As we already mentioned in Section 1.8, this theorem allows to retrieve the results of [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF], but it gives a stronger result: in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] we establish the leading order behaviour of the vorticity for minimizers: µ(u ε , A ε ) ∼ h ex µ λ ∼ µ ε,N , while here (1.50) gives (at least for small enough applied fields) for the whole class of "almost minimizers" the order of the fluctuations of the vorticity around the constant density µ ε,N : it is of order √ N , hence in particular the number of vortices is N with an error of order √ N .

-For energy minimizers it is possible to deduce from this result that N , now being defined as the total degree of the vortices, satisfies

G N ε = min N ∈N G N ε + o( N ). On the other hand min N ∈N G N ε is achieved at N - 0 or N + 0 .
From examining carefully the variations of N → G N ε one should be able to deduce that N = N - 0 or N + 0 with a smaller error than previously (at least for h ex ≤ H c 1 +O( |log ε|) we expect this error to be 0), see Remark 6.7 for more details. So we expect, for small enough fields, to be able to estimate exactly the total degree of the vortices of a minimizer, and for larger fields, to estimate it with an error which is at least better than √ N .

1.10 Use of the ergodic theorem for Ginzburg-Landau

As announced, the method consists in applying the framework of Section 1.3 to the Ginzburg-Landau energy.

We sketch the method in the case h ex = λ|log ε|, λ Ω < λ < +∞. The case of higher fields h ex ≫ |log ε| will reduce to this one by scaling.

Let (u ε , A ε ) -or (u ′ ε , A ′ ε ) in rescaled coordinates -denote a minimizer of G ε and let

µ ′ ε = µ(u ′ ε , A ′ ε )
. The splitting result of Section 1.8 combined with the blow up procedure reduces us (cf. Proposition 1.7) to bounding from below F ′ ε (u ′ ε , A ′ ε ). Thus we are in the setting of Example 2 in Section 1.3, i.e. the case where we want to bound from below the average over large domains ω ′ ε of some energy density e ε . Here the energy density is of the form

e ε (u, A) = 1 2 |∇ A u| 2 + h ex |curl A -m1 ω ′ ε | + (1 -|u| 2 ) 2 2ε 2 -ζ ′ dµ(u, A).
From the Jacobian estimate µ ′ ε is well approximated by ν ε = 2π i d i δ a i (where a i denotes the vortex center and d i its degree) and we should have

a i ∈ ω ′ ε . Using in addition that ζ ′ = 1 2 | log ε ′ |
there, we may thus formally replace the energy-density above by

e ε (u, A) = 1 2 |∇ A u| 2 + h ex |curl A -m1 ω ′ ε | 2 + (1 -|u| 2 ) 2 2ε 2 -π |log ε| i d i δ a i .
To apply the framework of Section 1.3 we need to check the coercivity and Γ-liminf properties of f ε (u, A) = e ε (u, A) χ for a cut-off function χ as in the example of Section 1.3. But the framework also requires that f ε be nonnegative (or bounded below by a constant will do) but this is obviously not the case, and one of the major difficulties of our analysis consists in getting around this problem. This part of the analysis was carried out in our companion paper [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF], where we introduced a method that consists in displacing the negative part of e ε so as to absorb it into the positive part and to obtain a density bounded below, called g ε . We

show there that e ε can be replaced by g ε ≥ -C with making only a small error (in average at the small scale). Note that since we are dealing with cancelling leading order terms, we need very precise estimates of the free-energy f ε , in fact we need to make errors which are at most o(1) per vortex (the total acceptable error is o(N )). Hence in [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF] we need refined estimates in the "ball construction methods" which are devised to obtain general lower bounds for the energy of the vortices even when their number is unbounded. We also show in [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF] the crucial fact that g ε controls the number of vortices.

Returning to the question of proving the coercivity of f ε , we may now heuristically replace e ε by g ε , and the coercivity requires proving 1.15 which in our case becomes

∀R > 0, lim sup ε→0 g ε (χ * 1 B R ) < ∞ =⇒ (u ε , A ε ) compact.
This is satisfied thanks to the fact that g ε controls the number of vortices, in other words we have roughly

∀R > 0, lim sup ε→0 g ε (χ * 1 B R ) < ∞ =⇒ a i ∈B R |d i | ≤ C R .
To prove the Γ-liminf relation on f ε , we may reduce to that setting, i.e. that where the number of vortices is bounded independently of ε on the compact support of χ. In that setting, it is now standard to retrieve very precise estimates for the Ginzburg-Landau energy, using an analysis of the type of [BBH]. This way, we obtain the precise Γ-liminf for f ε and can show it can be expressed as a function of j (limit of j(u ′ ε , A ′ ε ) and is equal (up to a constant) to f (j) = W (j, χ). But then, using the definition (1.20), the "cell-energy" is

f * (j) = lim R→+∞ - U R W (j(λ + •), χ) dλ.
One may immediately check, using Fubini, that this is equal to lim R→+∞

1 |U R | W (j, χ * 1 U R ),
which is in turn clearly equal to W U (j) (cf. (1.8) and the first item in Theorem 1). Combining all these elements, the result of Theorem 3 yields

lim inf ε→0 - ω ′ ε 1 2 |∇ A ′ ε u ′ ε | 2 + h ex |curl A ′ ε -m1 ω ′ ε | + (1 -|u ′ ε | 2 ) 2 2ε 2 -ζ ′ dµ(u ′ ε , A ′ ε ) ≥ W (j) dP (j),
which is essentially the desired lower bound.

The rigorous proof of this is detailed in Section 6.

Plan of the paper

The paper contains two parts. The first part is completely independent of the Ginzburg-Landau energy and can thus be read independently. It starts in Section 2 with the proof of Theorem 3. In Section 3, we prove Theorem 2 i.e. that W is minimized among lattice configurations by the triangular lattice. In Section 4, we study W more generally, and prove Theorem 1.

The second part is about the application of the tools of the first part to the Ginzburg-Landau energy and the derivation of W as its Γ-limit. In Section 5, we prove the new energy-splitting formula, in the same section we recall or prove some results on the splitting function that will be needed in the sequel and we also derive simple a priori bounds for energyminimizers. In Section 6, we show how to apply the abstract framework of Section 1.3 in the specific case of the Ginzburg-Landau energy, and this way we obtain the main energy lower bound, and prove Theorems 4 and 5 assuming the upper bound, which is proven in Section 7. This requires the improved lower bounds for the energy of vortices borrowed from [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF]. In Section 7, we prove the matching upper bound for the energy via an explicit construction, using the periodic minimizing sequence found in Theorem 1.

In the Appendix, we prove some additional qualitative results on the solutions to the obstacle problem which can be of independent interest, in particular estimates when the size of the coincidence set is small, which are needed in the regime h ex -H c 1 ≪ |log ε|.

Part I

The renormalized energy 2 Proof of Theorem 3

Before giving the proof of the theorem, we state a preliminary lemma.

Lemma 2.1. (E. Lesigne) Assume P n are Borel probability measures on a Polish metric space X and that for any δ > 0 there exists {K n } n such that P n (K n ) ≥ 1 -δ for every n and such that if {x n } n satisfies for every n that x n ∈ K n , then any subsequence of {x n } n admits a convergent subsequence (note that we do not assume K n to be compact).

Then {P n } n admits a subsequence which converges tightly, i.e. converges weakly to a probability measure P .

Proof. From Prohorov's Theorem, it suffices to show that the sequence of measures is tight. As a finite Borel measure on a Polish space, the measure P n is regular [Co], thus there is a compact subset

K ′ n ⊂ K n such that P n (K ′ n ) ≥ 1 -2δ. Then, letting K = n K ′ n ,
we have P n (K) ≥ 1 -2δ for every n and the assumption made on {K n } n implies that K is compact. Indeed, a sequence in K is either included in a finite union of the compact sets K n (then is compact) or has a subsequence which can be relabelled (x n ) and satisfied x ′ n ∈ K ′ n along a subsequence n ′ , hence compact by assumption. Therefore {P n } n is tight.

We now start proving the theorem. First we choose a sequence {ε n } n tending to 0 such that lim

n→+∞ F εn (u εn ) = lim inf ε→0 F ε (u ε ).
In this proof, ε will always be assumed to belong to this sequence, and lim ε→0 will mean the limit along this sequence.

Recall that P ε is the image of the normalized Lebesgue measure restricted to ω ε under the map λ → θ λ u ε . In particular for any positive measurable function Φ on X

(2.1) Φ(u) dP ε (u) = - ωε Φ(θ λ u ε ) dλ.
Step 1:

{P ε } ε is tight. Letting ω ε,R = {x ∈ ω ε | dist (x, ∂ω ε ) > R} we have ω ε,R B R f ε (θ λ+µ u ε ) dλ dµ = R 2 ×R 2 1 ω ε,R (λ)1 B R (µ)f ε (θ λ+µ u ε ) dλ dµ = R 2 1 ω ε,R * 1 B R (λ)f ε (θ λ u ε ) dλ ≤ |B R | ωε f ε (θ λ u ε ) dλ ≤ C|B R ||ω ε |. (2.2) Let us denote Y ε,R the image of ω ε,R by λ → θ λ u ε , and
X ε R,K = u ∈ X | - B R f ε (θ λ u) dλ > K . The left-hand side in (2.2) is larger than K|B R ||ω ε |P ε X ε R,K ∩ Y ε,R . In addition P ε (X ε R,K ) ≥ P ε X ε R,K ∩ Y ε,R -P ε (Y c ε,R ) ≥ P ε X ε R,K ∩ Y ε,R - |ωε\ω ε,R | |ωε| , so we deduce from (2.2) that P ε (X ε R,K ) ≤ C K + |ω ε \ ω ε,R | |ω ε | .
From (1.14) and for any δ > 0, there exists a subsequence {ε n } n such that

|ω εn \ ω εn,R | |ω εn | < δ2 -n
and then P εn ∪ n k=1 X εn k,2 k /δ ≤ Cδ. Now we have that the hypotheses of Lemma 2.1 are satisfied. Indeed, letting K n be the complement of

∪ n k=1 X εn k,2 k /δ , we have P εn (K n ) ≥ 1 -Cδ. Moreover, if u n ∈ K n for every n then ∀R > 0, ∀n > R we have u n / ∈ X εn R,2 R /δ , i.e. - B R f εn (θ λ u n ) dλ ≤ 2 R δ .
Then, from the coercivity assumption, a subsequence of {u n } converges. Applying Lemma 2.1 we can conclude that {P εn } is tight and then that a subsequence converges weakly to a probability measure P .

Step 2: P is θ-invariant. Let Φ be bounded continuous on X. Then from the definition of

P ε , Φ(u) dP (u) = lim ε→0 Φ(u) dP ε (u) = lim ε→0 - ωε Φ(θ λ u ε ) dλ. Moreover, ωε Φ(θ λ u ε ) dλ - ωε Φ(θ λ+λ 0 u ε ) dλ = ωε Φ(θ λ u ε ) dλ - ωε+λ 0 Φ(θ λ u ε ) dλ. Thus, - ωε Φ(θ λ u ε ) dλ -- ωε Φ(θ λ+λ 0 u ε ) dλ ≤ |ω ε △ (ω ε + λ 0 )| |ω ε | Φ L ∞ ,
where △ denotes the symmetric difference between sets, and follows from (1.14

) that |ω ε △ (ω ε + λ 0 )| = o(|ω ε |) as ε → 0. We deduce that Φ(u) dP (u) = lim ε→0 - ωε Φ(θ λ+λ 0 u ε ) dλ = Φ(θ λ 0 u) dP (u),
hence P is invariant under the action θ.

We state the proof of (1.18) as a lemma.

Lemma 2.2. Assume that X is a Polish metric space, that {P ε } ε>0 , P are Borel probability measures on X such that P ε → P as ε → 0, and that {f ε } ε>0 and f are positive measurable functions on

X such that lim inf ε→0 f ε (x ε ) ≥ f (x) whenever x ε → x.
Then,

(2.3) lim inf ε→0 f ε dP ε ≥ f dP. This is (1.18) since f ε dP ε = -ωε f ε (θ λ u) dλ.
Proof of the Lemma. It suffices to show that for any λ, δ > 0, we have

(2.4) lim inf ε→0 P ε ({f ε > λ -δ}) ≥ P ({f > λ}).
Indeed, using the standard expression for the integral of a positive function f

f (x) dµ(x) = ∞ 0 µ({f > λ}) dλ,
we find by applying it to f ε and P ε , and then to f and P , in view of (2.4) that lim inf

ε→0 f ε (u) dP ε (u) = lim inf ε→0 ∞ 0 P ε ({f ε > λ}) dλ ≥ ∞ 0 lim inf ε→0 P ε ({f ε > λ}) dλ ≥ ∞ 0 P ({f > λ + δ}) dλ ≥ ∞ 0 P ({f > λ}) dλ -δ = f (u) dP (u) -δ,
where we have used Fatou's lemma, and the fact that P is a probability measure. Since this is true for any δ > 0, we have (2.3). We prove (2.4). For any δ > 0 and u ∈ X we claim that there exists an open neighborhood V u of u and η > 0 such that,

(2.5) ∀ε < η, ∀v ∈ V u , f ε (v) > F (u) -δ.
Indeed, assume this were wrong, then there would exist δ > 0 and u ∈ X together with a sequence {ε} tending to 0 and a corresponding sequence {u ε } ε tending to u such that f ε (u ε ) ≤ f (u) -δ, thus contradicting the Γ-liminf assumption. Hence (2.5) holds.

We denote by V η the set of u's such that f ε (v) > F (u) -δ holds on V u for every ε < η. Clearly, {V η } η is decreasing and from the above ∪ η>0 V η = X. Thus, if we let for some λ > 0

E = {u ∈ X | f (u) > λ}, E η = E ∩ V η , then {E η } η is decreasing and E = ∪ η E η . Moreover, from the definition of V η , we have f ε > λ -δ on the open set O η = ∪ u∈Eη V u for every ε < η.
It follows that for any ε < η,

(2.6) P ε ({f ε > λ -δ}) ≥ P ε (O η ).
Then, since P ε → P and since O η is open and contains E η , we have

lim inf ε→0 P ε (O η ) ≥ P (O η ) ≥ P (E η ).
It then follows from (2.6) that lim inf

ε→0 P ε ({f ε > λ -δ}) ≥ P (E η ).
Since E η ր E as η ց 0 we deduce by monotone convergence that (2.4) holds.

Step 4: Proof of (1.19). Since P is invariant w.r.t. the action θ, we may apply the ergodic theorem as stated in [Be] to obtain that

E P (f (u)) = E P (f * (u)),
where f * is θ-invariant and P -a.e. equal to

(2.7) lim R→∞ - B R f (θ λ u) dλ.
It is also true (see [Be]) that this limit exists for P -a.e. u, and that balls may be replaced by any Vitali family satisfying (1.4). This proves (1.19).

Remark 2.3.

1. The lower bound of Theorem 3 implies in particular that lim inf ε→0 F ε (u ε ) ≥ min f * where f * is given by (2.7). If we assume in addition that for some family there is equality i.e. that lim sup ε→0 F ε (u ε ) ≤ min f * , then comparing with the lower bound obtained in Theorem 3 we deduce that we have

f * (u) = min f * for P -a.e.u.
2. We may apply the same reasoning as Theorem 3 in B(x ε , R) instead of ω ε , with R large, to the functional -B(xε,R) f ε (θ λ u) dλ, and we will obtain

lim inf ε→0 - B(xε,R) f ε (θ λ u ε ) dλ ≥ min f * + o R (1)
where o R (1) → 0 as R → ∞ (the o R (1) is due to near boundary errors). If in addition we know that for some x ε ,

(2.8) lim sup ε→0 - B(xε,R) f ε (θ λ u) dλ ≤ min f * + o R (1)
we deduce

(2.9) lim

R→∞ lim ε→0 - B(xε,R) f ε (θ λ u ε ) dλ = min f * .
But if we assume lim sup ε→0 F ε (u ε ) ≤ min f * , by a Fubini argument (2.8) holds for most x ε ∈ ω ε , so the local estimate (2.9) too. This says that when the upper and lower bounds match, the energy density is essentially uniformly distributed, at any scale ≫ 1.

3 Minimization of W in the periodic case : optimality of the triangular lattice

Calculation of W in the periodic case

In this section we study the minimization of the renormalized energy W (j) among lattices in the following sense: We assume that Λ = Z u + Z v, where the vectors ( u, v) form a basis of R 2 , that j is invariant w.r.t. translations by the vectors u, v and

curl j = 2π p∈Λ δ p -1, div j = 0.
It is not difficult to check that there exists such a current j if and only if det( u, v) = 2π. We will denote T Λ the torus T Λ = R 2 /Λ. We have Proposition 3.1. Assume Λ = Z u + Z v and j are as above. Then, letting {U R } be any family satisfying (1.4), (1.5), defining W U (j) as in (1.8) we have

(3.1) W U (j) = lim η→0 1 2π 1 2 T Λ \B(0,η) |j| 2 + π log η .
Moreover, letting H Λ be the unique solution (with mean zero) on T Λ to

(3.2) -∆H Λ = 2πδ 0 -1,
and denoting

(3.3) j Λ = -∇ ⊥ H Λ
we have that j -j Λ is a constant and

W U (j) ≥ W U (j Λ )
with equality if and only if j = j Λ .

We will denote W U (j Λ ) simply W (Λ): it is the minimum of W U (j) among Λ-periodic configurations, and using (3.1) it is given by

W (Λ) = lim η→0 1 2π 1 2 T Λ \B(0,η) |∇H Λ | 2 + π log η .
Proof. Since curl (j -j Λ ) = div (j -j Λ ) = 0 and j, j Λ are periodic, we have that j -j Λ is constant. We now prove (3.1). Let us denote by K the set of cells K of the form

{t u + s v, t ∈ (l -1 2 , l + 1 2 ), s ∈ (m -1 2 , m + 1 2 )}, where l, m ∈ Z. For K ∈ K, let c K be the center of the cell. The cells of K tile R 2 , hence for every R > 0, writing χ R for χ U R , W (j, χ R ) = lim η→0 K∈K 1 2 K\B(c K ,η) |j| 2 χ R + πχ R (c K ) log η.
There are only finitely many K's on which χ R is not identically zero, thus we may write

W (j, χ R ) = K∈K w K (j, χ R ) where w K (j, χ R ) = lim η→0 1 2 K\B(c K ,η) |j| 2 χ R + πχ R (c K ) log η.
Then, if χ R ≡ 1 on K, we have (by periodicity of j)

(3.4) w K (j, χ R ) = lim η→0 T Λ \B(0,η) |j| 2 + π log η.
On the other hand, there exists C > 0 such that,

(3.5) ∀R > 0, ∀K ∈ K, |w K (j, χ R )| ≤ C. Indeed, j = cst -∇ ⊥ H Λ with H Λ (x) = -log |x| + U (x)
where U is a C 2 function, so for any r > 0, we have

E(r) := 1 2 T Λ \B(0,r) |j| 2 ≤ C + π log 1 r .
From this we deduce first, by letting r = η, that

1 2 T Λ \B(0,η) |j| 2 χ R (0) + πχ R (0) log η ≤ C,
and second, that, since |∇χ R | ≤ C, for any 0 < η < r 0 ,

B(0,r 0 )\B(0,η) |j| 2 (χ R -χ R (0)) ≤ C B(0,r 0 )\B(0,η) |j| 2 |x| = -C r 0 η E ′ (t)t dt = -C E(r 0 )r 0 -E(η)η + r 0 η E(r) dr ≤ C.
Adding the two proves (3.5). To conclude, we note that by definition of χ R and (1.5)

#{K ∈ K|χ R ≡ 1 on K} ∼ R→∞ |U R | 2π #{K ∈ K|χ R ≡ 0 and χ R ≡ 1 on K} = o(|U R |) as R → +∞.
Together with (3.4)-(3.5) this yields

W (j, χ R ) = |U R | 2π lim η→0 T Λ \B(0,η) |j| 2 + π log η + o(|U R |).
Combining with (1.8), we deduce (3.1). It remains to show that W U (j) ≥ W U (j Λ ) with equality iff j = j Λ . Since j = j Λ + c, using (3.1), we have

4πW U (j) = lim η→0 T Λ \B(0,η) |j Λ | 2 + |c| 2 + 2c • j Λ + π log η = 4πW U (j Λ ) + |T Λ |c 2 since T Λ c • j Λ = -c • T Λ ∇ ⊥ H Λ = 0. The result follows.
We next express W (Λ) as a series using the Fourier decomposition of H Λ .

Lemma 3.2. For all Λ ∈ L we have

(3.6) W (Λ) = 1 2 lim x→0   p∈Λ * \{0} e 2iπp•x 4π 2 |p| 2 + log |x|   ,
where Λ * denotes the lattice dual to Λ.

Proof. Integrating by parts and using (3.2) we find

(3.7) 1 2 T Λ \B(0,η) |∇H Λ | 2 + π log η = 1 2 ∂(B(0,η)) H Λ ∂H Λ ∂ν - T Λ \B(0,η) H Λ + π log η.
But T Λ H Λ = 0 and H Λ (x) + log |x| is a C 1 function in a neighbourhood of 0, thus passing to the limit η → 0 above we find

(3.8) W (Λ) = 1 2 lim x→0 (H Λ (x) + log |x|).
Using the following normalisation of the Fourier transform

f (y) = R 2 f (x)e -2iπx•y dx, since -∆H Λ = 2π p∈Λ δ p -1 in R 2 and H Λ has zero mean we have ĤΛ (y) = 1 4π 2 |y| 2 p∈Λ * \{0} δ p (y),
where Λ * is the dual lattice of Λ, i.e. the set of vectors q such that p • q ∈ Z for every p ∈ Λ. By Fourier inversion formula, we obtain the expression of H Λ in Fourier series:

(3.9) H Λ (x) = p∈Λ * \{0} e 2iπp•x 4π 2 |p| 2 ,
and the result follows from (3.8).

We may now prove Lemma 1.3. For the more general periodic situation of Lemma 1.3 where Λ is assumed to be merely ( u, v) periodic instead of being itself a lattice, and j is also ( u, v) periodic with curl j = 2π p∈Λ δ p -n, we observe that Proposition 3.1 can be adapted with identical proofs. In this case, denoting by T ( u, v) the quotient R 2 /(Z u + Z v), the configuration Λ may be seen as a finite family of points (a 1 , . . . , a n ) in T ( u, v) , and the statements of Proposition 3.1 remain true, replacing

T Λ by T ( u, v) , replacing T Λ \ B(0, η) by T ( u, v) \ ∪ i B(a i , η) and replacing -∆H Λ = 2πδ 0 -1 by -∆H {a i } = 2π n i=1 δ a i -n, in particular W U (j {a i } ) = 1 2π W (j {a i } , 1 T ( u, v)
). Moreover, writing j {a i } as -∇ ⊥ H {a i } and using the translation invariance of the equation, we have

H {a i } (x) = n i=1 G(x -a i )
where G is the solution to -∆G(x) = 2πδ 0 -1 on the torus T ( u, v) . We may also define R(x) = G(x) + log |x|, which is known to be a continuous function. Next, integrating by parts exactly as in (3.7)-(3.8), we easily find that

W (j {a i } ) = 1 2   i =j G(a i -a j ) + n i=1 R(0)   ,
i.e. we deduce the result of Lemma 1.3. Note that we may also compute W in Fourier series just as above and we find (1.11) i.e.

W (j {a

i } ) = 1 2 i =j p∈(Z u+Z v) * \{0} e 2iπp•(a i -a j ) 4π 2 |p| 2 + n 2 lim x→0   p∈(Z u+Z v) * \{0} e 2iπp•x 4π 2 |p| 2 + log |x|   ,
which is formula (1.11). This may also be rewritten in the form

W (j {a i } ) = 1 2 i =j H Z u+Z v (a i -a j ) + nW (Z u + Z v)
where H Z u+Z v is as in (3.2) or (3.9) and W (Z u+Z v) as in (3.8). This way W is expressed as a sum of the form i =j f (a i -a j ). Note that the series defining H Z u+Z v is an Eisenstein series and can thus be expressed in terms of the Dedekind eta function via the "second Kronecker limit formula" as in the proof of Lemma 3.3 below, see also [BoS] for more computations.

Proof of Theorem 2

First, from (1.12) we may consider only the case m = 1. Thus the lattice Λ is in L iff its fundamental cell has area 2π. We return to the expression (3.6) for W (Λ), and, using standard functions and formulas from number theory, give a closed form for it in terms of the Dedekind eta function. One can also view the series expression of W (Λ) as a regularization, or renormalization, of the divergent series

p∈Λ * \{0} 1 8π 2 |p| 2 .
We also show that modulo a constant independent of Λ, this particular regularization is equal to the regularization which uses the Zeta functions

ζ Λ * (x) = p∈Λ * \{0} 1 8π 2 |p| 2+x ,
for which the minimizers w.r.t. Λ are known. Both results are the object of the following lemmas:

Lemma 3.3. We have

(3.10) W (Λ) = - 1 2 log( √ 2πb|η(τ )| 2 )
with η the Dedekind eta function, where the dual lattice Λ * to Λ has, up to rotation, a fundamental cell given by the vectors

1 √ 2πb (1, 0) and 1 √ 2πb (a, b)
, and τ denotes a + ib.

Proof. We may parametrize Λ * as { 1

√ 2πb (mτ + n), (m, n) ∈ Z 2 }. Then for x = (x 1 , x 2 ) we have (3.11) p∈Λ * \{0} e 2iπp•x 4π 2 |p| 2 = 1 2π (m,n)∈Z 2 \{0} e 2iπ √ 2πb (m(ax 1 +bx 2 )+nx 1 ) b |mτ + n| 2 .
One can recognize that this is an Eisenstein series i.e. of the form

E u,v (τ ) = (m,n)∈Z 2 \{0} e 2iπ(mu+nv) b |mτ + n| 2 with τ = a + ib, u = 1 √ 2πb (ax 1 + bx 2 ) and v = 1 √ 2πb
x 1 . But the "second Kronecker limit formula" (see [START_REF] Lang | Elliptic functions[END_REF]) states that

E u,v (τ ) = -2π log |f (u -vτ, τ )q v 2 /2 | where f (z, τ ) := q 1/12 (p 1/2 -p -1/2 ) n≥1 (1 -q n p)(1 -q n /p) with q = e 2iπτ , p = e 2iπz . Here z = u -vτ = b 2π (x 2 -ix 1 )
. As x → 0, we have p → 1, and then the expression above can be expressed in terms of the Dedekind eta function

η(τ ) := q 1/24 n≥1 (1 -q n ).
Inserting all the above in (3.11) we obtain that as |x| → 0,

p∈Λ * \{0} e 2iπp•x 4π 2 |p| 2 ∼ -log( √ 2πb|η(τ )| 2 |x|)
and combining with (3.6) we find the result (3.10).

Lemma 3.4. There exists C ∈ R such that for any Λ ∈ L we have

W (Λ) = C + lim x→0 ζ Λ * (x) - R 2 π 1 + 4π 2 |y| 2+x dy .
Two proofs can be given for this : one, which we give in the following subsection below, uses standard analysis methods. The other uses the closed form (3.10) and the fact that the Dedekind eta function is in turn related to the Epstein zeta function

Z(τ, s) = (m,n)∈Z 2 \0 b s |mτ + n| 2s = 1 (2π) s p∈ 1 √ 2πb (Z+τ Z)\0 1 |p| 2s
via the "first Kronecker limit formula"

Z(τ, s) = π s -1 + 2π(γ 0 -log 2 -log( √ b|η(τ )| 2 )) + O(s -1) as s → 1,
where γ 0 is this time the Euler constant. Putting these pieces together correctly yields a proof of Lemma 3.4. Then, in order to minimize W (Λ) over L, i.e. over lattices under the constraint

|T Λ | = 2π, one is reduced to the question of minimizing ζ Λ * (x). It is proven in [Mont] that 2 1+ x 2 8π 2 Γ(1 + x/2) (2π) 1+x/2 ζ Λ * (x) = 2 x - 1 1 + x 2 + ∞ 1 (θ Λ * (a) -1)(a -x/2 + a 1+x/2 ) da a where (3.12) θ Λ (a) = p∈Λ e -πa|p| 2
is the Jacobi Theta function and Γ is the Gamma function. Moreover, from [START_REF] Cassels | On a problem of Rankin about the Epstein zeta-function[END_REF][START_REF] Rankin | A minimum problem for the Epstein zeta function[END_REF][START_REF] Ennola | A remark about the Epstein zeta function[END_REF][START_REF] Diananda | Notes on two lemmas concerning the Epstein zeta-function[END_REF][START_REF] Montgomery | Minimal Theta functions[END_REF] and modulo rotation, the minimum of θ over L * is uniquely achieved by (Λ 0 ) * , where

Λ 0 = α Z(1, 0) + Z 1/2, √ 3/2 and the factor α is chosen such that |T Λ 0 | = 2π. But, from the above formula, (3.13) 2 1+ x 2 8π 2 Γ(1 + x/2) (2π) 1+x/2 (ζ Λ * (x) -ζ Λ * 0 (x)) = ∞ 1 (θ Λ * (a) -θ Λ * 0 (a))(a -x/2 + a 1+x/2 ) da a .
In view of (3.12) the integrand in (3.13) is dominated by an integrable function independently of x. Hence, letting x tend to 0, Lebesgue's dominated convergence theorem yields 8πΓ(1) lim

x→0 ζ Λ * (x) -ζ Λ * 0 (x) = ∞ 1 (θ Λ * (a) -θ Λ * 0 (a))(1 + a) da a .
Hence, using Lemma 3.4, we have

8πΓ(1)(W (Λ) -W (Λ 0 )) = ∞ 1 (θ Λ * (a) -θ Λ * 0 (a))(1 + a) da a .
We deduce that W (Λ) ≥ W (Λ 0 ) for any Λ, with equality if and only if θ Λ * (a) = θ Λ * 0 (a) for almost every a, i.e. if, modulo rotations, Λ = Λ 0 . This proves Theorem 2.

Analytic proof of Lemma 3.4

Denoting by G 0 the unique solution of -∆G 0 + G 0 = 2πδ 0 in R 2 , we may write

H Λ (x) + log |x| = U Λ + (G 0 (x) + log |x|) ,
where U Λ = H Λ -G 0 and G 0 (x) + log |x| are C 1 near the origin. Taking limits, we obtain from (3.8)

W (Λ) = γ 0 + 1 2 U Λ (0),
where

γ 0 = 1 2 lim x→0 (G 0 (x) + log |x|). Denote by ϕ(x) = (2π) -1 e -|x| 2 /2
the Gaussian distribution in R 2 and for any n ∈ N let ϕ n (x) = n 2 ϕ(nx), so that {ϕ n } n is an approximate identity. We have φn (y) = e -|y| 2 /2n 2 . Then, since U Λ is continuous at 0 and bounded in R 2 , we have

U Λ (0) = lim n→+∞ w(n, Λ), where w(n, Λ) = R 2 ϕ n (x)U Λ (x)dx = R 2 φn (y) ÛΛ (y) dy.
Also, it is standard that Ĝ0 = 2π/(1 + 4π 2 |y| 2 ). Then, since ÛΛ = ĤΛ -Ĝ0 , we get

w(n, Λ) = p∈Λ * \{0} e -|p| 2 /2n 2 4π 2 |p| 2 -2π R 2 e -|y| 2 /2n 2 1 + 4π 2 |y| 2 dy. Note that |T Λ * | = 1/|T Λ | -1 = 1/2π,
which accounts for the factor 2π in front of the integral.

The second step is to show that (3.14)

lim n→+∞ w(n, Λ) = lim x→0 v(x, Λ), where v(x, Λ) = p∈Λ * \{0} 1 4π 2 |p| 2+x - R 2 2π 1 + 4π 2 |y| 2+x dy.
First we truncate w(n, Λ) and v(x, Λ), letting for each N ∈ N

w N (n, Λ) = p∈Λ * \{0} |p|<N e -|p| 2 /2n 2 4π 2 |p| 2 -2π B(0,N ) e -|y| 2 /2n 2 1 + 4π 2 |y| 2 dy, wN (n, Λ) = w(n, Λ) -w N (n, Λ),
and defining similarly v N (x, Λ), ṽN (x, Λ). It is clear that for any N

(3.15) lim n→+∞ w N (n, Λ) = lim x→0 v N (x, Λ) = 1 4π 2 p∈Λ * \{0} |p|<N 1 |p| 2 -2π B(0,N ) 1 1 + 4π 2 |y| 2 dy.
- Kp e -|p| 2 /2n 2 4π 2 |p| 2 - e -|y| 2 /2n 2 1 + 4π 2 |y| 2 dy + δ(N, n, Λ),
where

δ(N, n, Λ) = 2π   p / ∈B(0,N ) Kp∩B(0,N ) e -|y| 2 /2n 2 1 + 4π 2 |y| 2 dy - p∈B(0,N ) Kp\B(0,N ) e -|y| 2 /2n 2 1 + 4π 2 |y| 2 dy   .
We can bound |δ| by the integral of (1 + 4π 2 |y| 2 ) -1 over those cells which intersect both B(0, N ) and its complement, the union of which is included in

B(0, N + C Λ ) \ B(0, N -C Λ ).
Thus we have for every Λ (3.18) lim

N →+∞ lim n→+∞ δ(N, n, Λ) = 0.
Then we consider the sum in (3.17

|M p | = 0.
If |p| ≤ n 2 , we use the fact that the gradient of y → e -|y| 2 /2n 2 is bounded by C/n on R 2 , and that e -|p| 2 /2n 2 ≤ 1 to write

|M p | ≤ - Kp 1 4π 2 |p| 2 - 1 1 + 4π 2 |y| 2 dy + C n - Kp dy 1 + 4π 2 |y| 2 .
The sum w.r.t p ∈ Λ * of the first term on the right-hand side is convergent, therefore the sum w.r.t p ∈ Λ * , n 2 ≥ |p| ≥ N tends to 0 as N → +∞, uniformly in n. For the second term, the corresponding sum may be compared with the integral

C n 2π N ≤|y|≤n 2 dy 1 + 4π 2 |y| 2 , which is O(log n/n) as n → +∞. Therefore lim N →+∞ lim n→+∞ p∈Λ * \{0} n 2 ≥|p|≥N |M p | = 0.
Together with (3.19) and (3.18), this proves that lim N →+∞ lim n→+∞ wN (n, Λ) = 0. The proof that lim N →+∞ lim x→0 ṽN (x, Λ) = 0 is similar, we omit it. Then (3.16) holds, which proves (3.14) and the lemma.

4 The renormalized energy in the general case: proof of Theorem 1

In this section we prove Theorem 1. We recall that we can reduce by scaling to studying the case of A m = A 1 i.e. where (see Definition 1.1)

(4.1) curl j = ν -1 div j = 0, where ν = 2π p∈Λ δ p is such that { ν(B R ) |B R | } R is bounded.
For convenience, once the class A 1 has been chosen, if p < 2 we may extend the definition of W U to all j ∈ L p loc (R 2 , R 2 ) by letting

W U (j) = +∞ if j / ∈ A 1 .
We now assume W has been likewise extended for a certain 1 < p < 2. In what follows we will show that the minimizers and the minimum of W U do not depend on U . However W U itself does. When a statement is independent of U , we will sometimes write W instead of W U . One of the difficulties about W is that it is not lower semi-continuous. Here is a hint as to why: consider a set of points Λ n which is equal to the square lattice (of density 1) in the ball B n and to the triangular lattice (of density 1) outside B n . Let j n be corresponding vector fields. Then, since W is insensitive to compact perturbations, we have for every n, in informal notation, W (j n ) = W (triangular). On the other hand, as n → ∞, j n → j square by construction. So if W was lower semi-continuous, we should have W (square) ≤ W (triangular), which is false by Theorem 2.

However, we will show that W is lower semi-continuous "up to translations". We split the results into several propositions:

Proposition 4.1. Let U refer to any family {U R } satisfying (1.4), (1.5). Let 1 < p < 2. We have: -The value of W U (j) is independent of the choice of cutoff functions {χ U R } R satisfying (1.3). -W U : L p loc (R 2 , R 2 ) → R ∪ {+∞} is a Borel function.
-The sublevel sets {j, W U (j) ≤ α} are "compact in L p loc (R 2 , R 2 ) up to translation", more precisely: for every j n such that lim sup n→∞ W (j n ) < +∞, after extraction of a subsequence, there exists a sequence

λ n ∈ R 2 and j ∈ L p loc (R 2 , R 2 ) such that j n (λ n + •) → j in L p loc (R 2 , R 2 ) and (4.2) W U (j) ≤ lim inf n→∞ W U (j n ). In particular inf L p loc (R 2 ,R 2 ) W U = inf A 1 W U is
achieved and is finite. -The minimizers and the value of the minimum of W U are independent of U .

We will also prove a result of uniform approximation by construction, which implies in particular that the minimum of W may be approximated by suitable periodic configurations. This construction will be crucial in particular in constructing test functions. In what follows K R denotes the square [-R, R] 2 , and W K denotes the renormalized energy relative to {K R } R . In the following results, squares could easily be replaced by arbitrary parallelograms.

Proposition 4.2. Let G ⊂ A 1 be such that there exist C > 0 such that for any j ∈ G, we have

(4.3) ∀R > 1, ν(K R ) |K R | < C
for the associated ν's and such that, uniformly with respect to j ∈ G,

(4.4) lim R→+∞ W (j, χ K R ) |K R | = W K (j) ≤ C.
Then for any ε > 0, there exists R 0 > 0 such that if R > R 0 and |K R | ∈ 2πN, for any j ∈ G there exists j R such that

(4.5) curl j R = 2π p∈Λ R δ p -1 in K R , j R • τ = 0 on ∂K R ,
where Λ R is a discrete subset of the interior of K R , and

-if x ∈ K R-2R 3/4 then j R (x) = j(x), - W (j R , 1 K R ) |K R | ≤ W K (j) + ε.
Remark 4.3. An inspection of the proof allows to see that the construction can alternatively be made in any rectangle which is a small perturbation of the square K R , i.e whose sidelengths are in [2R, 2R(1 + η)], where η depends on ε.

Note that this result is close to establishing that A 1 is "uniformly W -approximable" in the sense of [AM], Definition 4.14.

The following corollaries are proved at the end of this section.

Corollary 4.4. Given any j such that W K (j) < ∞, there exists a sequence {j R } R 2 ∈2πN in A 1 such that each j R is K R periodic (i.e. j R (x + 2Rke 1 + 2Rle 2 ) = j R (x) for k, l ∈ Z where (e 1 , e 2 ) is the canonical basis of R 2 ) and

(4.6) lim sup R→∞ W K (j R ) = lim sup R→∞ W (j R , 1 K R ) |K R | ≤ W K (j).
In particular there exists a minimizing sequence for min A 1 W consisting of periodic vectorfields.

Corollary 4.5. Let p < 2 and let P be a probability measure on L p loc (R 2 , R 2 ) which is invariant under the action of translations and concentrated in A 1 . Then there exists a sequence R → +∞ and a sequence {j R } R of vector fields defined over K R such that -There exists a finite subset Λ R of the interior of K R such that

(4.7) curl j R = 2π p∈Λ R δ p -1 in K R j R • τ = 0 on ∂K R .
-Letting P R be the probability measure on L p loc (R 2 , R 2 ) which is defined as the image of the normalized Lebesgue measure on K R by x → j R (x + •) where j R is extended periodically to the whole R 2 , we have P R → P weakly as R → ∞.

-lim sup R→∞ W (j R , 1 K R ) |K R | ≤ W K (j) dP (j).

A mass displacement result

In this subsection, we show that even though the integrand in the definition of W is not bounded below, we can somehow reduce to that case by a "mass displacement" method, which is an adaptation of that of [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF]. The situation is much simpler here due to the fact that all degrees are +1. It follows the same steps however, consisting in ball construction combined with mass displacement. We begin with a ball construction argument à la Sandier and Jerrard [Sa, Je]. For the sake of generality, we prove the result for an average density m which may depend on x.

Proposition 4.6. Assume curl j = ν -m in the sense of distributions in some open set U , where m ∈ L ∞ (U ) and ν = 2π p∈Λ δ p for some finite subset Λ of U , and that j ∈ L 2 loc (U \Λ). Let n = #Λ and η 0 > 0 be the minimal distance between points of Λ. Then there exists for any r ∈ (0, 1] a family of disjoint closed balls B r of total radius r covering Λ such that:

-If Λ = {p} then B r = {B(p, r)}. -If r/n ≤ η 0 , then B r = {B(p, r n )} p∈Λ .
-The set ∪ B∈Br B is increasing as a function of r. Moreover, for any η ≤ r/n, and every

B ∈ B r such that B ⊂ U we have, letting d B = #(Λ ∩ B), 1 2 B\∪ p∈Λ B(p,η) |j| 2 ≥ πd B log r nη -m ∞ r 2 2 .
-If B ∈ B r and χ is a positive function with support in U , then

B\∪ p∈Λ B(p,η) χ|j| 2 -2π log r nη -m ∞ r 2 2 p∈B∩Λ χ(p) ≥ -2rν(B) ∇χ ∞ .
Proof of the first three items. We let M = m ∞ . The first items are obtained by a standard ball construction argument à la Jerrard/Sandier. Since curl j = ν -m we have for any circle C = ∂B of radius r(B) not intersecting Λ, and letting

d B = #(Λ ∩ B), C j • τ ≥ ν(B) -M πr B 2 = 2πd B -M πr B 2 .
Using the Cauchy-Schwarz inequality and the fact that d B is a nonnegative integer, we deduce that

(4.8) C |j| 2 ≥ 2π r B (d B ) 2 -2πM d B r B ≥ 2πd B 1 r B -M r B . Define F(x, r) = B(x,r) |j| 2 . The above yields ∂F ∂r ≥ 2πd B 1 r B -M r B .
In order to define B r , we first fix a reference family of balls produced via a ball-growth. Set η 1 = min{η 0 /3, 1/(n + 1)} and let B 0 = { B(p, η 1 )} p∈Λ . According to the definition of η 0 , we have that B 0 is a finite, disjoint collection of closed balls of total radius nη 1 < 1. We then apply Theorem 4.2 of [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] to B 0 to produce a family of collections {B(s)} s∈[0,log 1 nη 1 ] , such that the total radius of the balls in B(s) is r = nη 1 e s . Then, given nη 1 < r < 1 we may choose s = log r nη 1 and write B r for B(s). We then extend this reference family "backward" to radii smaller than nη 1 by letting B r = { B(p, r/n)} p∈Λ for any 0 < r ≤ nη 1 .

Since the balls in these collections never become tangent when r < nη 1 , for any η > 0 we may trivially view {B(s)} s , where B(s) = B r , with r = nηe s as having been generated by a ball-growth from B(0) = { B(p, η)} p∈Λ , i.e. satisfying all the results of Theorem 4.2 in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF]. Then each B r has total radius r and covers Λ and from Proposition 4.1 in [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] for every

B ∈ B r = B(s), with r = nηe s , B\∪ p∈Λ B(p,η) |j| 2 ≥ s 0 B ′ ∈B(t) B ′ ⊂B 2πd B ′ (1 -M r B ′ 2 ) dt.
Since the sum of the d B ′ 's is #(B ∩ Λ) and since the sum of the r B ′ 's is less than nηe t , we find

B\∪ p∈Λ B(p,η) |j| 2 ≥ 2πd B s 0 (1 -M (nηe t ) 2 ) dt = 2πd B log r nη -M r 2 -n 2 η 2 2 .
Since this is true for any 0 < η < r/n, the first three items are satisfied.

Proof of the last item.

Let B η = B \ ∪ p∈Λ B(p, η).
Then by the "layer-cake" theorem (4.9)

Bη χ|j| 2 = +∞ 0 Bη∩{χ>t} |j| 2 dt.
Now if a ∈ Λ ∩ B, then for any s ∈ (0, r] there exists a closed ball B a,s ∈ B s containing a. For t > 0 we call s(a, t) = sup{s ∈ (0, 1], B a,s ⊂ {χ > t}} if this set is nonempty, and let s(a, t) = 0 otherwise, i.e. if χ(a) ≤ t. Then we let B t a = B a,s(a,t) . Note that a is not necessarily the center of B t a , note also that s(a, t) bounds from above the radius of B t a , but is not necessarily equal to it. As noted above s(a, t) = 0 iff χ(a) ≤ t while if s(a, t) ∈ (0, r) then B t a ⊂ {χ > t}, otherwise there would exist s ′ > s(a, t) such that B a,s ′ ⊂ {χ > t}, contradicting the definition of s(a, t). Thus, choosing y in B t a \ {χ > t}, we have (4.10)

χ(a) -t ≤ χ(a) -χ(y) ≤ 2s(a, t) ∇χ ∞ .
Also, for any t ≥ 0 the collection {B t a } a , where a ∈ Λ and the a's for which s(a, t) = 0 have been excluded, is disjoint. Indeed if a, b ∈ Λ and s(a, t) ≥ s(b, t) then, since B s(a,t) is disjoint, the balls B a,s(a,t) and B b,s(a,t) are either equal or disjoint. If they are disjoint we note that s(a, t) ≥ s(b, t) implies that B b,s(b,t) ⊂ B b,s(a,t) and therefore B t b = B b,s(b,t) and B t a = B a,s(a,t) are disjoint. If they are equal, then B b,s(a,t) ⊂ E t ∩ B and therefore s(b, t) ≥ s(a, t), which implies s(b, t) = s(a, t) and then B t b = B t a . Now assume that B ′ ∈ {B t a } a and let s be the common value of s(a, t) for a's in B ′ ∩ Λ and n = #Λ. Then the previous item of the proposition yields for any η < min(η 0 , r/n) (but the inequality is trivially true if η > r/n),

B ′ \∪ p∈Λ B(p,η) |j| 2 ≥ ν(B ′ ) log s nη -M s 2 2 + .
We may rewrite the above as

B ′ \∪ p∈Λ B(p,η) |j| 2 ≥ 2π a∈B ′ ∩Λ log s(a, t) nη -M s(a, t) 2 2 +
and summing over B ′ ∈ {B t a } a we deduce, noting that the a's for which s(a, t) = 0 do not contribute to the sum,

Bη∩{χ>t} |j| 2 ≥ 2π a∈B∩Λ log s(a, t) nη -M s(a, t) 2 2 + .
Integrating the above in view of (4.9) yields, using (4.10) and the fact that s(a, t) ≤ r,

Bη χ|j| 2 ≥ 2π a∈B∩Λ χ(a) 0 log s(a, t) nη -M s(a, t) 2 2 + dt ≥ 2π a∈B∩Λ χ(a) 0 log r nη + log χ(a) -t 2r ∇χ ∞ ∧ 1 -M r 2 2 dt.
The right-hand side is greater than

2π a∈B∩Λ χ(a) log r nη -M r 2 2 -2r ∇χ ∞ ,
which proves the result.

We deduce a control of j by W , which we point out is substantially improved in [ST] into a control in a critical Lorentz space. where ν = 2π p∈Λ δ p for some finite subset Λ of U . Then, there exists C > 0 universal and for any p ∈ [1, 2), C p > 0 depending only on p, such that

U χ p/2 |j| p ≤ C(|U | + C p ) 1-p/2 (W (j, χ) + n(log n + m ∞ ) χ ∞ + n ∇χ ∞ ) p/2 .
where n = ν( U )/2π = #Λ.

Proof. We use an argument of M. Struwe [St]. Let M = m ∞ . We construct as above balls B r in U and define for k ≥ 1 the set

U k = B 2 -k \ B 2 -(k+1) and U 0 = U \ B 1/2 .
Then since χ is supported in U , we have by Hölder's inequality, (4.11)

U χ p/2 |j| p = b U χ p/2 |j| p = ∞ k=0 U k χ p/2 |j| p ≤ ∞ k=0 |U k | 1-p/2 U k χ|j| 2 p/2
.

For any given k ∈ N and if η is small enough we have

U k ⊂ U η \ (B 2 -(k+1) ) η ,
χ|j| 2 ≥ 2π log 2 -(k+1) nη -CM p∈B∩Λ χ(p) -Cν(B) ∇χ ∞ .
Summing over balls in B 2 -(k+1) and subtracting from Uη χ|j| 2 we find

U k χ|j| 2 ≤ Uη χ|j| 2 -2π log 2 -(k+1) nη -CM p∈U ∩Λ χ(p) + Cν(U ) ∇χ ∞ .
Taking the limit η → 0 and plugging into (4.11) we find, recalling that n = #Λ,

U χ p/2 |j| p ≤ ∞ k=0 |U k | 1-p/2 (W (j, χ) + Cn (M + k + log n) χ ∞ + Cn ∇χ ∞ ) p/2 .
We have

|U 0 | ≤ |U | while for k ≥ 1 we have |U k | ≤ C2 -k . It follows easily that for some C p , C > 0, U χ p/2 |j| p ≤ C(|U | + C p ) 1-p/2 (2W (j, χ) + n(log n + M ) χ ∞ + n ∇χ ∞ ) p/2 . Lemma 4.8. Assume that m ∈ L ∞ (B R ) and {(j n , ν n )} n is a sequence in L p loc (R 2 , R 2 ) × M such that ν n restricted to B R
is of the form 2π i δ a i for every n, with div j n = 0 and curl j n = ν n -m, and which converges in the distributional sense to (j, ν) on B R .

Then div j = 0 and curl j = ν -m on B R , where ν is a locally finite sum of the form 2π p∈Λ d p δ p , where Λ is a discrete subset of B R and

d p ∈ N * . Moreover, if χ ∈ C ∞ c (B R
) is positive and sup n W (j n , χ) < +∞, then d p = 1 for every p such that χ(p) = 0. In addition, for any smooth function ξ compactly supported in {χ > 0} we have

lim inf n→+∞ W (j n , ξ) = W (j, ξ).
Proof. This is essentially a consequence of the type of analysis of [BBH].

First, since ν n is positive for each n, the distributional convergence of {ν n } n is in fact a weak convergence of the measures, and {ν n } n is locally bounded on B R , and we deduce that ν is of the form 2π p∈Λ d p δ p with d p ∈ N * . That div j = 0 and curl j = ν -m on B R follows by passing to the limit in the corresponding equations satisfied by j n , ν n . Now assume sup n W (j n , χ) < +∞, let U be an open set compactly included in B R and containing the support of χ, and choose η > 0. Let

U η = U \ ∪ p∈Λ B(p, η). If n is large enough depending on η we have Λ n ∩ U η = ∅ hence div (j -j n ) = curl (j -j n ) = 0 in U η . Elliptic regularity then implies that j n → j in C k (U 2η ) for any k ∈ N and thus we have convergence in C k loc (B R \ Λ).
In particular we have for any ρ > 0

lim n→∞ 1 2 Uρ χ|j n | 2 = 1 2 Uρ χ|j| 2 .
We choose ρ small enough so that the balls B(p, ρ) for p ∈ Λ ∩ U are disjoint. Then for each p ∈ Λ ∩ U and n large enough, there are exactly d p ≥ 1 points p 1 n , . . . , p dp n in Λ n ∩ B(p, ρ), and these points converge to p. We apply Proposition 4.6 to (j n , ν n ) in B(p, ρ), with r = ρ 2 . We deduce the existence of a family of balls B n of total radius ρ 2 containing the points p k n and such that for η > 0 small enough

Bn\∪ k B(p k n ,η) χ|j n | 2 -2π log ρ 2 d p η -CM dp k=1 χ(p k n ) ≥ -Cd p ρ 2 ∇χ ∞ .
Moreover, since the p k n 's converge to p as n → +∞, if n is chosen large enough then they are at distance less than ρ 2 from p hence B n ⊂ B(p, 2ρ 2 ). Using (4.8) to bound from below

∂B(p,t) |j n | 2 for 2ρ 2 < t < ρ, we find 1 2 B(p,ρ)\Bn χ|j n | 2 ≥ 1 2 min B(p,ρ) χ B(p,ρ)\Bn |j n | 2 ≥ π min B(p,ρ) χ d p 2 log ρ 2ρ 2 -Cd p ρ.
Also, since on B(p, ρ) we have min B(p,ρ) χ ≥ χ -2ρ ∇χ ∞ , we may write min

B(p,ρ) χ d p 2 ≥ k χ(p k n ) + χ(p) d p 2 -d p -Cρd p 2 ∇χ ∞ .
Putting together the above lower bounds, replacing χ(p k n ) by χ(p) -Cρ ∇χ ∞ , and summing with respect to k and p, we deduce

1 2 U \∪ p∈Λn B(p,η) χ|j n | 2 ≥ 1 2 Uρ χ|j n | 2 + π log ρ 2 η p∈Λn χ(p) + π log ρ 2ρ 2 p∈Λn χ(p) + π log ρ 2ρ 2 p∈Λ (d p 2 -d p )χ(p) -C p∈Λ d p (log d p + 1)C(χ) -Cρ log 1 2ρ p∈Λ d p 2 C(χ),
where C(χ) denotes a constant depending only on the function χ and possibly on M . Adding π log η p χ(p) on both sides and passing to the limit η → 0 we find

W (j n , χ) ≥ 1 2 Uρ χ|j n | 2 + π log ρ 2 p∈Λn χ(p) + π log 1 2ρ p∈Λ (d p 2 -d p )χ(p) -R,
where the error term R is bounded independently of ρ ∈ (0, 1], n. This inequality is true for n large enough, but the left-hand side was assumed to be bounded above independently of n, hence the right-hand side is bounded above independently of ρ, n, which can only be true if (d p 2 -d p -1)χ(p) ≤ 0 for every p ∈ Λ, i.e. (since d p is a nonzero integer) if d p = 1 for every p ∈ Λ such that χ(p) = 0.

We now prove that lim inf n→+∞ W (j n , ξ) = W (j, ξ) if Supp(ξ) ⊂ {χ > 0}. For this we note that since j n → j in C k loc (B R \ Λ), then for any ρ > 0 we have

I(j n , ρ) := 1 2 Uρ ξ|j n | 2 + π p∈Λn ξ(p) log ρ n→+∞ -----→ I(j, ρ).
Then, the convergence of W (j n , ξ) to W (j, ξ) will follow if we may reverse the limits n → ∞ and ρ → 0, which is the case provided for instance that I(j n , ρ) tends to W (j n , ξ) as ρ → 0 uniformly with respect to n.

We prove this fact. Denote by {p 1 , . . . , p k } the set Λ ∩ Supp(ξ). We know already that the points are distinct hence there exist a neighborhood V of Supp(ξ) and k sequences

{p n i } n , 1 ≤ i ≤ k such that Λ n ∩ V = {p n i } i and such that p n i → p i as n → +∞. There exists N 0 such that if n > N 0 then |p n i -p i | < ρ 0 when i = j, where ρ 0 = 1 4 min i =j |p i -p j |. Now choose ρ 1 < ρ 2 < ρ 0 and n > N 0 . We have j n = -∇ ⊥ H n , where -∆H n = 2πδ p n i -m in B(p n i , 2ρ 0 ) with m ∞ < M . Then H n = log | • -p n i | + H i,n
, where H i,n is bounded by a constant independent of n in C 1 (B(p n i , ρ)). It follows straightforwardly by writing

|j n | 2 = |∇ log +∇H i,n | 2 , expanding and estimating each term, that 1 2 B(p n i ,ρ 2 )\B(p n i ,ρ 1 ) ξ|j n | 2 -πξ(p n i ) log ρ 2 ρ 1 ≤ Cρ 2 ∇ξ ∞ + C √ ρ 2 ξ ∞ ,
where C is independent of n ≤ N 0 . The left-hand side is nothing but |I(j n , ρ 2 ) -I(j n , ρ 1 )| thus we have proved the uniform convergence w.r.t. n of I(j n , ρ) as ρ → 0, and then it follows that W (j n , ξ) → W (j, ξ).

The energy density defining W , |j| 2 +log η p δ p has no sign, which makes it impossible to apply directly Theorem 3 to it. We now show that it can be modified into a density bounded below by a constant, using again the mass displacement method introduced in [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF] (but in a simpler setting), that is by absorbing the negative part into the positive part while making a controlled error.

The next proposition summarizes the properties of the modified density g.

Proposition 4.9. Assume U ⊂ R 2 is open and (j, ν) are such that ν = 2π p∈Λ δ p for some finite subset Λ of U and curl j = ν -m, div j = 0 in U , where m ∈ L ∞ ( U ). Then there exists a measure g supported on U and such that -g ≥ -C( m 2 ∞ + 1) on U , where C is a universal constant.

-For any function χ compactly supported in U we have

(4.12) W (j, χ) -χ dg ≤ Cn(log n + m ∞ ) ∇χ ∞ , where n = #{p ∈ Λ | B(p, C) ∩ Supp(∇χ) = ∅} and C is universal. -For any E ⊂ U , (4.13) #(Λ ∩ E) ≤ C 1 + m 2 ∞ | E| + g( E) ,
where C is universal.

Proof. The proof follows the method of [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF]. Throughout M = m ∞ . We cover R 2 by the balls of radius 1/4 whose centers are in Z 4 × Z 4 . We call this cover {U α } α and {x α } α the centers. In each U α ∩ U and for any r ∈ (0, 1/4) we construct disjoint balls B α r using Proposition 4.6. Then choosing a small enough ρ ∈ (0, 1/4) to be specified below, we may extract from ∪ α B α ρ a disjoint family which still covers Λ as follows: Denoting by C a connected component of ∪ α B α ρ , we claim that there exists α 0 such that C ⊂ U α 0 . Indeed if x ∈ C and letting ℓ be a Lebesgue number of the covering {U α } α (in our case ℓ ≤ 1/4), there exists α 0 such that B(x, ℓ) ⊂ U α 0 . If C intersected the complement of U α 0 , there would exist a chain of balls connecting x to (U α 0 ) c , each of which would intersect U α 0 . Each of the balls in the chain would belong to some B β ρ with β such that dist (U β , U α 0 ) ≤ 2ρ < 1/2. Thus, calling k the maximum number of β's such that dist (U β , U α 0 ) < 1/2, the length of the chain is at most 2kρ and thus ℓ ≤ 2kρ. If we choose ρ < ℓ/2k, this is impossible and the claim is proven. Let us then choose ρ = ℓ/4k. By the above, each C is included in some U α .

Then, to obtain a disjoint cover of Λ from ∪ α B α ρ , we let C run over all the connected components of ∪ α B α ρ and for a given C such that C ⊂ U α 0 , we remove from C the balls which do not belong to B α 0 ρ . We will still denote B α ρ the family with deleted balls, and let

B ρ = ∪ α B α ρ .
Then B ρ covers Λ and is disjoint. We then proceed to the mass displacement. Note that by construction every ball in

B α ρ is included in U α .
From the last item of Proposition 4.6 applied to a ball B ∈ B α ρ , if η is small enough then, letting B η = B \ ∪ p∈Λ B(p, η), for any function χ vanishing outside B ∩ U we have

Bη χ|j| 2 -2π log ρ n α η -CM p∈B∩Λ χ(p) ≥ -Cν(B) ∇χ L ∞ (B) ,
where n α = ν(U α ) and M = m ∞ . Then applying Lemma 3.1 of [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF] to

f B,η = 1 2   |j| 2 1 Bη -log ρ n α η -CM 2π p∈B∩Λ δ p  
we deduce the existence of a positive measure g B,η such that f B,η -g B,η ≤ Cν(B), where the norm is that of the dual of the space Lip of Lipschitz functions in B which vanish outside B ∩ U . Now we let η → 0. Since g B,η ≥ 0, it subsequentially converges to a positive measure g B and for any χ ∈ Lip,

(4.14) χ dg B -W B (j, χ) ≤ Cν(B) ∇χ L ∞ (B) , where W B (j, χ) = lim η→0 χ df B,η .
Next we note that, letting W ′ (j, χ) = W (j, χ) -B∈Bρ W B (j, χ), we have

W ′ (j, χ) = χ df ′ , where f ′ = 1 2 |j| 2 1 U \Bρ + π p∈Λ log ρ n αp -CM δ p ,
where, denoting Λ α the set of p ∈ Λ belonging to a ball of B α ρ , we define α p as the index α such that p belongs to Λ α .

We define a set C α as follows: recall that ρ was assumed equal to ℓ/4k, where ℓ ≤ 1/4 and k bounds the number of β's such that dist (U β , U α ) < 1/2 for any given α. Therefore the total radius of the balls in B ρ which are at distance less than 1 2 from U α is at most kρ < 1/16. In particular, letting T α denote the set of t ∈ ( 1 4 , 1 2 ) such that the circle of center x α (where we recall x α is the center of U α ) and radius t does not intersect B α ρ , we have

|T α | ≥ 3/16. We let C α = {x | |x -x α | ∈ T α } and D α = U α ∪ C α . If U α ∩ U = ∅ then d(x α , U ) ≤ 1/4 hence B(x α , 1
2 ) ⊂ U . In particular, we have C α ⊂ D α ⊂ U . Then there exists universal constants c > 0 and C such that (4.15)

Cα |j| 2 ≥ cn α 2 -CM 2 .
To see this, apply the lower bound (4.8) on the circle S t = {|x -x α | = t}, i.e. with r B = t and d B = #(Λ∩B(x α , t)). Using d B ≥ n α and t ∈ ( 1 8 , 1 2 ) we deduce that St |j| 2 ≥ π(n α ) 2 -πM n α and integrating with respect to t ∈ T α yields (4.15).

The overlap number of the sets {C α } α , defined as the maximum number of sets to which a given x belongs is bounded by the overltap number of {B(x α , 3/4)} α , call it k ′ . Then, letting

f α = 1 2k ′ |j| 2 1 Cα + π p∈Λα log ρ n α -CM δ p ,
we have

(4.16) f ′ - α f α ≥ |j| 2 1 U \Bρ - 1 2k ′ |j| 2 1 Cα ≥ 1 2 |j| 2 1 U \Bρ ≥ 0
and, from (4.15), (4.17)

f α (D α ) = 1 2k ′ Cα |j| 2 + πn α log ρ n α -CM ≥ cn α 2 -C(M 2 + 1),
where the constants may have changed. We then apply [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF] Lemma 3.2 over

D α to f α + C(M 2 +1)
|Dα| , where C is the constant in the right-hand side of (4.17). We deduce for any α such that U α ∩ U = ∅ the existence of a measure g α supported in D α such that g α ≥ -C(M 2 +1) |Dα| and such that, for every Lipschitz function χ

(4.18) χ d(f α -g α ) ≤ C ∇χ L ∞ (Dα) (f α ) -(D α ) ≤ Cn α (log n α + C(M + 1)) ∇χ L ∞ (Dα) .
In particular, taking χ = 1,

(4.19) g α (D α ) = f α (D α ) ≥ cn α 2 -C(M 2 + 1). Now we let g = B∈Bρ g B + α s.t. Uα∩U =∅ g α + f ′ - α f α .
The term g B is positive for every B, f ′α f α is bounded below by (4.16), and α g α is bounded below by -k ′ C(M 2 + 1) since g α ≥ -C(M 2 + 1). Thus g is bounded below by

1 2 |j| 2 1 U \Bρ -C(M 2 + 1) ≥ -C(M 2 + 1
), and we have proved the first item. In addition, if χ has support in U then

α χ df α = α s.t. Uα∩U =∅ χ df α hence χ dg = χ d    B g B + α s.t. Uα∩U =∅ (g α -f α ) + f ′    = B χ dg B + α s.t. Uα∩U =∅ χ d(g α -f α ) + W ′ (j, χ)
hence in view of the definition of W ′ , (4.14) and (4.18),

χ dg -W (j, χ) = B∈Bρ χ dg B -W B (j, χ) + α s.t. Uα∩U =∅ χ d(g α -f α ) ≤ C   B∈Bρ ν(B) ∇χ L ∞ (B) + α n α (log n α + C(M + 1)) ∇χ L ∞ (Dα)   . (4.20)
Then if we denote by A the set of α's such that ∇χ L ∞ (Dα) = 0, and k is the overlap number of the U α 's, 2π

α∈A

n α ≤ kν (∪ α∈A U α ) ,
and x ∈ U α with α ∈ A implies that ∇χ(y) = 0 for some y ∈ D α hence B(x, 5/4) ∩ Supp∇χ = ∅. It follows that α∈A n α ≤ kn, where n is defined after (4.12). Similarly, the sum of ν(B) for B's in B ρ such that ∇χ L ∞ (B) = 0 is less then 2πn. Thus (4.20) may be rewritten as (4.12). Finally, summing (4.19) for α's such that U α ∩ E = ∅ and recalling that gα g α ≥ 0, we easily deduce (4.13).

Remark 4.10. We have in fact proved the following stronger property on g: There exists ρ > 0 and a family B ρ of disjoint closed balls covering Λ, such that the sum of the radii of the balls in B ρ intersected with any ball of radius 1 is bounded by Cρ < 1 2 , and such that on

Û g ≥ -C( m 2 ∞ + 1) + 1 2 |j| 2 1 U \Bρ .
4.2 Application: proof of Proposition 4.1

We start with Lemma 4.11. Assume j ∈ A 1 . Then, for any family {U R } R satisfying (1.5) and R > C, for any p > 1,

max |ν(U R ) -|U R || , ν( U R ) -ν(U R ) ≤ C R θ + R θ(1-1 p ) j L p (U R+C ) + 1 ,
where θ < 2 is the exponent in (1.5), and C only depends on the constants in (1.5).

Proof. Let d R denote the signed distance to ∂U R , i.e. d R (x) = -d(x, U R c ) if x ∈ U R and d R (x) = d(x, U R ) otherwise. Let U R t = {x | d R (x) ≤ t.}. Integrating (4.1) over U R
t and using Hölder's inequality we find

(4.21) ν(U R t ) -|U R t | = ∂U R t j • τ ≤ |∂U R t | 1-1 p j L p (∂U R t ) .
Then, using the coarea formula,

1 0 |∂U R t | dt = |U R 1 \ U R |, 1 0 j p L p (∂U R t ) dt = j p L p (U R 1 \U R )
and from (1.5) there exists C > 0 such that U R 1 ⊂ U R+C . Using a mean-value argument and (1.5) again there exists t ∈ (0, 1) such that

|∂U R t | ≤ CR θ , j p L p (∂U R t ) ≤ 2 j p L p (U R 1 ) ≤ 2 j p L p (U R+C ) ,
and therefore

(4.22) ν(U R t ) -|U R t | ≤ CR θ(1-1 p ) j L p (U R+C ) .
A similar mean value argument yields the existence of s ∈ (-1, 0) such that

|ν(U R s ) -|U R s || ≤ CR θ(1-1 p ) j L p (U R ) . Since t → ν(U R t ) is increasing we deduce, since U R-C ⊂ U R s , ν(U R s ) -|U R s | ≤ ν(U R ) -|U R | + |U R \ U R-C | ≤ ν(U R t ) -|U R t | + |U R+C \ U R-C |,
( U R ) -ν(U R ) follows easily since U R ⊂ U R ⊂ U R+C .
Corollary 4.12. Assume that j ∈ A 1 , that {U R } R , {V R } R satisfy (1.5) and that c > 0 is such that U cR ⊂ V R for any R ≥ 1 (it is easy to show from (1.5) that such a c exists). We assume that either W U (j) < +∞, or W V (j) < +∞.

Then, denoting g U R the result of applying Proposition 4.9 to (j, ν) in U R , we have

(4.23) lim R→+∞ 1 R 2 U cR χ U cR dg V R -W (j, χ U cR ) = 0. In particular if {V R } R = {U R } R , we may take c = 1 to obtain (4.24) lim R→+∞ 1 R 2 U R χ U R dg U R -W (j, χ U R ) = 0.
As a consequence, W U does not depend on the particular choice of χ U R satisfying (1.3).

Remark 4.13. Let G ⊂ A 1 satisfy the hypothesis of Proposition 4.2 and let

{U R } R = {K R } R .
Then the convergence in (4.24) is uniform with respect to j ∈ G.

Proof. Since W U (j) < +∞ and

j ∈ A 1 , both W (j, χ U R ) and ν(U R ) are O(R 2 ) as R → ∞. Applying Lemma 4.7 in U R+C we find, choosing some p ∈ (1, 2), that U R |j| p = O(R 2 log R).
Then, using Lemma 4.11, we have |ν( U R ) -ν(U R )| = O(R θp ) for some exponent θ p ∈ (0, 2), and thus the same holds for |ν(U R ) -ν(U R-C )|. Inserting into (4.12) yields (4.25)

W (j, χ U cR ) |U cR | -- U cR χ U cR dg V R R→+∞ -----→ 0,
hence (4.23), and (4.24) easily follows.

In view of (4.24), proving that the definition of W U is independent of the choice of {χ U R } R satisfying (1.3) now reduces to proving the same statement for lim sup

R→+∞ - U R χ U R dg U R . But, since g U R ≥ -C and since U R-C ⊂ {χ U R = 1}, g U R (U R-C ) -C|U R \ U R-C | ≤ U R χ U R dg U R ≤ g U R (U R ) + C|U R \ U R-C |.
Dividing by |U R | and in view of (1.5) we obtain that lim sup

R→+∞ - U R χ U R dg U R = lim sup R→+∞ g U R (U R ) |U R | ,
which is clearly independent of χ U R and finite thanks to (4.24). The proof of Remark 4.13 follows from the fact that under the hypothesis of Proposition 4.2 we clearly have bounds ν(K R ) < CR 2 and W (j, χ K R ) < CR 2 which are uniform with respect to j ∈ G and thus that the convergence in (4.25) is uniform with respect to j ∈ G as well, when

{U R } R = {K R } R .
We may now give the proof of Proposition 4.1, in several steps.

W U is measurable. First we show that A 1 (recall Definition 1.1) is a Borel subset of X := L p loc (R 2 , R 2 ). For R, ε > 0 we let A R,ε be the set of j ∈ L p loc (R 2 , R 2
) such that first curl j = ν-1 and div j = 0, where the restriction of ν to B 2R is of the form 2π i δ a i with a i ∈ B R and |a i -a j | ≥ ε (in particular the sum is finite), and second j L p (B 2R ) ≤ 1/ε. We also let

A R,ε,C = {j ∈ A R,ε | ν(B R ) ≤ CR 2 }.
Clearly both A R,ε , and A R,ε,C are closed. Noting that

A 1 = (∪ C>1 ∩ R>1 ∪ ε>0 A R,ε,C ) , we find that A 1 is a Borel subset of X. For j ∈ A 1 we have W (j) = lim sup R W (j, χ U R )/|U R |, hence proving that W is Borel reduces to proving that W χ : j → W (j, χ) if j ∈ A 1 +∞ otherwise
is Borel for any smooth, positive χ with compact support.

This follows from Lemma 4.8. Choosing R > 0 such that Supp(χ) ⊂ B R the lemma implies that W χ restricted to the closed set A R,ε is continuous, therefore {j ∈ A R,ε | W χ ≤ t} is closed for any t, and

{j ∈ A 1 | W χ (j) ≤ t} = A 1 ∩ (∩ R>0 ∪ ε>0 {j ∈ A R,ε | W χ ≤ t}) is Borel. inf A 1 W U is finite.
The results of Section 3 for example show that inf W U < +∞. The fact that inf W U > -∞ is a direct consequence of Corollary 4.12 and the fact that for any R > 0 we have g U R ≥ -C, where g U R is the result of applying Proposition 4.9 on U R .

Sub-level sets are compact up to translation, min W U is achieved. Consider a family {V R } R satisfying (1.4)-(1.5) and consider a sequence { n } n such that W V ( n ) ≤ α. In particular n ∈ A 1 . Let νn be curl n + 1. Then since νn (B R ) = O(R 2 ) and from the definition of W V as a lim sup, for any fixed arbitrary C > 0 and any θ > 1 there exists a sequence

R n → +∞ such that lim inf n→+∞ W V ( n ) - W ( n , χ V Rn ) |V Rn | ≥ 0, νn (V Rn+C \ V Rn-C ) ≤ R n θ .
Indeed, given n, the second relation is satisfied by arbitrarily large R's, using a mean value argument. Now, letting V n = V Rn , we apply Proposition 4.9 to ( n , νn ) with U = V n and deduce the existence of a measure ḡn ≥ -C satisfying the properties described there. The choice of

{R n } n ensures that (4.26) lim inf n→+∞ W V ( n ) ≥ lim inf n→+∞ W ( n , χ Vn ) |V n | = lim inf n→+∞ - Vn χ Vn dḡ n ,
using (4.12). Now we apply the abstract scheme described in Theorem 3. Let X = L p loc (R 2 , R 2 ) × M 0 , where M 0 is the space of Radon measures on R 2 bounded below by twice the fixed constant given by item 1 of Proposition 4.9 (this means that we are considering measures µ such that µ + C is a positive Radon measure), and the topology is that of convergence in L p loc (R 2 , R 2 ) and weak convergence on M 0 . X is a Polish space, and on it we have the natural continuous action θ λ (j, g) = (j(λ + •), g(λ + •)). We may check the hypotheses of Section 2 are satisfied. Then we choose a smooth positive χ with compact support in B(0, 1) and integral 1, we let

V ′ n = V Rn-C where C is chosen (according to (1.5)) such that (4.27) χ Vn ≥ χ * 1 V ′ n everywhere and χ Vn = χ * 1 V ′ n = 1 in V Rn-2C and define f n (j, g) = χ(y) dg(y) if there exists λ ∈ V ′ n such that (j, g) = θ λ ( n , ḡn ), +∞ otherwise.
We also let

F n (j, g) = - V ′ n f n (θ λ (j, g)) dλ = 1 |V ′ n | χ * 1 V ′ n dḡ n if (j, g) = ( n , ḡn ), +∞ otherwise.
Since ḡn ≥ -C, the property (4.27) implies that (4.28)

χ Vn dḡ n ≥ |V ′ n |F n ( n , ḡn ) -O(R n ).
Then we check the coercivity and Γ-lim inf properties of {f n } n as in (1.15)-(1.16). The latter is the trivial observation that if (j n , g n ) → (j, g) then for any subsequence (not relabeled) such that {f n (j n , g n )} n is bounded we have

lim n→∞ f n (j n , g n ) = lim n→+∞ χ dg n = f (j, g), where f (j, g) = χ dg.
To prove coercivity, assume as in (1.15) that for every R

(4.29) lim sup n→+∞ B R f n (θ λ (j n , g n )) dλ < +∞.
Then for every R, if n is large enough the integrand is finite a.e. hence there exists

λ n ∈ V ′ n such that (j n , g n ) = θ λn ( n , ḡn ) and λ + λ n ∈ V ′ n for almost every λ ∈ B R , i.e. λ n + B R ⊂ V ′ n . Then (4.29) reads lim sup n→+∞ B R χ(x -λ n -λ) dḡ n (x) dλ = χ * 1 λn+B R dḡ n < +∞, which is equivalent to saying that {g n = ḡn (λ n + •)} n is bounded in L 1 (B R ) for every R. This implies that a subsequence converges in M 0 .
Then, in view of (4.13) this proves that {ν n (λ n + •)} n is locally bounded. Inserting this information into (4.12) we find that {W ( n (λ n + •), χ R )} n is bounded and then using Lemma 4.7 we deduce that { n (λ n + •)} n is bounded in L p (B R ) for any R. Thus going to a further subsequence {j n = n (λ n + •)} n converges to j locally weakly in L p . Moreover div j n = 0 and by the above curl j n is locally bounded in the sense of measure, hence weakly compact in W -1,p loc . By elliptic regularity it follows that the convergence of j n to j is strong in L p loc (R 2 , R 2 ). This proves coercivity.

We may now apply Theorem 3. Letting P n be the image of the normalized Lebesgue measure on B ′ n by the map λ → θ λ ( n , ḡn ), there is a subsequence such that P n → P , where P is a probability measure on X and

lim inf n→+∞ F n ( n , ḡn ) ≥ f * U (j, g) dP (j, g), where f * U (j, g) = lim R→+∞ - U R χ(x -λ) dg(x) dλ,
for any family of open sets {U R } R>0 satisfying (1.4). We claim that if (j, g) ∈ Supp(P ) and f * U (j, g) ≤ +∞, then (4.30) j ∈ A 1 and f * U (j, g) = W U (j).

Assuming this, and choosing (j, g) ∈ Supp(P ) such that f * U (j, g) ≤ lim inf n F n ( n , ḡn ) we obtain, using (4.26) and (4.28),

(4.31) lim inf n W V ( n ) ≥ lim inf n F n ( n , ḡn ) ≥ W U (j).
Choosing V = U shows that inf A 1 W U is achieved. Taking for n a minimizing sequence for W V , it follows from (4.31) that min

A 1 W V ≥ min A 1 W U , hence the value of min A 1 W U is independent of U .
We prove the claim (4.30). Since f * U (j, g) ≤ +∞, we have g(U R ) < C|U R | ≤ CR 2 and thus ∀R > 1 there exists N R ∈ N such that n ≥ N R implies g n (U R ) ≤ CR 2 . Using (4.13) this in turn implies that if n ≥ N R then ν n (U R ) ≤ CR 2 and then, passing to the limit n → ∞, that ν(U R ) ≤ CR 2 , so that in particular j ∈ A 1 .

Moreover, still if n > N R , from g n (U R ), ν n (U R ) ≤ CR 2 and using (4.12) we deduce that W (j n , χ U R ) ≤ CR 2 log R and then, as in the proof of Corollary 4.12, that

W (j n , χ U R ) -χ U R dg n ≤ o(R 2 ),
for some θ < 2. Passing to the liminf n → ∞ we obtain in view of Lemma 4.8 the same relation for j, g. Dividing by |U R | -which is bounded below by cR 2 for some c > 0 -and letting R → +∞ we find that W U (j) = f * U (j, g), which finishes the proof of (4.30).

Independence w.r.t. the shape. We have just seen that if U and V refer to two families of sets, the infimum of W V and W U are both achieved and are equal. There remains to show that minimizers are also the same. Consider j V a minimizer of W V . By Corollary 4.12 we have for any

{U R } R satisfying (1.5), (4.32) lim R→∞ W (j V , χ U R ) |U R | -- U R χ U R dg U R = 0,
where g U R is the result of applying Proposition 4.9 in U R . We deduce that

(4.33) W U (j V ) = lim sup R→∞ - U R χ U R dg U R = lim sup R→∞ f (j, g) dP U R (j, g) ≥ lim inf R→∞ f (j, g) dP U R (j, g) ≥ f (j, g) dP U (j, g),
where f (j, g) = χ dg, χ U R = 1 U R-C * χ, and where P U R is the image of the normalized Lebesgue measure on U R-C by λ → θ λ (j V , g U R ). Moreover we have chosen a subsequence {R} such that {P U R } R converges to a probability measure P U . Since P U is θ-invariant and using the ergodic theorem and (4.30) we get

f (j, g) dP U (j, g) = f * U (j, g) dP U (j, g) = W U (j) dP U (j, g) ≥ min A 1 W,
where W can be defined using any family of sets satisfying (1.4), (1.5), not necessarily the family U . Together with (4.33) we get

(4.34) W U (j V ) = lim sup R→∞ f (j, g) dP U R (j, g) ≥ f (j, g) dP U (j, g) = W U (j) dP U (j, g)
and this is bounded below by min A 1 W . From the minimality of j V and since min W U = min W V , applying (4.34) to {U R } R = {V R } R implies that there is equality everywhere and therefore

(i) lim sup R f dP V R = f dP V , (ii) P V -almost every j minimizes W .
Since f is continuous and bounded below on the support of P V R independently of R,

lim sup R f dP V R = f dP V ⇐⇒ sup R {f >M } f dP V R M →+∞ -----→ 0. Now choose c > 0 such that U cR ⊂ V R-C
for every R ≥ 1 and let P ′ V R be the image of the normalized Lebesgue measure on U cR by λ → θ λ (j V , g V R ). Then (4.35)

P V R ≥ |U cR | |V R-C | P ′ V R ≥ δP ′ V R ,
for some δ > 0 independent of R (this follows from (1.5)). It follows that sup

R {f >M } f dP ′ V R M →+∞ -----→ 0
and then that, choosing a subsequence {R} such that [Bi, Du]). Now we claim that the support of

P ′ V R → P ′ V , that lim sup R f dP ′ V R = f dP ′ V (cf.
P ′ V is included in the support of P V . Indeed if ϕ ≥ 0 is continuous with compact support in (Supp P V ) c , then ϕ dP V R → ϕ dP V = 0 hence from (4.35) δ ϕ dP ′ V R R→∞ ----→ 0 and thus ϕ dP ′ V = lim R ϕ dP ′ V R = 0.

It follows that P ′

V -almost every j minimizes W U and that (4.36)

f dP ′ V R R→∞ ----→ min W U . But f dP ′ V R = -U cR χ U cR dg V R
by definition of f and P ′ V R and using Corollary 4.12, we have that -U cR χ U cR d(g V R -g U cR ) tends to 0 as R → +∞. Therefore f d(P ′ V R -P U cR ) tends to 0 as well, which together with (4.36) and (4.34) yields

W U (j V ) = lim sup R→+∞ f dP U cR = min A 1 W U .
It follows that j V minimizes W U .

Proof of Proposition 4.2 and Corollaries 4.4, 4.5

The first lemma (whose proof is postponed to the end of the subsection) serves to extract a good boundary. We denote by W K the renormalized energy relative to the family

{K R = [-R, R] 2 } R .
Lemma 4.14. Let G satisfy the assumptions of Proposition 4.2. Then for any γ ∈ (0, 1), any R large enough depending on γ, and any p ∈ [1, 2) there exists, for any j ∈ G, some

t ∈ [R -2R γ , R -R γ ] such that (4.37) ∂Kt |j| p ≤ C p R 2-γ (4.38) lim R→∞ W (j, 1 Kt ) |K t | = W K (j), |ν(K t ) -|K t || ≤ CR 2-γ ,
where C, C p do not depend on j ∈ G, and where the convergence in (4.38) is uniform with respect to j ∈ G.

The next steps consist in modifying j in K R \K t so that j • τ = 0 on ∂K R , and so that the ensuing modification of W (j, 1 K R ) is negligible compared to R 2 as R → +∞. This relies on the following two lemmas.

Lemma 4.15. Let R be a rectangle with sidelengths in

[ L 2 , 3L 2 ]. Let p ∈ (1, 2). Let g ∈ L p (∂R)
be a function which is 0 except on one side of the rectangle R. Let m be a constant such that (m -1)|R| = -∂R g. Then the mean zero solution to

(4.39) -∆u = m -1 in R ∂u ∂ν = g on ∂R satisfies for every q ∈ [1, 2p] (4.40) R |∇u| q ≤ C p,q L 2-q p g q L p (∂R)
.

Proof. We write the solution u of (4.39) as u = u 1 + u 2 where (4.41)

-∆u 1 = m -1 in R ∂u 1 ∂ν = ḡ on ∂R
where ḡ is equal to the average of g on the side where g is supported and is 0 on the other sides; and

-∆u 2 = 0 in R ∂u 2 ∂ν = g -ḡ on ∂R. Assume that R = [0, ℓ 1 ] × [0, ℓ 2 ], with ℓ i ∈ [ L 2 , 3L 2 ],
and that ḡ is supported on the side x 2 = 0. Then, up a constant, the solution of (4.41) is u

1 (x 1 , x 2 ) = m-1 2 (x 2 -ℓ 2 ) 2 . Therefore R |∇u 1 | q = (m -1) q ℓ 1 ℓ 2 0 (x 2 -ℓ 2 ) q dx 2 = (m -1) q ℓ 1 ℓ q+1 2 q + 1 ≤ C(m -1) q L 2+q .
Then, m -1 = -1 ℓ 1 ℓ 2 ∂R g and using Hölder's inequality |m -1| ≤ CL -2 g L p (∂R) L 1-1 p . Inserting above we are led to

(4.42) R |∇u 1 | q ≤ C p,q L 2-q p g q L p (∂R)
.

For u 2 , note that the conjugate harmonic function u * 2 has trace ϕ which satisfies ∂ τ ϕ = g -ḡ, hence ∇ϕ L p (∂R) ≤ g -ḡ L p (∂R) . Then from the Sobolev imbedding W 1,p (∂R) ֒→ W 1-1 2p ,2p (∂R), which is the trace space of W 1,2p (R), and elliptic regularity it follows that

∇u 2 L 2p (R) = ∇u * 2 L 2p (R) ≤ C R g -ḡ L p (∂R) ,
and it is easy to check that the constant C R may be chosen to depend only on p, L and not on ℓ i , as long as

ℓ i ∈ [ L 2 , 3L 2 ]
. Scaling arguments then show that for some C p,q independent of L we have

(4.43) R |∇u 2 | q ≤ C p,q L 2-q p g q L p (∂R)
.

Combining (4.42) and (4.43), we obtain the result (4.40).

Lemma 4.16. Let R be a rectangle of barycenter 0, and sidelengths ∈ √ 2π[ 1 2 , 3 2 ], and let m be a constant such that m|R| = 2π. Then the solution to

-∆f = 2πδ 0 -m in R ∂f ∂ν = 0 on ∂R satisfies (4.44) lim η→0 R\B(0,η) |∇f | 2 + 2π log η ≤ C
where C is universal, and for every 1 ≤ q < 2 (4.45)

R |∇f | q ≤ C q ,
where C q depends only on q.

Proof. This is a standard computation, of the type of [BBH], Chap. 1, observing that f = -log |x| + S(x) where S is a C 1 function.

Proof of Proposition 4.2

Let j ∈ G. Apply Lemma 4.14 with p ∈ ( 3 2 , 2) and γ = 3 4 . For any R large enough it provides us with a square K t , where t depends on j ∈ G but satisfies

(4.46) R 3 4 ≤ R -t ≤ 2R 3 4 .
We wish to extend j in K R \K t , keeping j and Λ unchanged in K R \K t , to obtain a j R satisfying j R • τ = 0 on ∂K R and curl j R = 2π p∈Λ R δ p -1 -while the extension's contribution to the renormalized energy remains negligible compared to R 2 , uniformly with respect to j ∈ G.

Below, the notations o(•), ∼ and O(•) are understood with respect to R → +∞, and uniform with respect to j ∈ G.

We divide each side of K t into [ √ R] intervals, so that there are a total of 4[ √ R] intervals, which we label

{I i } i , of length 2t [ √ R]
, which is equivalent to 2 √ R as R → +∞, uniformly with respect to t satisfying (4.46). For each i we consider the square K i ⊂ K R \ K t with one side equal to I i . By perturbing the length of the other side of an amount which is O(1/ √ R) as R → ∞ we may obtain a rectangle R i whose aspect ratio tends to 1. and such that (4.47)

|R i | - I i j • τ ∈ 2πN.
We let g i denote the restriction of j • τ to I i (and extend it by 0 on the rest of ∂R i ), and let

m i = 1 - R ∂R i g i |R i |
. Using Hölder's inequality, we have

|m i -1| ≤ C |R i | I i |j • τ | p 1 p R 1 2 (1-1 p ) ≤ CR 1 2 (-1-1 p ) ∂Kt |j| p 1 p .
In view of (4.37), we deduce

|m i -1| = O(R -1 2 -1 2p + 2 p -γ p ) = o(1),
since γ = 3 4 and p > 3 2 . By (4.47) and by choice of m i we also have m i |R i | ∈ 2πN. We may thus tile R i by an integer number of rectangles R ik , whose sidelengths are in √ 2π[ 1 2 , 3 2 ] and such that for each i, k, we have

m i |R ik | = 2π. Since m i ∼ 1, the number of rectangles inside each R i is equivalent to |R i |/2π ∼ R/2π as R → +∞.
On each of these rectangles, we may apply Lemma 4.16, which yields a function f ik satisfying (4.44)-(4.45). We then define the vector field j 1 in ∪ i R i by j 1 = -∇f ik in each R ik . We can check that j 1 satisfies (4.48)

curl

j 1 = 2π p∈Γ δ p -i m i 1 R i in ∪ i R i j 1 • τ = 0 on ∂(∪ i R i )
where Γ is the union over i, k, of the centers of the rectangles R ik . Indeed since ∂f ij ∂ν = 0 no curl is created at the interfaces between the R ik 's, and no curl is created either at the interfaces between the R i 's.

Moreover, by (4.44)-(4.45), since the number of R ik for each i is of order |R i |, and since the number of R i 's is O( √ R), j 1 satisfies (4.49) lim

η→0 ∪ i R i \∪B(p,η) |j 1 | 2 + π#Γ log η ≤ CR 3 2 ,
and for q < 2 (4.50)

∪ i R i |j 1 | q ≤ C q R 3 2 .
Since (m i -1)|R i | = -∂R i g i , we may also apply Lemma 4.15 in each R i with g = g i for boundary data. It yields a function u i satisfying (4.40). We then define the vector field j 2 as

-∇ ⊥ u i in each R i . It satisfies (4.51) curl j 2 = i m i 1 R i -1 in ∪ i R i j 2 • τ = g on ∂ ∪ i R i
where g = j • τ on ∂K t and 0 on the rest of ∂(∪ i R i ). Indeed, g i is only supported on the I i i.e. on the sides of the R i which are in ∂K t , so j 2 • τ = 0 on all the boundaries of the R i which intersect, therefore again no curl is created there. For every 1 ≤ q ≤ 2p we have, since

|I i | ∼ √ R, R i |j 2 | q ≤ C p,q R 1-q 2p g i q L p (∂R i ) .
Adding these relations, we obtain

∪ i R i |j 2 | q ≤ C p,q R 1-q 2p i ∂R i |g i | p q p .
But when q/p > 1 we have i x q/p i

≤ ( i max(1, x i )) q/p and number of R i 's is O( √ R) hence, ∪ i R i |j 2 | q ≤ C p,q R 1-q 2p ∂Kt |g| p + √ R q p .
Using (4.37), we deduce, for all 1 ≤ q ≤ 2p (4.52)

∪ i R i |j 2 | q ≤ C p,q R 1-q 2p + q p (2-3 4 ) ≤ C p,q R 1+ 3q 4p .
From now on we choose p ∈ ( 3 2 , 2) and we have for every q < 4p 3 , (4.53)

∪ i R i |j 2 | q ≤ C q R σ , for some σ < 2.
We can now define j R more precisely. In ∪ i R i we let j R = j 1 + j 2 and ν R = 2π p∈Γ δ p . By summing (4.48) and (4.51) we have

curl j R = 2π p∈Γ δ p -1 in ∪ i R i j R • τ = g on ∂(∪ i R i ),
where g = j • τ on ∂(∪ i R i ) ∩ ∂K t and g = 0 elsewhere. Also, (4.54)

∪ i R i \∪B(p,η) |j R | 2 = ∪ i R i \∪B(p,η) |j 1 | 2 + |j 2 | 2 + 2j 1 • j 2 .
We have

∪ i R i \∪B(p,η) j 1 • j 2 ≤ ∪ i R i |j 1 | q ′ 1/q ′ ∪ i R i |j 2 | q 1/q
where q > 2 and 1 q ′ = 1 -1 q . Using (4.50) and (4.53) where we can choose q > 2 since p > 3 2 , we find

∪ i R i \∪B(p,η) j 1 • j 2 ≤ CR 3 2q ′ + σ q = o(R 2 ),
since σ < 2 and 1 q ′ + 1 q = 1. Inserting into (4.54) and combining with (4.49) and (4.52) we obtain lim

η→0 ∪ i R i \∪B(p,η) |j R | 2 + π#Γ log η = O(R 3 2 ) + o(R 2 ) = o(R 2 ).
There remains to define j R in A := K R \ (K t ∪ i R i ). First we note that |A| ∈ 2πN. Indeed, from curl j = ν -1 and (4.47), we have

2π#(Λ ∩ K t ) -|K t | = ∂Kt j • τ = i |R i | (mod 2π), thus |A| = |K R | -i |R i | -|K t | is in 2πN if |K R | ∈ 2πN.
The set A can be described as follows: It is the union of four squares of sidelength R -t positioned at the corners of K R , and a union ∪ i R ′ i , where R ′ i is a rectangle having a side of length |I i | in common with R i , and such that their union is isometric to

I i × [0, R -t].
Since both dimensions of these rectangles as well as those of the four squares tend to +∞ as R → +∞, and since |A| ∈ 2πN, it is possible to tile A by rectangles of area 2π and aspect ratio close to 1. Applying Lemma 4.16 in each of them yields a current j A which satisfies curl j A = 2π p∈Γ ′ δ p -1 in A and j A • τ = 0 on ∂A -where Γ ′ is the set of centers of the rectangles tiling A.

The cardinal of Γ ′ is |A|/2π, which is O(R 1+ 3 4 ), and therefore from (4.44) we deduce (4.55) lim

η→0 K R \(Kt∪ i R i ) |j A | 2 + π#Γ ′ log η = O(R 7 4 ),
and from (4.45), for all 1 ≤ q < 2, (4.56)

K R \(Kt∪ i R i ) |j A | q ≤ C q R 7 4 . Letting j R = j A in A and Λ R = (Λ ∩ K t ) ∪ Γ ∪ Γ ′ , ν R = 2π p∈Λ R δ p we have j = j R in K t , ν R = ν in K t , curl j R = ν R -1 in K R , j R •τ = 0 on ∂K R ,
W (j R , 1 K R ) |K R | = W K (j) + o(1) as R → +∞,
where the o(1) is uniform with respect to j ∈ G. This completes the proof of Proposition 4.2. We also note that from (4.50), (4.53) and (4.56), for every 1 ≤ q < 2, we have (4.58)

K R \Kt |j R | q ≤ R σ for some σ < 2.

Proof of Lemma 4.14

Let G satisfy the assumptions of Proposition 4.2, and R > 2 with |K R | ∈ 2πN, 0 < γ < 1 be given. Assume j ∈ G. In this proofs the constants and limits as R → +∞ are understood to be uniform with respect to j ∈ G.

Step 1: Denote by g R the result of applying Proposition 4.9 in K R to (j, ν). We apply (4.12) to functions of the form χ(x) = ρ( x ∞ ), i.e. whose level sets are squares, with the additional assumption that ρ ′ (t) = 0 outside [R -2, R -1] and ρ = 0 on [R -1, +∞]. Since for any Radon measure µ on K R we have

χ dµ = - R-1 0 ρ ′ (t)µ(K t ) dt, we deduce that (4.59) R-1 R-2 (W (j, 1 Kt ) -g R (K t )) ρ ′ (t) dt = -W (j, χ) + χ dg R ≤ Cn(log n + 1) ρ ′ ∞ ,
where the last inequality is (4.12), and

n = #{p ∈ Λ, B(p, C) ∩ Supp ∇χ = ∅}, so that 2πn ≤ ν(K R+C ) -ν(K R-C ).
Here C denotes a universal constant, hence independant of j ∈ G. We deduce by duality that for some universal C > 0 we have

(4.60) R-1 R-2 |W (j, 1 Kt ) -g R (K t )| dt ≤ Cn(log n + 1).
On the other hand, Lemma 4.7 yields that for any p ∈ [1, 2) and R > 0,

j L p (K R ) ≤ CR 2/p log 1/2 R,
where C depends only on p and on the constants in (4.3) and (4.4). Arguing as in the proof of Lemma 4.11 this implies that

(4.61) n ≤ 1 2π |ν(K R+C ) -ν(K R )| ≤ CR θ(1-1 p ) j L p (K R+2C ) ≤ CR θ(1-1 p )+ 2 p log 1 2 R ≤ CR β
for some β < 2, where we have chosen p < 2 close enough to 2 and used θ < 2 in (1.5). It then follows from (4.60), (4.61) that

(4.62) 1 R 2 R-1 R-2 |W (j, 1 Kt ) -g R (K t )| dt ≤ CR β-2 log R.
Now denote by {χ R } R a family of functions satisfying (1.3) relative to the family {K R } R . We also assume χ R ≤ 1. Since g R ≥ -C and since χ R = 1 on K R-1 we have for any

t ∈ [R -2, R -1] the inequalities χ R-2 dg R -C|K R-1 \ K R-3 | ≤ g R (K t ) ≤ χ R dg R + C|K R \ K R-2 | ≤ χ R dg R + CR. and thus (4.63) χ R-2 dg R -CR ≤ g R (K t ) ≤ χ R dg R + CR.
Step 2: For any integer k ≥ 1 let ξ k = χ k+1 -χ k , and let

ξ 0 = χ 1 . Then ξ k ≥ 0, since χ k+1 = 1 on K k and since χ k ≤ 1 and is supported in K k . Moreover ξ k is supported in C k := K k+1 \ K k-1 . Since (4.3) holds, the number of integers k in [R -2R γ + 1, R -R γ ] such that ν(K k+2 \ K k-2 ) ≤ CR 2-γ is greater than R γ /2 if C is chosen large enough. Similarly, [R-R γ ] k=[R-2R γ ] ξ k dg R = (χ [R-R γ ]+1 -χ [R-2R γ ] ) dg R ≤ CR 2 ,
where we use Corollary 4.12, Remark 4.13 and (4.4). Since g R ≥ -C we have ξ k dg R ≥ -CR and therefore the number of integer k's between [R -2R γ ] + 1 and [R -R γ ] such that ξ k dg R ≤ CR 2-γ is larger than R γ /2 if C and R are chosen large enough, and thus such a k satisfying

ν(K k+2 \ K k-2 ) ≤ CR 2-γ , ξ k dg R ≤ CR 2-γ ,
for some C uniform with respect to j ∈ G.

Applying Proposition 4.9 in C k to ξ k , we have

W (j, ξ k ) -ξ k dg R ≤ CR 2-γ log R hence W (j, ξ k ) ≤ CR 2-γ log
R and applying Lemma 4.7 we find that for p < 2,

C k |ξ k | p 2 |j| p ≤ CR 1-p 2 R 2-γ log R p 2 ≤ CR 2-γ . But ξ k (x) = 1 if x ∞ = k and thus from the gradient bound ξ k ≥ 1/2 if k -1/C ≤ x ∞ ≤ k + 1/C. Therefore K k-1 C \K k+ 1 C |j| p ≤ CR 2-γ .
By a mean value argument on this integral as well as on (4.62) (applied with R = k + 1), we deduce the existence t

∈ [k -1, k] -hence t ∈ [R -2R γ , R -R γ ] -such that, one the one hand ∂Kt |j| p ≤ CR 2-γ ,
proving (4.37) since C is uniform with respect to j ∈ G -and on the other hand

(4.64) g R (K t ) -W (j, 1 Kt ) ≤ CR β log R.
Now, from (4.63) applied to R = k + 1 and using (4.12) in Proposition 4.9 together with (4.61) we obtain

W (j, χ R-2 ) -CR β log R ≤ g R (K t ) ≤ W (j, χ R ) + CR β log R,
Which together with (4.64) and in view of (4.4) yields (4.38). Finally,

|ν(K t ) -|K t || = ∂Kt j • τ ≤ j L p (∂Kt) |∂K t | 1-1 p ≤ CR 2-γ ,
using (4.37). Lemma 4.14 is proved.

Proof of Corollary 4.4

Let j ∈ A 1 be such that W K (j) < +∞. Let R be such that R 2 ∈ 8πN, and j R be obtained by applying Proposition 4.

2 over K = [0, R] × [0, R].
We let R = j R -∇ζ, where ∆ζ = div j R on K and ζ = 0 on ∂K. Then R = -∇ ⊥ H R in K since div R = 0 there, and we have

∂ ν H R = 0 on ∂K since R • τ = j R • τ = 0 there. Thus defining H R on K R = [-R, R] × [-R, R] by letting H R (±x, ±y) = H R (x, y) we have -∆H R = p∈Λ R δ p -1
, where Λ R is obtained from the restriction of curl j + 1 to K by reflections across the coordinate axis. Moreover

H R (-R, y) = H R (R, y) and H R (x, -R) = H R (x, R
) so that we may periodize H R to have it defined on R 2 . Then j R := -∇ ⊥ H R belongs to A 1 and since everything is periodic W can be computed through the results of Section 3.1:

W K (j R ) = W (j R , 1 K R ) |K R | ≤ W K (j) + o(1) as R → ∞.
The last assertion of the Corollary follows by taking j to minimize W K over A 1 (a minimizer exists by Proposition 4.1), and remembering that the minimum does not depend on the choice of shapes used.

Proof of Corollary 4.5

The proof of Corollary 4.5 consists in constructing a sequence from a Young measure on micropatterns, to use the terminology of [AM], while retaining an energy control. We thus assume P is a probability measure on L p loc (R 2 , R 2 ) which is invariant under the action of translations and concentrated in A 1 .

First we choose distances which metrize the topologies of L p loc (R 2 , R 2 ) and B(L p loc ), the set of finite Borel measures on

L p loc (R 2 , R 2 ). For j 1 , j 2 ∈ L p loc (R 2 , R 2 ) we let d p (j 1 , j 2 ) = ∞ k=1 2 -k j 1 -j 2 L p (B(0,k)) 1 + j 1 -j 2 L p (B(0,k))
.

On B(L p loc ) we define a distance by choosing a sequence of bounded continuous functions {ϕ k } k which is dense in C b (L p loc ) and we let, for any µ 1 , µ 2 ∈ B(L p loc ),

d B (µ 1 , µ 2 ) = ∞ k=1 2 -k | ϕ k , µ 1 -µ 2 | 1 + | ϕ k , µ 1 -µ 2 | ,
where we have used the notation ϕ, µ = ϕ dµ.

We have the following general facts.

Lemma 4.17. For any ε > 0 there exists η 0 > 0 such that if P, Q ∈ B(L p loc ) and P -Q < η 0 , then d(P, Q) < ε. Here P -Q denotes the total variation of the signed measure P -Q, i.e. the supremum of ϕ, P -Q over measurable functions ϕ such that |ϕ| ≤ 1.

In particular, if

P = ∞ i=1 α i δ x i and Q = ∞ i=1 β i δ x i with i |α i -β i | < η 0 , then d B (P, Q) < ε. Lemma 4.18. Let K ⊂ L p loc (R 2 , R 2 ) be compact. For any ε > 0 there exists η 1 > 0 such that if x ∈ K, y ∈ L p loc (R 2 , R 2 ) and d p (x, y) < η 1 then d B (δ x , δ y ) < ε. Lemma 4.19. Let 0 < ε < 1. If µ is a probability measure on a set A and f, g : A → L p loc (R 2 , R 2 ) are measurable and such that d B (δ f (x) , δ g(x) ) < ε for every x ∈ A, then d B (f # µ, g # µ) < Cε(|log ε| + 1).
Proof. Take any bounded continuous function ϕ k defining the distance on B(L p loc ). Then if

d B (δ f (x) , δ g(x) ) < ε for any x ∈ L p loc (R 2 , R 2 ) we have in particular |ϕ k (f (x)) -ϕ k (g(x))| 1 + |ϕ k (f (x)) -ϕ k (g(x))| ≤ 2 k ε.
It follows that

d B (f # µ, g # µ) ≤ k 2 -k min(ε2 k , 1) ≤ ε ([log 2 ε] + 1) + ∞ k=[log 2 ε]+1 2 -k ≤ Cε(|log ε| + 1).
Selection of a good subset of L p loc . We must restrict to a compact subset of L p loc (R 2 , R 2 ) in a suitable way. This is not surprising when constructing an approximation: note that the set of micropatterns in [AM] is assumed to be compact, and we need to reduce to this case. This is the aim of the following Lemma.

Lemma 4.20. Given P as above and ε, R > 0 there exist subsets

H ε ⊂ G ε in L p loc (R 2 , R 2
) with G ε compact and such that: i) η 0 being given by Lemma 4.17 we have

(4.65) P (G ε c ) < min(η 0 2 , η 0 ε), P (H c ε ) < min(η 0 , ε).
ii) For every j ∈ H ε there is a subset Γ(j) ⊂ K R such that

(4.66) |Γ(j)| < CR 2 η 0 and λ / ∈ Γ(j) =⇒ θ λ j ∈ G ε .
iii) (4.67) W K (j) and ν(K t ) t 2 are bounded uniformly with respect to j ∈ G ε and t > 1,

where curl j = ν -1, and the convergence in the definition of W K (j) is uniform. iv) We have (4.68) d B (P, P ′′ ) < Cε(|log ε| + 1), where

P ′′ = Hε 1 |K R | K R \Γ(j)
δ θ λ j dλ dP (j).

Moreover, there exists a partition

H ε = ∪ Nε i=1 H i ε such that diam (H i ε ) < η 3 , where η 3 is such that (4.69) j ∈ H ε , d p (j, j ′ ) < η 3 , λ ∈ K R \Γ(j) =⇒ d B (δ θ λ j , δ θ λ j ′ ) < ε;
and for all 1 ≤ i ≤ N ε there exists

J i ∈ H i ε such that (4.70) W K (J i ) < inf H i ε W K + ε. Proof. Choice of G ε . Since L p loc (R 2 , R 2
) is Polish we can always find a compact set G ε satisfying (4.65) and P (G ε c ) < η 0 . Then from Lemma 4.17, P G ε (the restriction of P to G ε ) satisfies d B (P, P G ε ) < ε.

From the translation-invariance of P , we have for any λ that P (θ λ G ε ) > 1 -η 0 and therefore that d B (P, P θ λ G ε ) < ε. It follows that for any λ ∈ R 2 we have P -P λ < η 0 and then d B (P, P λ ) < ε, where

P λ = θ λ Gε δ j dP (j) = Gε δ θ λ j dP (j).
Then using Lemma 4.19 we deduce that if A ⊂ R 2 is any measurable set of positive measure, then (4.71) d B (P, P ′ ) < Cε(|log ε| + 1), where

P ′ = Gε - A δ θ λ j dλ dP (j).
Moreover, since P is invariant, choosing χ to be a smooth positive function with integral 1 supported in B(0, 1), the ergodic theorem (as in [Be]) ensures that for P -almost every j the limit lim

t→+∞ 1 |K t | Kt W (j(λ + •), χ(λ + •)) dλ
exists. Then 1 Kt * χ is a family of functions which satisfies (1.3) with respect to the family of squares {K t } t , and from the definition of the renormalized energy relative to {K t } t we may rewrite the limit above as

(4.72) W K (j) = lim t→+∞ 1 |K t | W (j, 1 Kt * χ).
By Egoroff's theorem we may choose the compact set G ε above to be such that, in addition to (4.71), the convergence in (4.72) is uniform on G ε . In fact, since W K (j) < +∞ and lim sup t ν(K t )/t 2 < +∞ for P -a.e. j, where curl j = ν -1, we may choose G ε such that (4.67) holds.

The difficulty we have to face next is that θ λ j need not belong to G ε if j does.

Choice of H ε . For j ∈ G ε , let Γ(j) be the set of λ's in K R such that θ λ j ∈ G ε . Since, from (4.65) and the translation-invariance of P , for any λ ∈ R 2 we have

P (θ λ (G ε ) c ) < η 0 2 , it follows from Fubini's theorem that Gε |Γ(j)| dP (j) = K R P ((θ λ G ε ) c ) dλ < 4R 2 min(η 0 2 , η 0 ε).
Therefore, letting (4.73)

H ε = {j ∈ G ε : |Γ(j)| < 4R 2 η 0 },
we have that (4.65) holds. Combining (4.65) and (4.73) with Lemma 4.17, we deduce from (4.71) that (4.68) holds.

Then we use the fact that G ε is compact and Lemma 4.18 to find that there exists η 4 > 0 such that

(4.74) ∀j ∈ G ε , ∀j ′ ∈ L p loc (R 2 , R 2 ), d p (j, j ′ ) < η 4 =⇒ d B δ j , δ j ′ < ε.
Moreover, from the continuity of (λ, j) → θ λ j, there exists η 3 > 0 such that

∀j ∈ G ε , ∀λ ∈ K R , d p (j, j ′ ) < η 3 =⇒ d p θ λ j, θ λ j ′ < η 4 . Now if j ∈ H ε and λ ∈ K R this implies that if d p (j, j ′ ) < η 3 then d p (θ λ j, θ λ j ′ ) < η 4 . But if λ /
∈ \Γ(j) we have θ λ j ∈ G ε hence applying (4.74) to θ λ j, θ λ j ′ , we get (4.69).

Choice of J 1 , . . . , J Nε . Now we cover the relatively compact H ε by a finite number of balls B 1 , . . . , B Nε of radius η 3 /2 and derive from it a partition of H ε by sets with diameter less than η 3 by letting

H 1 ε = B 1 ∩ H ε and H i+1 ε = B i+1 ∩ H ε \ (B 1 ∪ • • • ∪ B i ) .
We then have (4.75)

H ε = Nε i=1 H i ε , diam (H i ε ) < η 3 (R),
where the union is disjoint. Then we may choose J i ∈ H i ε such that (4.70) holds.

Completion of the construction. First we apply Proposition 4.2 with G = G ε . The proposition yields R 0 > 1 such that for any j ∈ G ε and any R > R 0 such that |K R | ∈ 2πN there exists j R defined in K R such that (4.5) is satisfied and such that, if

x ∈ K R-2R 3/4 , then j R (x) = j(x). Moreover, (4.76) W (j R , 1 K R ) |K R | ≤ W K (j) + ε.
We choose R ε > R 0 such that |K Rε | ∈ 2πN and large enough so that (4.77)

K Rε(1-η 0 ) ⊂ {x : d(x, K Rε c ) > R ε 3 4 }.
where η 0 is the constant in Lemma 4.17.

If λ ∈ K Rε(1-η 0 ) and since j(x) = j Rε (x) if d(x, K Rε c ) > R ε 3 4 , we deduce from (4.77) that θ λ j Rε = θ λ j in B(0, R ε 3 4 ) as soon as η 0 R ε > 2R ε 3 4 , so that from the definition of d p , taking R ε larger if necessary, (4.78) ∀j ′ ∈ L p loc (R 2 , R 2 ), ∀λ ∈ K Rε(1-η 0 ) , j ′ = j Rε on K Rε =⇒ d p (θ λ j, θ λ j ′ ) < η 1 ,
where η 1 comes from Lemma 4.18 applied on G ε , i.e. is such that

(4.79) j ∈ G ε , j ′ ∈ L p loc (R 2 , R 2 ) and d p (j, j ′ ) < η 1 =⇒ d B (δ j , δ j ′ ) < ε.
Having chosen R ε , we get from Lemma 4.20 a set H ε and a partition H ε = ∪ Nε i=1 H i ε and in each H i ε a current J i satisfying (4.70). We also choose an arbitrary

J 0 ∈ A 1 such that W K (J 0 ) < +∞.
Second we choose an integer q ε large enough so that (4.80)

N ε q ε 2 < η 0 , N ε q ε 2 × max 0≤i≤Nε W K (J i ) < ε. Now if j ∈ H i ε then d p (j, J i )
< η 3 and we deduce from (4.69) that for every λ ∈ K Rε \ Γ(j) we have

(4.81) d B (δ θ λ j , δ θ λ J i ) < ε.
Using (4.81) together with Lemmas 4.18, 4.17, and the bound (4.66), we deduce from (4.68) that d B (P, P ′′′ ) < Cε(|log ε| + 1), where (4.82)

P ′′′ = 1≤i≤Nε p i - K Rε δ θ λ J i dλ, where p i = P H i ε .
We now replace p i in the definition (4.82) by (4.83)

n i q ε 2 , where n i = q ε 2 p i .
Then Nε i=1 n i ≤ q ε 2 and (4.84)

1≤i≤Nε n i q ε 2 -p i < N ε q ε 2 < η 0 .
Then Lemma 4.17 implies that d B (P, P (4) ) < Cε(|log ε| + 1) where (4.85)

P (4) = 1≤i≤Nε n i q ε 2 - K Rε δ θ λ J i dλ. Now we let K ε = [-q ε R ε , q ε R ε ] 2 , and n 0 := q ε 2 -Nε i=1 n i , so that Nε i=0 n i = q ε 2 . Then we divide K ε in a collection L ε of q ε
2 identical subrectangles which are translates of K Rε , and for each 0 ≤ i ≤ N ε we choose n i subrectangles in an arbitrary way and call the collection of these subrectangles L ε,i , so that {L ε,i } 0≤i≤Nε is a partition of L ε .

Let us call J i,Rε the currents obtained from J i using Proposition 4.2. They satisfy (4.76) and (4.78). We claim that, as a consequence of the latter, we have for any L ∈ L ε,i that (4.86)

j ′ = J i,Rε on K Rε =⇒ d B - K Rε δ θ λ J i dλ, - K Rε δ θ λ j ′ dλ < Cε(|log ε| + 1).
This goes as follows: (i) Using (4.66) and Lemma 4.17, integrating on K (1-η 0 )Rε \Γ ( J i ) instead of K Rε induces an error of Cε. (ii) From (4.78), and (4.79) applied to θ λ J i by θ λ j ′ we have d B (δ θ λ J i , δ θ λ j ′ ) < ε and thus in view of Lemma 4.19 we may replace θ λ J i by θ λ j ′ in the integral with an error of Cε|log ε| at most. (iii) Using (4.66) and Lemma 4.17 again, we may integrate back on K Rε rather than on K (1-η 0 )Rε \ Γ ( J i ), with an additional error of Cε. this proves (4.86).

Then combining (4.86) with (4.85) and d B (P, P (4) ) < Cε(|log ε| + 1), using Lemma 4.19 we find d B (P, P (5) ) < Cε(|log ε| + 1), where (4.87)

P (5) = 1≤i≤Nε n i q ε 2 - K Rε δ θ λ Ji,Rε dλ,
and Ji,Rε is an arbitrary field in L p loc (R 2 , R 2 ) such that Ji,Rε = J i,Rε on K Rε , the constant C being independent of the choice of Ji,Rε .

We chose above an arbitrary J 0 in A 1 such that W K (J 0 ) < +∞. Let the sum in (4.87) range over 0 ≤ i ≤ N ε instead of 1 ≤ i ≤ N ε , this defines a measure P (6) such that, by (4.80), (4.88)

P (5) -P (6) ≤ n 0 q ε 2 ≤ η 0 ,
where we have used (4.84) and the fact that 1i p i = P (H ε c ) < η 0 , from (4.82), (4.65). Hence using Lemma 4.17 we have d B (P (5) , P (6) ) < ε and then d B (P, P (6) ) < Cε(|log ε| + 1).

We now define the vector field j ε : R 2 → R 2 by letting j ε (x) = J i,Rε (x -x L ) on every L ∈ L ε,i , 0 ≤ i ≤ N ε -where x L is the center of L and thus L = x L +K Rε -and by requiring j ε to be K ε -periodic. For every L ∈ L ε,i we have j ε (x L + •) = J i,Rε on K Rε , therefore we may choose Ji,Rε = j ε (x L + •) in (4.87) and then

Kε δ θ λ jε dλ = 0≤i≤Nε L∈L ε,i L δ θ λ jε dλ = 0≤i≤Nε L∈L ε,i K Rε δ θ λ jε(x L +•) dλ = 0≤i≤Nε n i K Rε δ θ λ Ji,Rε dλ.
Therefore we may summarize the discussion concerning P (5) , P (6) by writing (4.89) d B (P, P (6) ) < Cε(|log ε| + 1),

P (6) = - Kε δ θ λ jε dλ.
Note that since J i,Rε = 0 outside K Rε , and J i,Rε • τ = 0 on ∂K Rε we have, in K ε , (4.90)

j ε = 1≤i≤Nε L∈L ε,i J i,Rε (• -x L ), curl j ε = 2π p∈Λε δ p -1, j ε • τ = 0 on ∂K ε .
where Λ ε is a finite subset of the interior of K ε . This completes the construction of j ε .

Estimate of the energy. We have

W (j ε , 1 Kε ) = 0≤i≤Nε L∈L ε,i W (J i,Rε (• -x L ), 1 L ) = 0≤i≤Nε L∈L ε,i W (J i,Rε , 1 K Rε ).
From (4.76) applied to the J i 's we deduce that (4.91)

W (j ε , 1 Kε ) ≤ |K Rε | Nε i=0 n i (W K (J i ) + Cε) .
Now from (4.84) and the fact that 1i p i = P (H ε c ) < ε we deduce that n 0 ≤ N ε + q ε 2 ε, and then from (4.80) that n 0 ≤ Cεq ε 2 , where we have included W K (J 0 ) in the constant. This and (4.91), together with the estimates (4.84), (4.80) and the definition of p i in (4.82), implies that

W (j ε , 1 Kε ) ≤ q ε 2 |K Rε | Nε i=1 P (H i ε )W K (J i ) + Cε ,
and then from (4.70), and since

|K ε | = q ε 2 |K Rε |, that (4.92) W (j ε , 1 Kε ) ≤ |K ε | Hε W K (j) dP (j) + Cε ≤ |K ε | W K (j) dP (j) + Cε ,
using (4.65) and the fact from Proposition 4.1 that W K is bounded below. Choosing a sequence {ε} → 0 we thus obtain a sequence {R} tending to +∞, where R = q ε R ε and a sequence of currents {j R }, where j R = j ε such that (4.7) is satisfied -this is (4.90) -and lim sup

R→∞ W (j R , 1 K R ) |K R | ≤ W K (j) dP (j)
-this is (4.92). Moreover from (4.89)

we have P R → P , using the notations of Corollary 4.5. Thus Corollary 4.5 is proved.

Part II

From Ginzburg-Landau to the renormalized energy 5 The energy-splitting formula and the blow up

In this section, we return to the Ginzburg-Landau energy and prove an algebraic splitting formula on it already discussed in Section 1.8, as well as results on the splitting function. We recall that if lim ε→0 h ex /|log ε| > λ Ω , then h 0,ε may be used as the splitting function. Only when the limit is equal to λ Ω does one need to use h ε,N with N = N 0 instead. We recall h 0,ε is the minimizer of (1.35) and h ε,N is given by (1.40). We also introduce the notation of the appendix: for m ∈ (-∞, 1], H m denotes the minimizer of (5.1) min

h-1∈H 1 0 (Ω) (1 -m) Ω | -∆H + H| + 1 2 Ω |∇H| 2 + |H -1| 2 .
H m is the solution of an obstacle problem and its properties are studied in the appendix. In particular H m ≥ m and -∆H m + H m = m1 ωm , where ω m = {H m = m} is the so-called coincidence set. We have Proof. The minimization problem (1.40) has a unique minimizer h ε,N by convexity. On the other hand, by the theory of Lagrange multipliers, h ε,N is the minimizer of min

h-hex∈H 1 0 (Ω) 1 2 Ω |∇h| 2 + |h -h ex | 2 + λ Ω | -∆h + h|
for some number λ characterized by the fact that the unique minimizer satisfies

Ω | -∆h + h| = 2πN . But this minimizer is precisely h ex H m with m such that 2πN = h ex m|ω m | hence h ε,N = h ex H m . From Proposition A.1, m → |ω m | is continuous increasing and one to one from I = [1 -1 2λ Ω , 1] to [0, |Ω|],
hence if 2πN is between 0 and h ex |Ω|, then there exists a unique m ∈ I such that h ε,N = h ex H m , and m is a continuous increasing function of 2πN/h ex -hence of N -characterized by m|ω m | = 2πN h ex , and obviously |ω m | is too.

We will denote by m ε,N the m corresponding to h ε,N , and note that

(5.2) 0 < 1 - 1 2λ Ω ≤ m ε,N ≤ 1.
Replacing in (1.22) and integrating by parts the term ∇ ⊥ h µ • j(u, A 1 ), using the fact that h µ = h ex on ∂Ω, we find

G ε (u, A) = 1 2 h µ -h ex 2 H 1 (Ω) + 1 2 Ω |u| 2 -1 |∇h µ | 2 + + 1 2 Ω |∇ A 1 u| 2 + (curl A 1 -µ) 2 + (1 -|u| 2 ) 2 2ε 2 + Ω (h µ -h ex ) (curl j(u, A 1 ) + curl A 1 -µ) .
This yields (5.6), using the fact that µ(u, A 1 ) = curl j(u, A 1 ) + curl A 1 .

Using the particular choice of splitting function h µ = h ε,N in Lemma 5.2, we obtain the following result:

Proposition 5.3. Let 0 ≤ N ≤ hex|Ω| 2π
and let h ε,N be the corresponding minimizer of (1.40), then for any (u, A), denoting A 1,ε = A -∇ ⊥ h ε,N , and using the notation (1.41) we have

(5.7) G ε (u, A) = G N ε + F ε (u, A 1,ε ) - 1 2 Ω (1 -|u| 2 )|∇h ε,N | 2 ,
where

(5.8) F ε (u, A) = 1 2 Ω |∇ A u| 2 + (curl A -µ ε,N ) 2 + (1 -|u| 2 ) 2 2ε 2 + Ω (h ε,N -h ex -c ε,N )µ(u, A) + c ε,N Ω (µ(u, A) -µ ε,N ).
Proof. In (5.6), we replace Ω (h ε,N -h ex ) (µ(u, A 1,ε ) -µ ε,N ) by using the fact that by definition of h ε,N , on the support of µ ε,N , h ε,N -

h ex = h ex m ε,N -h ex = -1 2 | log ε ′ | + c ε,N and Ω µ ε,N = 2πN .
Note that the functional F ε depends on N but for simplicity we will not denote that dependence.

Dependence on N

We have the following Lemma 5.4. For N ∈ [0, |Ω|hex 2π ] we have

(5.9) dG N ε dN = 2πh ex (m -m 0,ε )
where m is the one-to-one function of N given by Lemma 5.1. Moreover, G N ε is minimized uniquely at N 0 and minimized among integers at N - 0 or N + 0 (or both) where N - 0 is the largest integer below N 0 and N + 0 the smallest integer above.

Proof. From (1.41) and Lemma 5.1 we have (5.10)

G N ε = πN | log ε ′ | + 1 2 h ex 2 H m -1 2 H 1 (Ω)
where m and N are related by 2πN = h ex m|ω m |. On the other hand, since H m minimizes (5.1) we have

(5.11) (1 -m) 2πN h ex + 1 2 H m -1 2 H 1 (Ω) ≤ (1 -m) 2πN ′ h ex + 1 2 H m ′ -1 2 H 1 (Ω) where 2πN ′ = h ex m ′ |ω m ′ |.
Reversing the roles of m and m ′ and taking m ′ = m + δ with δ → 0, we obtain

d dm H m -1 2 H 1 (Ω) = 4π h ex (m -1) dN dm
(in the sense of BV derivatives). Inserting into (5.10) we deduce

dG N ε dN = π| log ε ′ | + 2πh ex (m -1) = 2πh ex m -1 + | log ε ′ | 2h ex
hence the result (5.9) in view of (1.36). But m 0,ε is the m that corresponds to N 0 . It immediately follows with the monotonicity of

N → m that G N ε is decreasing in [0, N 0 ], increasing in [N 0 , |Ω|hex 2π ]
, hence minimized at N 0 , and minimized among integers at N - 0 or N + 0 .

For the rest of Section 5 and Section 6,

N ∈ {N - 0 , N + 0 }. Since 2πN 0 = Ω µ 0,ε ≤ |Ω| h ex -1 2 | log ε ′ | , see (1.37), it is clear that N - 0 ≤ N + 0 < |Ω|hex 2π
so Lemma 5.1 applies and the corresponding h ε,N are well-defined. We also record the following Lemma 5.5.

If N 0 ≫ 1 and N ∈ {N - 0 , N + 0 } then (5.12) lim ε→0 c ε,N | log ε ′ | = 0 lim ε→0 m ε,N = m λ ,
where λ = lim ε→0 hex |log ε| and m λ = 1 -1 2λ .

Proof. First we recall that m 0,ε = 1 -| log ε ′ | 2hex , hence by definition of λ, we have lim ε→0 m 0,ε = m λ .

Assume that N = N + 0 . We use the fact, seen in Lemma 5.1, that m and |ω m | are increasing functions of N . Thus, since

N 0 ≤ N ≤ N 0 + 1, we have |ω ε,N | ≥ |ω 0,ε | and, using 0 ≤ m ε,N -m 0,ε = 2πN + 0 h ex |ω ε,N | - 2πN 0 h ex |ω 0,ε | ≤ 2πN + 0 h ex |ω 0,ε | - 2πN 0 ≤ |ω 0,ε | ≤ 2π h ex |ω 0,ε | = m 0,ε N 0 ≤ 1 N 0 ,
because we always have m ≤ 1. Since we are in the regime N 0 ≫ 1, we find that m ε,N and m 0,ε have the same limit, that is m λ . The case N = N - 0 is treated analogously, and we deduce in both cases (5.13)

|m ε,N -m 0,ε | ≤ O( 1 N 0 ).
Assume then that h ex ≤ O(|log ε|), from (5.3) and (5.13), we deduce c ε,N = o(h ex ), which proves (5.12). If h ex ≫ |log ε|, then m 0,ε ∼ 1 and 2πN 0 ∼ h ex |Ω| hence combining with (5.3) and (5.13), we find c ε,N = O(1) which also implies the result in this case.

Blow-up procedure

We write for simplicity ω ε instead ω ε,N and m ε instead of m ε,N , when the precise value of N ∈ {N - 0 , N + 0 } does not need to appear explicitly. Assume (1.48) holds. Let

ℓ ε = 1 √ h ex ,
and, assuming 0 ∈ ω ε ⊂ Ω, write x = ℓ ε x ′ . Under this change of coordinates the domain Ω becomes Ω ′ ε and the subdomain ω ε becomes ω ′ ε , both becoming infinitely large as ε → 0this is obvious for Ω ′ ε and for ω ′ ε , it is proven to be a consequence of (1.48) in Proposition 5.6 below. We call

F ′ ε the expression of F ε (u, A 1,ε ) (see (5.8)) in terms of the rescaled unknowns u ′ (x ′ ) = u(x) and A ′ (x ′ ) = ℓ ε A 1,ε (x). It is given by (5.14) F ′ ε (u ′ , A ′ ) = 1 2 Ω ′ ε |∇ A ′ u ′ | 2 + 1 ℓ ε 2 |curl A ′ -m ε 1 ω ′ ε | 2 + (1 -|u ′ | 2 ) 2 2(ε ′ ) 2 - Ω ′ ε ζ ε ′ µ(u ′ , A ′ ) + c ε,N Ω ′ ε (µ(u ′ A ′ ) -m ε 1 ω ′ ε ), where ε ′ , ζ ε ′ are given by (5.15) ε ′ = ε ℓ ε , ζ ε ′ (x ′ ) = h ex -h ε,N (x) + c ε,N .
We also define the blown-up current in Ω ′ ε (5.16)

j ′ ε = curl (iu ′ , ∇ A ′ u ′ )
and the blow-up measure

µ ′ ε = µ(u ′ ε , A ′ ε ) and extend them by 0 outside Ω ′ ε . Note that j ′ ε (x ′ ) = ℓ ε j 1,ε (ℓ ε x ′ ) if j 1,ε denotes the current (iu ε , ∇ A 1,ε u ε ) in the original variables. The function h ε,N is in C 1,1
(Ω) and equal to h ex on ∂Ω. It attains its minimum h ex m ε,N on ω ε,N . From (5.3), (5.15)

c ε,N = min Ω (h ε,N -h ex ) + 1 2 | log ε ′ |, ζ ε ′ (x ′ ) = 1 2 | log ε ′ | -h ε,N (x) + min Ω h ε,N , thus ζ ε ′ ∈ C 1,1 (Ω ′ ε ), ζ ε ′ = c ε,N on ∂Ω ′ ε and ζ ε ′ attains its maximum on ω ′ ε . Moreover
(5.17) max

Ω ′ ε ζ ε ′ = 1 2 | log ε ′ |, ∇ζ ε ′ ∞ ≤ C | log ε ′ |,
this last assertion following from (5.20) in Proposition 5.6 below.

Additional results on the splitting function

In this subsection, we adapt some results from the appendix that we will need below. In the appendix we introduce an ellipse E Q of measure 1 and a function

U Q ≥ 0 defined in R 2 such that ∆U Q = ∆Q 2 1 R 2 \E Q , {U Q = 0} = E Q ,
where Q is the quadratic form D 2 h 0 (x 0 ) as introduced in (1.31).

The following proposition can be applied to N ∈ {N 0 , N - 0 , N + 0 }. In either case we denote by ω ε the associated coincidence set, and by ω ′ ε its blow-up.

Proposition 5.6. Assume (1.48) holds. Let h ε,N be as above with |N -N 0 | ≤ 1.

1. We have N 0 ≫ 1, and if h ex -H c 1 ≪ |log ε| then N 0 ≪ h ex and

(5.18)

h ex -H c 1 ∼ λ Ω N 0 log h ex N 0 . 2. For every x ∈ Ω (5.19) d(x, ω ε ) ≤ C | log ε ′ | h ex , |Ω\ω ε | ≤ C | log ε ′ | h ex . Moreover (5.20) ∇h ε,N L ∞ (Ω) ≤ C h ex | log ε ′ |. 3. If K is any compact subset of (λ Ω , +∞), (5.21) lim δ→0 |{x | d(x, ∂ω ε ) < δ}| |ω ε | = 0, lim δ→0 |{h ex m ε,N h ε,N < h ex m ε,N + δh ex }| |ω ε | = 0
uniformly with respect to ε such that hex |log ε| ∈ K.

4. If h ex /|log ε| → λ Ω then there exists {L ε } ε such that for any δ, M > 0 and if ε is small enough

h ε,N -min Ω h ε,N h ex L ε 2 ≥ M ⊂ x 0 + L ε {U Q ≥ M -δ}, h ε,N -min Ω h ε,N h ex L ε 2 ≤ M ⊂ x 0 + L ε {U Q ≤ M + δ}, {d(x, ω Q c ) > δL ε } ⊂ ω ε ⊂ {d(x, ω Q ) < δL ε }, (5.22)
where

ω Q = x 0 + L ε E Q and x 0 is defined in (1.31).
Moreover there is a constant C Ω depending only on Ω such that

(5.23) L 2 ε | log L ε | ∼ C Ω h ex -λ Ω |log ε| h ex .
In particular log |log ε|

|log ε| ≪ L 2 ε | log L ε | and ℓ ε ≪ L ε , and from (5.22) L 2 ε |ω Q | ∼ |ω ε |, {x | d(x, ∂ω ε ) ≤ δℓ ε } ≪ |ω ε |. 5. For any R > 0, (5.24) {x | d(x, (ω ′ ε ) c ) > R} ∼ |ω ′ ε | as ε → 0, i.e. ω ′ ε satisfies (1.14).
6. We have

(5.25) G N ε = G N 0 ε + O h ex N 0 ≤ Ch ex | log ε ′ |. If |log ε| 4 ≪ h ex ≪ 1 ε 2 , (5.26) G N ε = 1 2 h ex |Ω|| log ε ′ | + o(h ex ).
Proof. Proof of item 1: We apply Proposition A.

1 to m 0,ε = 1 -| log ε ′ | 2hex .
We denote by L ε the L m corresponding to m 0,ε given by Proposition A.1. In particular we have |ω 0,ε | ∼ L 2 ε . In that proposition h 0 denotes min h 0 and is equal to 1 -

1 2λ Ω . Moreover h ex -λ Ω |log ε| ≫ log |log ε| is equivalent to 1 2(1-m 0,ε ) - 1 2(1-h 0 ) ≫ log |log ε| |log ε| which in turn is equivalent to m 0,ε -h 0 ≫ log |log ε| |log ε| . From (A.7) (5.27) L 2 ε | log L ε | ∼ 2π(m 0,ε -h 0 )/h 0 = 2π h 0 1 2λ Ω - | log ε ′ | 2h ex . Then, since |ω 0,ε | ∼ L 2 ε and N 0 = h ex m 0,ε |ω 0,ε | we have L 2 ε | log L ε | ∼ 1 2 |ω 0,ε || log |ω 0,ε || ∼ 1 2 N 0 m 0,ε hex | log N 0 m 0,ε hex
| and inserting into (5.27) we find (5.28) 1 2

N 0 m 0,ε h ex | log N 0 m 0,ε h ex | ∼ 2π h 0 1 2λ Ω - | log ε ′ | 2h ex = π λ Ω h 0 h ex h ex -λ Ω |log ε| + 1 2 log h ex .
It follows, since m 0,ε ≥ h 0 > 0, that if (1.48) holds, we must have N 0 ≫ 1, for otherwise the left-hand side is O(log |log ε|) while the right-hand side is ≫ log |log ε|. Moreover, if h ex -H c 1 ≪ |log ε| then m 0,ε ∼ 1 -1 2λ Ω = h 0 and inserting into (5.28) and rearranging terms, we obtain N 0 ≪ h ex and (5.18).

Proof of item 2. From (1.36), m 0,ε = 1

-| log ε ′ | 2hex hence using Proposition A.1 we have ∇h 0,ε ∞ = h ex ∇H m 0,ε ∞ ≤ C h ex | log ε ′ |. If |N -N 0 | ≤ 1 then from (5.12) we have |m 0,ε -m ε,N | ≪ | log ε ′ |/h ex and therefore m ε,N ≤ 1 -| log ε ′ | hex
for ε small enough. Proposition A.1 applied to m ε,N yields the same bound for ∇h ε,N , and (5.19) follows similarly from Proposition A.1 using But,from (1.36), if this is the case then m 0,ε belongs to a compact subset K of (h 0 , 1). Then from (5.12) and if ε is small enough the same is true of m ε,N , and the result follows from Proposition A.1 applied to m ε,N .

m ε,N ≤ 1 -| log ε ′ | hex . Proof of item 3. First note that if h ex /|log ε| is bounded above as ε → 0 then |log ε| ∼ | log ε ′ | hence h ex /| log ε ′ | is bounded as well. Moreover, since | log ε ′ | ≤ |log ε|, if h ex /|log ε| ≥ α > λ Ω then h ex /| log ε ′ | ≥ α hence if h ex /|log ε| belongs to a compact subset K of (λ Ω , +∞), then h ex /| log ε ′ | belong to a (different) compact subset K ′ of (λ Ω , +∞).
Proof of items 4,5. This is again Proposition A.1. Indeed

h ex /|log ε| → λ Ω implies that m 0,ε → h 0 = 1 -1
2λ Ω hence lim ε→0 m ε,N = h 0 and (A.7) holds, and for any δ, M > 0, (A.8) also if ε is small enough. It is easy to check using (5.12), (1.36) and (A.3) that (A.7) implies (5.23), and (5.22)

is (A.8) since h ε,N -min Ω h ε,N = h ex (H m ε,N -m ε,N ).
To prove (5.24) we distinguish the case m ε,N → h 0 from the case where m ε,N is bounded away from h 0 . In the latter and using Proposition A.1, |ω ε,N | is bounded away from 0 and

|{x | d(x, (ω ′ ε ) c ) > R}| |ω ′ ε | = |ω R | |ω ε | , where ω R = {x | d(x, ω ε c ) > Rℓ ε }. Since Rℓ ε → 0 we have |ω R | -|ω ε | → 0, proving (5.24) in this case. If m ε,N → h 0 then m 0,ε → h 0 , i.e. h ex /| log ε ′ | → λ Ω , or equivalently h ex /|log ε| → λ Ω . Then L ε ≫ ℓ ε and therefore, using (5.22), {d(x, ω Q c ) > δL ε } ⊂ ω R ⊂ {d(x, ω Q ) < δL ε } holds for any R, δ > 0 if ε is small enough. It follows, since |ω Q | = L 2 ε , that |ω R | ∼ |ω Q |
as ε → 0 for any R > 0, which implies in particular (5.24).

Proof of item 6. Combining (5.9) with (5.13) it follows that

|G N ε -G N 0 ε | = N N 0 2πh ex (m -m 0,ε ) ≤ 2πh ex N 0
where we have used the fact that N → m is increasing and

|N -N 0 | ≤ 1.
The upper bound part of (5.26) follows from (5.25) by noting that

| log ε ′ | µ 0,ε + h 0,ε -h ex 2 H 1 ≤ |Ω|| log ε ′ |,
which follows from the minimality of H m 0,ε in (5.1), using 1 as a test function. Finally, for the lower bound part of (5.26) we note that from (5.19)

| log ε ′ | µ 0,ε + h 0,ε -h ex 2 H 1 ≥ | log ε ′ ||ω 0,ε |m 0,ε ≥ | log ε ′ |   |Ω| -C | log ε ′ | h ex   1 - | log ε ′ | 2h ex ≥ | log ε ′ ||Ω| + o(h ex ), if h ex ≫ |log ε| 4 .
Remark 5.7. In (5.18), we have recovered the formula (9.88) from [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF].

A priori bounds

Lemma 5.8.

If (u ε , A ε ) minimizes G ε then (5.29) G ε (u ε , A ε ) ≤ Ch ex | log ε ′ |.
Moreover, for any (u ε , A ε ) satisfying (5.29) we have, if

N ∈ {N - 0 , N + 0 }, (5.30) G ε (u ε , A ε ) = G N ε + F ε (u ε , A 1,ε ) + o(N ),
where F ε is defined in (5.8).

Mass displacement

There are two problems which arise when trying to apply the abstract scheme described in Section 2 to the Ginzburg-Landau energy. The first, and less problematic one, is that F ε is not translation invariant. Indeed 1 ω ′ ε and ζ ε ′ are constant only in the subdomain ω ′ ε . Therefore we need to reduce to integrating on ω ′ ε rather than Ω ′ ε . The second, more delicate problem, is that the integrand in F ε is not positive, and not even expected to be bounded below uniformly as ε → 0: Each vortex creates in F ε two terms which get infinitely large as ε → 0 and which balance each other. To capture the difference in the limit we need to absorb the negative part in the positive part to obtain an essentially positive integrand (this will by the way solve in essence the first issue). Note that the cancellation is not a pointwise cancellation of the different terms: While the negative contribution is very concentrated near each vortex, the positive one is more spread out.

The method, introduced in [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF] is the same that we used in Proposition 4.9 in a simpler setting, so the reader can refer to that section for an idea of it.

Let us denote the free energy functional (6.1)

G 0 ε (u, A) = Ωε e ε , e ε = 1 2 |∇ A u| 2 + 1 2 (curl A) 2 + 1 4ε 2 1 -|u| 2 2 ,
where Ω ε is a smooth domain depending on ε and large as ε → 0. We now state the result of [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF] for the sake of completeness. It contains (in a slightly different form) Theorems 1 and 2, as well as Corollary 1.1 of [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF].

First, f + and f -will denote the positive and negative parts of a function or measure, both being positive functions or measures. If f and g are two measures, we will write f ≤ g in the sense of measures to mean that g -f is a positive measure.

For any set E in the plane, E will denote the 1-tubular neighborhood of E in Ω ε i.e.

E = {x ∈ Ω ε , dist (x, E) ≤ 1}.
This way

∂Ω ε = {x ∈ Ω ε , dist (x, ∂Ω ε ) ≤ 1}.
For any function v on Ω ε we denote (notice the absolute value)

v(x) = sup y∈B(x,1)∩Ωε |v(y)|.
Note that here the choice of the number 1 is arbitrary.

Theorem 6 ([SS3]). Let {Ω ε } ε>0 be a family of bounded open sets in R 2 . Assume that {(u ε , A ε )} ε , where (u ε , A ε ) is defined over Ω ε , satisfies for some 0 < β < 1 small enough (6.2) G 0 ε (u ε , A ε ) ≤ ε -β .
Then for any small enough ε, there exists a measure g ε defined over Ω ε and a measure ν ε depending only on u ε of the form 2π i d i δ a i for some points a i ∈ Ω ε and some integers d i such that, C denoting a generic constant independent of ε:

1. We have

(6.3) µ(u ε , A ε ) -ν ε (C 0,1 0 (Ωε)) * ≤ C √ εG ε (u ε , A ε ),

The following inequality holds

-C ≤ g ε ≤ e ε + 1 2 |log ε|(ν ε ) -. 3. For any measurable set E ⊂ Ω ε , (6.4) (g ε ) -(E) ≤ C e ε ( E) |log ε| , (g ε ) + (E) ≤ Ce ε ( E).

Letting

f ε = e ε - 1 2 |log ε|ν ε ,
for every Lipschitz function ξ vanishing on ∂Ω ε and every 0 < η ≤ 1 we have

(6.5) Ωε ξ d(f ε -g ε ) ≤ C Ωε ∇ξ d|ν ε | + (β + η) d(g ε ) + + | log η| 2 η dx + C d ∂Ωε ξe ε .
5. For any measurable set E ⊂ Ω ε and every 0 < η ≤ 1 we have

(6.6) |ν ε |(E) ≤ C η(g ε ) + ( E) + 1 η | E| + e ε ( E ∩ ∂Ω ε ) |log ε| , |ν ε |(E) ≤ C e ε ( E) |log ε| . 6. Assuming |u ε | ≤ 1 in Ω ε , then for every ball B R of radius R such that B R+C ⊂ Ω ε and every p < 2, B R |j ε | p ≤ C p (g ε ) + (B R+C ) + R 2 .
7. Assume |u ε | ≤ 1, that dist (0, ∂Ω ε ) → +∞ as ε → 0 and that for any R > 0 (6.7) lim sup

ε→0 g ε (U R ) < +∞,
where U R is any family satisfying (1.4)-(1.5).

Then, up to extraction of a subsequence and for any p < 2, the vorticities

{µ(u ε , A ε )} ε converge in W -1,p loc (R 2 ) to a measure ν of the form 2π p∈Λ δ p , where Λ is a discrete subset of R 2 , the currents {j(u ε , A ε )} ε converge weakly in L p loc (R 2 , R 2 ) to j, and the induced fields {curl A ε } ε converge weakly in L 2 loc (R 2 ) to h which are such that curl j = ν -h, in R 2 .
8. If we replace the assumption (6.13) by the stronger assumption

lim sup ε→0 g ε (U R ) < CR 2 ,
where C is independent of R, then the limit j of the currents satisfies, for any p < 2, (6.8) lim sup

R→+∞ - U R |j| p dx < +∞.
Moreover for every family χ U R satisfying (1.3) we have

(6.9) lim inf ε→0 R 2 χ U R |U R | dg ε ≥ W (j, χ U R ) |U R | + 1 2 - U R h 2 + γ 2π - U R h + o R (1),
where γ is the constant in (1.52) and o R (1) is a function tending to 0 as R → +∞.

The main point is that g ε is a modification of f ε = e ε -1 2 |log ε|ν ε with a small error (measured by (6.5)) which, contrarily to f ε , is bounded below according to item 2. Now, using the notation of Section 5, we blow up at the scale

ℓ ε = √ h ex , letting x ′ = ℓ ε x, ε ′ = ε/ℓ ε , and (6.10) u ′ ε (x ′ ) = u ε (x), A ′ ε (x ′ ) = ℓ ε (A ε (x) -∇ ⊥ h ε,N (x)
). Then we deduce from Theorem 6 applied in

{Ω ′ ε } ε to {(u ′ ε , A ′ ε )} ε Proposition 6.1. Assume that as ε → 0 h ex |log ε| → λ ∈ [λ Ω , +∞], log |log ε| ≪ h ex -λ Ω |log ε| and h ex ≤ 1 ε β ,
where β is small enough, and that

G ε (u ε , A ε ) ≤ Ch ex | log ε ′ |.
We also assume that (1.47) holds. Then there exists N ∈ {N - 0 , N + 0 } such that there exist measures {g ′ ε } ε defined on Ω ′ ε satisfying the following four properties, for any family {U R } R of sets satisfying (1.4), (1.5) and any family of functions

{χ U R } R satisfying (1.3). 1. g ′ ε is bounded below by a -not necessarily positive -constant C independent of ε. 2. Defining F ′ ε as in (5.14) we have, writing ω ′ ε for ω ′ ε,N and letting ω′ ε = {x | d(x, ω ′ ε c ) ≥ 2}, (6.11) lim inf ε→0 F ′ ε (u ′ ε , A ′ ε ) -g ′ ε (ω ′ ε ) |ω ′ ε | ≥ 0.
and for any 1 ≤ p < 2, (6.12)

Ω ′ ε |j ′ ε | p ≤ C p F ′ ε (u ′ ε , A ′ ε ) + |ω ′ ε | , 1 ℓ ε 2 Ω ′ ε |curl A ′ ε -m ε 1 ω ′ ε | 2 ≤ C F ′ ε (u ′ ε , A ′ ε ) + |ω ′ ε | , where j ′ ε := (iu ′ ε , ∇ A ′ ε u ′ ε ). 3. If {x ′ ε } ε satisfies dist (x ′ ε , (ω ′ ε ) c ) → +∞ and (6.13) ∀R > 0, lim sup ε→0 g ′ ε (x ′ ε + U R ) < +∞,
then, up to extraction of a subsequence, the translated vorticities {µ

′ ε (x ′ ε + •)} ε -where µ ′ ε := µ(u ′ ε , A ′ ε ) -converge in W -1,p loc (R 2
) to a measure ν of the form 2π p∈Λ δ p , where Λ is a discrete subset of R 2 , and the translated currents

{j ′ ε (x ′ ε + •)} ε converge in L p loc (R 2 , R 2 )
for any p < 2 to j such that div j = 0 and curl j = ν -m λ . 4. If, in addition, we assume that there exists C > 0 such that for any R > 0 (6.14) lim sup

ε→0 g ′ ε (x ′ ε + U R ) |U R | < C, then j ∈ A m λ (A m λ is defined in Definition 1.1) and (6.15) lim sup R→+∞ - U R |j| p dx < +∞, and
(6.16) lim inf R→∞ lim inf ε→0 1 |U R | χ U R (x -x ′ ε ) dg ′ ε (x) ≥ lim sup R→∞ W (j, χ U R ) |U R | + γ 2π m λ .
Proof of Proposition 6.1. We apply Theorem 6 to

{(u ′ ε , A ′ ε )} on Ω ′ ε . First, we check that (6.2) holds. Since A ′ ε (x ′ ) = ℓ ε (A ε -∇ ⊥ h ε,N )(x), G 0 ε ′ (u ′ ε , A ′ ε ) = 1 2 Ω |∇ Aε-∇ ⊥ h ε,N u ε | 2 + ℓ 2 ε (curl A ε -∆h ε,N ) 2 + (1 -|u ε | 2 ) 2 2ε 2 ≤ 2G ε (u ε , A ε ) + Ω |∇h ε,N | 2 + 1 h ex |h ex + µ ε,N -h ε,N | 2 , using the fact that -∆h ε,N = µ ε,N -h ε,N . If N ∈ {N - 0 , N + 0 }, then (5.25) implies that h ex -h ε,N 2 H 1 ≤ Ch ex | log ε ′ |, while Ω µ 2 ε,N h ex = Ω h ex m ε 2 ≤ Ch ex . Therefore, if h ex ≤ ε -β and G ε (u ε , A ε ) ≤ Ch ex | log ε ′ |, then G 0 ε ′ (u ′ ε , A ′ ε ) ≤ Cε -2β if ε is small enough. We conclude by noting that if h ex ≤ ε -1 2 then ε ′ ≤ ε 3/4 hence if β is small enough, Theorem 6 applies.
It gives us a spread out density (g ′ ε ) 0 of e ′ ε and a measure ν ′ ε = 2π i d i δ a i depending only on u ′ ε , hence on u ε , but not on our choice of N . Noting that 1 2π ν ′ ε (Ω ′ ε ) is an integer, we let (6.17)

N = N + 0 if 1 2π ν ′ ε (Ω ′ ε ) ≥ N + 0 N = N - 0 if 1 2π ν ′ ε (Ω ′ ε ) ≤ N - 0
and this is the choice of N we make from now on.

Testing

µ(u ′ ε , A ′ ε ) -ν ′ ε against ζ ε ′ -c ε,N which is in C 0,1 0 (Ω ′ ε )
, we find, in view of (5.14) (6.18)

F ′ ε (u ′ ε , A ′ ε ) = 1 2 Ω ′ ε |∇ A ′ ε u ′ ε | 2 + 1 ℓ ε 2 |curl A ′ ε -m ε 1 ω ′ ε | 2 + (1 -|u ′ ε | 2 ) 2 2(ε ′ ) 2 - Ω ′ ε ζ ε ′ dν ′ ε + c ε,N Ω ′ ε (dν ′ ε -m ε 1 ω ′ ε ) + o(1). Note that m ε |ω ′ ε | = µ ε,N (ω ε,N ) = 2πN
. But, from the choice (6.17), if 1 2π ν ′ ε (Ω ′ ε ) ≥ N + 0 then we have N = N + 0 ≥ N 0 and thus c ε,N ≥ 0 by (5.4), hence

c ε,N Ω ′ ε (dν ′ ε -m ε 1 ω ′ ε ) ≥ 2πc ε,N (N + 0 -N ) = 0.
If on the other hand

1 2π ν ′ ε (Ω ′ ε ) ≤ N - 0 then by (5.4) we have c ε,N ≤ 0 hence c ε,N Ω ′ ε (dν ′ ε -m ε 1 ω ′ ε ) ≥ 2πc ε,N (N - 0 -N ) = 0.
So in both cases, the last term in (6.18) is nonnegative and we are led to

(6.19) F ′ ε (u ′ ε , A ′ ε ) ≥ 1 2 Ω ′ ε |∇ A ′ ε u ′ ε | 2 + 1 ℓ ε 2 |curl A ′ ε -m ε 1 ω ′ ε | 2 + (1 -|u ′ ε | 2 ) 2 2(ε ′ ) 2 - Ω ′ ε ζ ε ′ dν ′ ε + o(1).
1

To the spread out density (g ′ ε ) 0 we add

(g ′ ε ) 1 = 1 ℓ ε 2 curl A ′ ε -m ε 1 ω ′ ε 2 - 1 2 (curl A ′ ε ) 2
and call the result g ′ ε . Using the notation (6.1) we may rewrite (6.19) as (6.20)

F ′ ε (u ′ ε , A ′ ε ) ≥ Ω ′ ε e ′ ε + (g ′ ε ) 1 - Ω ′ ε ζ ε ′ dν ′ ε + o(1).
We now check that g ′ ε satisfies the required properties. Since m ε ∈ (0, 1] we have that (g ′ ε ) 1 is bounded below by a universal constant if, for instance, ℓ ε < 1/2, which is true for small ε since lim ε→0 ℓ ε = 0.

1. Since (g ′ ε ) 0 and (g ′ ε ) 1 are both bounded below by a constant independent of ε ′ , so is g ′ ε .

2. Item 2 will be proven in full below. We prove here the case λ Ω < λ < +∞, which is technically simpler. Using

f ε ′ = e ′ ε -1 2 | log ε ′ |ν ′ ε , rewrite (6.20) as (6.21) F ′ ε (u ′ ε , A ′ ε ) ≥ Ω ′ ε ξ df ′ ε + Ω ′ ε (1 -ξ)e ′ ε + Ω ′ ε (g ′ ε ) 1 + o(1)
,

where ξ = 2ζ ε ′ /| log ε ′ |.
Then, from (6.5) applied to -ξ and since from (5.17), we have ξ = 1 on ω ′ ε , it follows that (6.22)

Ω ′ ε ξ df ε ′ ≥ Ω ′ ε ξ d(g ′ ε ) 0 -∇ξ ∞ Ω ′ ε \ω ′ ε d(|ν ′ ε | + (g ′ ε ) + 0 ).
Since again ξ = 1 on ω ′ ε , we may write

(6.23) ξ d(g ′ ε ) 0 = (g ′ ε ) 0 (ω ′ ε ) + Ω ′ ε \ω ′ ε ξ d(g ′ ε ) + 0 - Ω ′ ε \ωε ξ d(g ′ ε ) - 0 . Let A = Ω ′ ε \ ω ′ ε ∩ {ξ > 1/2}
. Since, from (5.17), ∇ξ ∞ ≤ Cℓ ε , we have A ⊂ Ã := {ξ > 1/4} \ ω′ ε if ε is small enough. Using (6.6) with η = 1 we deduce that (6.24

) |ν ′ ε | Ω ′ ε \ ω ′ ε ∩ {ξ > 1/2} ≤ C(g ′ ε ) + 0 ( Ã) + C|Ω ′ ε | ≤ C Ω ′ ε \ω ′ ε ξ d(g ′ ε ) + 0 + Ch ex .
The same bound is trivially true for (g

′ ε ) + 0 Ω ′ ε \ ω ′ ε ∩ {ξ > 1/2} .
On the other hand, letting B = Ω ′ ε \ ω ′ ε ∩ {ξ ≤ 1/2}, we have ξ < 2/3 on B if ε is small enough, therefore using (6.6) we find that (6.25)

|ν ′ ε | Ω ′ ε \ ω ′ ε ∩ {ξ ≤ 1/2} ≤ Ce ε ({ξ < 2/3}) ≤ C (1 -ξ)e ′ ε ,
1 At this point we could also choose N to be 1 2π ν ′ ε (Ω ′ ε ) i.e. the total degree of uε. Then the term in factor of cε,N in (6.18) is 0, and we still have (6.19). We may then proceed with an unchanged proof of the lower bound with that N . Alternatively, we may analyse further the positive term cε,N R

Ω ′ ε (dν ′ ε -mε1 ω ′ ε ) that has been discarded.
and the same bound is true for (g , using (6.4). We have thus obtained that the negative terms in the right-hand side of (6.22) can be absorbed (since ∇ξ ∞ ≤ Cℓ ε = o(1)) in the positive terms of (6.21) and (6.23). There remains to absorb the negative term of the right-hand side of (6.23).

′ ε ) + 0 Ω ′ ε \ ω ′ ε ∩ {ξ > 1/2}
Moreover for any n > 0 and letting A n = {ξ > 1 -1/n} we have if ε is small enough that A c n ⊂ A c 2n and C/|log ε| < 1/4n. Thus, splitting into A n and A c n , and using the fact that (g ′ ε ) - 0 ≤ C and (6.4) we have (6.26)

Ω ′ ε \ωε ξ d(g ′ ε ) - 0 ≤ C |log ε| e ′ ε (A c 2n ) + (g ′ ε ) - 0 (A n \ ωε ) ≤ 1 2 (1 -ξ)e ′ ε + C|A n \ ωε |.
Using the fact that ∇ξ ∞ = o(1), and the fact that (g ′ ε ) 1 ≥ 0 outside ω ′ ε and is bounded below by -C in ω ′ ε \ ω′ ε we deduce from (6.21)-(6.26) that for any n > 0

F ′ ε (u ′ ε , A ′ ε ) ≥ (g ′ ε ) 0 (ω ′ ε ) + (g ′ ε ) 1 (ω ′ ε ) -C|A n \ ωε | + o(|log ε|).
Then (6.11) follows by dividing by |ω ′ ε | and taking the limit first as ε → 0 and then as n → +∞, noting that from (5.15), (5.3), (1.36), we have

A n \ ωε = {x ∈ ω ′ ε | d(x, ω ′ ε c ) < 2} ∪ x ∈ ω ′ ε | m ε,N < h ε,N (ℓ ε x) h ex < m ε,N + | log ε ′ | 2n
and thus -using Lemma 5.6, (5.21) -that lim sup ε→0 |An\ωε| |log ε| tends to 0 as n → +∞.

3. For notational simplicity, we assume x ′ ε = 0. Since (g ′ ε ) 1 is bounded below, U R g ′ ε being bounded above independently of ε for any R > 0 implies that the same is true for U R (g ′ ε ) 0 hence (6.7) is satisfied and from item 7 in Theorem 6 we deduce the convergence of the currents (locally weak L p ), vorticities (W -1,p loc ) and fields (weak L 2 loc ) to j, ν and h satisfying curl j = ν -h. Since we assume (1.47) we have div j(u ε , A ε ) = 0. But a direct computation (see Lemma 5.9) gives div j 1,ε = div j(u ε , A ε ) + div ((1 -|u ε | 2 )∇ ⊥ h ε,N ) = 0 in ω ε,N since ∇h ε,N = 0 in ω ε,N . At the blown-up scale this means that when d(0, (ω ′ ε ) c ) → +∞, we have for any R > 0 and ε small enough div j ′ ε = 0 in B R . We deduce that div j ′ ε → 0 strongly in W -1,p loc (R 2 ) and thus div j = 0 . Since curl j

′ ε = µ ′ ε + curl A ′ ε we also have that curl j ′ ε is compact in W -1,p loc (R 2 ) for p < 2. It follows that j ′ ε is compact in L p loc and the convergence of j ′ ε is strong. Moreover we have (6.27) lim sup ε→0 - U R 1 ℓ ε 2 curl A ′ ε -m ε 1 ω ′ ε 2 - 1 2 (curl A ′ ε ) 2 < +∞
for every R > 0, from which we easily deduce that the limit h of {curl A ′ ε } ε is equal to m λ , where λ ∈ [λ Ω , +∞] is the limit of h ex /|log ε|. Indeed under the hypothesis d(0, (ω ′ ε ) c ) → +∞, we have m ε 1 ω ′ ε (x ′ ε + •) → m λ locally from (5.12). We thus have curl j = ν -m λ .

4. Again we assume x ′ ε = 0. The hypothesis that lim sup ε→0 -U R g ′ ε is bounded above independently of R implies as above that the same is true for lim sup ε→0 -U R (g ′ ε ) 0 .

Then item 8 of Theorem 6 applies and we obtain (6.8), hence (6.15), and from (6.9) we get (6.28) lim inf

ε→0 χ U R |U R | d(g ′ ε ) 0 ≥ W (j, χ U R ) |U R | + 1 2 - U R m λ 2 + γ 2π - U R m λ + o R (1),
where lim R→+∞ o R (1) = 0. On the other hand (6.27) implies that curl

A ′ ε → m λ locally strongly in L 2 , hence lim inf ε→0 χ U R |U R | d(g ′ ε ) 1 ≥ - 1 2 m λ 2 .
Together with (6.28), this proves (6.16).

To prove that j ∈ A m λ we integrate curl j = ν -m λ over B R+t and B R-t to obtain

πm λ (R -t) 2 + ∂B R-t j • τ = ν(B R-t ) ≤ ν(B R ) ≤ ν(B R+t ) = πm λ (R + t) 2 + ∂B R+t j • τ.
Then, a mean-value argument and (6.15), with p = 1, allow to deduce the existence of

t ∈ [0, √ R] such that ∂B R-t ∪∂B R-t |j| ≤ CR 3/2 ,
and we deduce that ν(B R ) ∼ πm λ R 2 as R → +∞, and so j ∈ A m λ .

It remains to prove item 2 in all generality using Theorem 6.

Proof of item 2 in Proposition 6.1 in the general case. Recall that from its definition (5.15), ζ ′ ε achieves its maximum 1 2 | log ε ′ | on ω ′ ε , and is equal to c ε,N on ∂Ω ′ ε . Thus

ξ = 2 ζ ′ ε | log ε ′ |
achieves its maximum 1 on ω ′ ε and its minimum c ε,N /| log ε ′ |, which from (5.12) is o(1), on ∂Ω ′ ε .

We let

E 1 = {x ∈ Ω ′ ε , ξ > 1 -δ}, E 2 = {x ∈ Ω ′ ε , ξ > 1 -2δ}, and recall that ω′ ε = {x ∈ ω ′ ε , dist (x, (ω ′ ε ) c ) ≥ 2}.
We will need the following lemma. Lemma 6.2. For any M > 0 and ε small enough there exist δ > 0 such that

(6.29) δ > M | log ε ′ | , E 1 ⊂ E 2 , |ω ′ ε | > M | E 2 \ ω′ ε |.
Proof. First we treat the case where h ex /|log ε| → λ ∈ (λ Ω , +∞]. In this case, using (5.21) in Proposition 5.6 we find

lim δ→0 |{1 -2δh ex /| log ε ′ | < ξ < 1}| |ω ′ ε | = 0,
uniformly with respect to ε ≤ ε 0 , if ε 0 is small enough. Since h ex /| log ε ′ | has the same limit as h ex /|log ε|, it is bounded away from 0 as ε → 0. We deduce easily that δ may be chosen small enough so that (6.30)

|{1 -3δh ex /| log ε ′ | < ξ < 1}| |ω ′ ε | ≤ 2 M ,
for any ε ≤ ε 0 .

Then we note, since from (5.15) we have |∇ξ| ≤ | log ε ′ | -1 2 , that E 1 ⊂ E 2 holds for ε small enough, as well as E 2 ⊂ {1 -3δh ex /| log ε ′ | < ξ}. It follows, in view of (6.30) and since ω ′ ε = {ξ = 1}, that for ε small enough (6.31)

| E 2 \ ω ′ ε | |ω ′ ε | ≤ 2 M .
To conclude we note that

|ω ′ ε \ ω′ ε | ≤ |{d(x, ∂ω ′ ε ) ≤ 2ℓ ε }| = o(|ω ′ ε |)
, where we have used (5.21) and scaled. Thus if ε is small enough and using (6.31) we find

|ω ′ ε | > M | E 2 \ ω′ ε |. If h ex /|log ε| → +∞ then we choose δ = 1/2. As above, ∇ξ ∞ → 0 implies that E 1 ⊂ E 2 for ε small enough, and of course δ > M/| log ε ′ | is satisfied for ε small enough depending on M . Moreover (5.19) implies that d(ω ε , Ω c ) → 0 as ε → 0, and d(ω ε , (ω ε ) c ) = 2ℓ ε → 0 as well. Therefore |Ω \ ωε | = o(|ω ε |) and after scaling |Ω ′ ε \ ω′ ε | = o(|ω ′ ε |), and in particular | E 2 \ ω′ ε | = o(|ω ′ ε |). If h ex /|log ε| → λ Ω , we choose δ = cL 2
ε with c > 0 independent of ε to be chosen small enough depending on M . From (5.22) applied with M = c

2λ Ω and δ = η 2λ Ω , and rewritten in terms of

ξ(x) = 1 -2 h ε,N (ℓ ε x) -min Ω h ε,N | log ε ′ | we deduce that 1 -ξ(x) ≤ cL 2 ε h ex λ Ω | log ε ′ | =⇒ U Q ℓ ε x -x 0 L ε ≤ c + η 2λ Ω .
Therefore, since δ = cL 2 ε and λ Ω = lim ε h ex /| log ε ′ |, for any η > 0 and if ε is small enough then

1 -ξ(x) ≤ 2δ =⇒ U Q ℓ ε x -x 0 L ε ≤ c + η 2λ Ω .
Since ℓ ε /L ε → 0 from Proposition 5.6 this in turn implies that if ε is small enough and |y| < 1 then

U Q ℓ ε x -x 0 L ε + ℓ ε L ε y < c + 2η 2λ Ω .
Using (5.22) again, this implies that ξ(x + y) > 1 -2(c + 3η)L 2 ε . Choosing η = c/6 we have proven that E 2 ⊂ {ξ > 1 -3δ} if ε is small enough. Similarly, we may prove that E c 2 ⊂ {ξ < 1 -δ} = E c 1 . We further deduce that, as ε → 0, (6.32)

| E 2 | < L ε ℓ ε 2 U Q < c 2λ Ω + o L m ℓ ε 2 .
Moreover from item 4 in Proposition 5.6 we have |ω| ∼ Lm ℓε 2

. With (6.32), and since

|{U Q < 2c λ Ω }| → |E Q | = 1
as c → 0, and ω ⊂ E 2 , we deduce that | E 2 \ ω|/|ω| can be made arbitrarily small by choosing c, and then ε, small enough. This proves (6.29) and the lemma.

Returning to our proof, we start from (6.20), and we bound from below

Ω ′ ε e ′ ε -ζ ε ′ dν ′ ε .
Let C 0 be the constant in (6.6) i.e. such that for any set

E ⊂ Ω ′ ε (6.33) |ν ε |(E) ≤ C 0 |log ε| e ε ( E)
and assume that C 0 > 2.

For notational simplicity we now write Ω, e, ν, g, Ω, ωε , 1) on ∂Ω and ∇ξ ∞ = o(1) (recall (5.12), (5.17)), for any η > 0 and ε small enough there exists a smooth positive cut-off function χ ≤ 1 such that |∇χ| ≤ η, such that χ = 0 on {ξ(x) ≤ 1 8C 0 } and such that χ = 1 on {ξ

ζ instead of Ω ′ ε , e ′ ε , ν ′ ε , (g ′ ε ) 0 , ω ′ ε , ω′ ε , ζ ′ ε and we let f = e -1 2 | log ε ′ |ν. Since ξ = o(
(x) ≥ 1 4C 0 }. Moreover, {x | d(x, ∂Ω) ≤ 2} ⊂ {ξ ≤ 1 8C 0 }. We then note that since f = e -1 2 | log ε ′ |ν and ζ = 1 2 | log ε ′ |ξ, we may write (6.34) e -ζν = (1 -χ) e - 1 2 ξ| log ε ′ |ν + χ (ξ(f -g) + ξg + (1 -ξ)e) .
Step 1: We first study (6.35)

Ω χ (ξ d(f -g) + ξ dg + (1 -ξ)e) = ω χξ dg + (ω) c χξ dg + + Ω χ(1 -ξ)e + Ω χξ d(f -g) - (ω) c χξ dg -.
The first parenthesis on the right-hand side contains positive terms, while the second one contains negative terms. We use (6.4), (6.6) to bound from below the negative terms, and use the positive terms to balance them. From (6.35) and (6.5) (applied with η = 1), and since χ = ξ = 1 on ω by construction and χ = 0 on {dist (x, ∂Ω) ≥ 2}, we have (6.36)

Ω χ(ξ d(f -g) + ξ dg + (1 -ξ)e) ≥ ω dg + 3 4 Ω χ(1 -ξ)e + 1 2 (ω) c χξ dg + + I 1 + I 2 ,
where

I 1 = -C Ω ∇(χξ) dg + + 1 4 (ω) c χξ dg + + 1 8 Ω χ(1 -ξ)e, I 2 = -C Ω ∇(χξ) d|ν| - (ω) c χξ dg -+ 1 4 (ω) c χξ dg + + 1 8 Ω χ(1 -ξ)e.
We first study I 1 . Since ∇(χξ) = 0 in ω, we have { ∇(χξ) = 0} ⊂ (ω) c . In addition |∇(χξ)| ≤ η if ε is small enough since |χ| ≤ 1, |ξ| ≤ 1, |∇χ| ≤ η and |∇ξ| = o(1). Thus, noting that χ = 1 on {ξ ≥ 1 2 },

2 Ω ∇(χξ) dg + ≤ η (ω) c ∩{ξ≥ 1 2 } dg + + η (ω) c ∩{ξ≤ 1 2 } dg + ≤ η (ω) c ∩{ξ≥ 1 2 } ξχ dg + + Cη Ω (1 -ξ)e
where we have used (6.4). We thus obtain, choosing η small enough,

I 1 ≥ - 1 8 Ω (1 -χ)(1 -ξ)e.
We turn to I 2 and split the negative contributions over E 2 and E c 2 . Using Lemma 6.2, choosing M large enough and using (6.4), (6.6) and the above, we have

E c 2 ∇(χξ) d|ν| + χξ dg -≤ Ce( E c 2 ) | log ε ′ | ≤ 1 8 c E c 2 (1 -ξ)e, since E c 2 ⊂ E c 1 and δ > M/| log ε ′ |.
On the other hand, using (6.6) with η = 1 and the fact that ∇(ξχ) = 0 in ω and |∇(χξ)| ≤ η + o(1) we have, since E 2 \ ω ⊂ E 2 \ ω and by choosing η small enough,

E 2 ∇(χξ) d|ν| ≤ Cη c E 2 \ω (dg + + 1) ≤ 1 4 ωc χξ dg + + Cη| E 2 \ ω|,
while using the fact that g is bounded below (hence g -≤ C), we have

ωc ∩E 2 χξ dg -≤ C|E 2 \ ω|.
Combining all the above, we deduce that choosing η small enough we have, if ε is small, that

I 2 ≥ - 1 8 Ω (1 -χ)(1 -ξ)e -C| E 2 \ ω|.
From Lemma 6.2, it follows that ∀M > 0 and ε small enough,

I 2 ≥ -C M |ω|-1 8 Ω (1-χ)(1-ξ)e.
Inserting the bounds for I 1 and I 2 into (6.36) we find (6.37)

Ω χ (ξ d(f -g) + ξ dg + (1 -ξ)e) ≥ g(ω) + 3 4 Ω χ(1 -ξ)e + 1 2 (ω) c χξ dg + - 1 4 Ω (1 -χ)(1 -ξ)e - C M |ω|.
Step 2: We examine Ω (1 -χ)(e -1 2 ξ| log ε ′ | dν). Since |ξ| ≤ 1 4C 0 on the support of 1 -χ we have there e -

1 2 ξ| log ε ′ |ν ≥ e - 1 8C 0 | log ε ′ ||ν|.
We may also bound this term below by g(ω). Combining with (6.39) and (6.38) we are led either to (6. 40)

Ω e -ζ dν ≥ g(ω) - C M |ω|,
where M can be arbitrarily large, or, using g -≤ C and keeping 1 6 (1 -ξ)e, to (6. 41)

Ω e -ζ dν ≥ 1 16C 0 g + (Ω) -C|ω| + 1 6 Ω (1 -ξ)e.
Step 4: Conclusion. We return to (6.20) and recall that the letter g stood for (g ′ ε ) 0 . We deduce from (6.40)

F ′ ε (u ′ ε , A ′ ε ) ≥ (g ′ ε ) 0 (ω ′ ε ) + (g ′ ε ) 1 (Ω ′ ε ) - C M |ω ′ ε | + o(1). Since (g ′ ε ) 1 ≥ 0 on Ω ′ ε \ ω ′ ε , since (g ′ ε ) 1 ≥ -C on ω ′ ε and since |ω ′ ε \ ω′ ε | = o(|ω ′ ε |), we have (g ′ ε ) 1 (Ω ′ ε ) ≥ (g ′ ε ) 1 (ω ′ ε ) + o(ω ′ ε )
. Then we deduce from (6.40), letting ε → 0 and then M → +∞, that proving (6.11).

F ′ ε (u ′ ε , A ′ ε ) ≥ g ′ ε (ω ′ ε ) + o(|ω ′ ε |),
To prove (6.12) we combine (6.41) with item 6 in Theorem 6 to obtain

F ′ ε (u ′ ε , A ′ ε ) ≥ 1 C p {d(x,(Ω ′ ε ) c )≥C} |j ′ ε | p -C|ω ′ ε | + 1 6 Ω ′ ε (1 -ξ)e ′ ε .
This proves the first inequality in (6.12), noting that (1 -ξ) ≥ 1/2 on {d(x, (Ω ′ ε ) c ) ≥ C} if ε is small enough and that e ′ ε ≥ 1 2 |j ′ ε | 2 . Finally, from (6.40), (6.42)

F ′ ε (u ′ ε , A ′ ε ) ≥ (g ′ ε ) 0 (ω ′ ε ) + (g ′ ε ) 1 (Ω ′ ε ) + o(|ω ′ ε |),
and it is straightforward to check that

(g ′ ε ) 1 ≥ 1 2ℓε 2 (curl A ′ ε -m ε ) 2 on ω ′ ε 1 2ℓε 2 (curl A ′ ε ) 2 on Ω ′ ε \ ω ′ ε . Therefore (g ′ ε ) 1 (Ω ′ ε ) ≥ Ω ′ ε (curl A ′ ε -m ε ) 2 2ℓ ε 2 -C|ω ′ ε |,
which together with (6.42) finishes the proof of (6.12).

The case of small applied field

Here, by small applied field we mean fields h ex that satisfy the assumptions of Proposition 6.1. We are ready to define the appropriate space and Γ-converging functions to apply the scheme described in Section 2. We choose X = L p loc (R 2 , R 2 ) × M 0 , where M 0 denotes the set of measures µ such that µ + C is a positive locally bounded measure on R 2 , -C being the constant bounding from below g ε in item 1 of Proposition 6.1. We consider on L p loc (R 2 , R 2 ) the strong topology and on M 0 that of weak convergence. The space of positive measures equipped with the weak convergence is metrizable hence the space X is a Polish space. We will typically denote by x an element of X.

There is a natural action θ of R 2 on X by translations:

θ λ (j(•), g(•)) = (j(λ + •), g(λ + •))
which is clearly continuous with respect to the couple (j, g) as well as with respect to λ. Denoting as above ω ′ ε the rescaled coincidence set, we know from Lemma 5.6 that it satisfies (1.14). Now consider {(u ε , A ε )} ε satisfying the hypothesis of Proposition 6.1, then the proposition provides us with measures g ′ ε defined on Ω ′ ε . We also have the rescaled current

j ′ ε = curl (iu ′ ε , ∇ A ′ ε u ′ ε )
where (u ′ ε , A ′ ε ) are as in Proposition 6.1. We define functions {f ε } ε on X as follows.

Having chosen a smooth positive function χ : R 2 → R with support in the unit ball and integral equal to 1, we let

f ε (x) =    χ(y -x) dg ′ ε (y) if ∃x ∈ ω ′ ε s.t. x = (j ′ ε (x + •), g ′ ε (x + •)) +∞ otherwise.
Note that since j ′ ε and g ′ ε vanish outside a compact set, there is at most one x such that x = (j ′ ε (x + •), g ′ ε (x + •)) unless x = 0, in which case we let f ε (x) = +∞. The third statement of Proposition 6.1 implies that {f ε } ε satisfies the requirement of coercivity (1.15). Indeed assume that {x ε } ε is a sequence in X such that (6.43) lim sup

ε→0 B R f ε (θ λ x ε ) dλ < +∞
for every R > 0, then in particular this integral is finite if ε is small enough, which implies that f ε (θ λ x ε ) < +∞ for almost every λ ∈ B R . Thus there exists {x ε } ε such that x ε = (j ′ ε (x ε + •), g ′ ε (x ε + •)) and λ + x ε ∈ ω ′ ε for almost every λ ∈ B R , when ε is small enough. In particular the distance of x ε to R 2 \ ω ′ ε is larger than R for ε small enough, and (6.43) reads

∀R > 0, lim sup ε→0 B R χ(y -λ -x ε ) dg ′ ε (y) dλ < +∞.
Integrating first w.r.t. λ we find lim sup

ε→0 χ R (y -x ε ) dg ′ ε (y) < +∞, where χ R = χ * 1 B R .
Since g ′ ε is bounded from below independently of ε, and since χ R is a positive function equal to 1 on B R-1 , this implies that the second part of (6.13) is satisfied. Thus, up to a subsequence, the currents j

′ ε (x ε + •) converge in L p loc (R 2 , R 2 ). Going to a further subsequence, {g ′ ε (x ε + •)} ε converges in M.
Finally, χ was chosen smooth, it is clear that the Γ-liminf requirement (1.16) is satisfied if we define the function f on X by (6.44) f (j, g) = χ dg. Now Theorem 3 applies and combined with Proposition 6.1 gives:

Proposition 6.3. Assume that {(u ε , A ε )} ε satisfy (1.47) and G ε (u ε , A ε ) ≤ min N ∈{N - 0 ,N + 0 } G N ε + CN 0
for some constant C, and assume h ex satisfy the hypothesis of Proposition 6.1 and (5.29).

Let (u ′ ε , A ′ ε ) and N be as in Proposition 6.1. Using the notation above, and letting P ε be the probability measure on L p loc (R 2 , R 2 ) which is the push-forward of the normalized uniform measure on ω ′ ε by the map x → j ′ ε (x + •), we have the following. 1. A subsequence of {P ε } ε weakly converges to a translation-invariant probability measure P on L p loc (R 2 , R 2 ) such that P -a.e. j ∈ A m λ . 2. For any family {U R } R>0 satisfying (1.4), (1.5), we have

(6.45) lim inf ε→0 F ′ ε (u ′ ε , A ′ ε ) |ω ′ ε | ≥ W U (j) dP (j) + m λ γ 2π and (6.46) G ε (u ε , A ε ) ≥ G N ε + N 2π m λ W U (j) dP (j) + γ + o(1) .
Remark 6.4. We claim that under the hypothesis of this proposition, the limit of P ε which is the image of the normalized Lebesgue measure on ω ′ ε by the map

ϕ : x → j ′ ε (• -x) is unchanged if, in the definition of ϕ, we replace A ′ ε (y) = ℓ ε (A ε (x + ℓ ε y) -∇ ⊥ h ε,N (x + ℓ ε y)) by ℓ ε A ε (x + ℓ ε y), i.e. define P ε as in Theorem 5.
Indeed, denote by ψ the corresponding modification of ϕ and by Q ε the ensuing modification of P ε . From (1.14), there exists ω′ ε such that ω′

ε + B R ⊂ ω ′ ε for any R > 0 if ε is small enough, and such that |ω ′ ε | ∼ |ω ′ ε |. Since ω′ ε ⊂ ω ′ ε and |ω ′ ε | ∼ |ω ′ ε |, replacing ω ′
ε by ω′ ε in the definition of either P ε or Q ε does not change their limit. Then, since ∇ ⊥ h ε,N (x + ℓ ε y) = 0 for any x ∈ ω′ ε and y ∈ B R and if ε is small enough, we find that P ε and Q ε , seen as measures on L p (B R ), coincide if ε is small enough depending on R. It follows that their limits in L p loc (R 2 , R 2 ) are equal, proving the claim.

Proof of Proposition 6.3. It is immediate that (u ε , A ε ) satisfies the hypotheses of Proposition 6.1. Let N be given by Proposition 6.1. By Lemma 5.8, we have, for that N

, G ε (u ε , A ε ) = G N ε + F ε (u ε , A 1,ε ) + o(N ). But from the upper bound assumption, G ε (u ε , A ε ) ≤ G N ε + CN 0 so we deduce F ε (u ε , A 1,ε ) ≤ CN 0 ≃ CN = O(h ex |ω ε,N |).
In blown-up coordinates, this means

F ′ ε (u ′ ε , A ′ ε ) ≤ C|ω ′ ε |.
On the other hand, from (6.11) in Proposition 6.1, we have

F ′ ε (u ′ ε , A ′ ε ) ≥ g ′ ε (ω ′ ε ) -o(|ω ′ ε |),
where we recall ω′

ε = {d(x, (ω ′ ε ) c ) ≥ 2}. Also, since χ = 1, since χ has support in B 1 and since g ′ ε is bounded below, g ′ ε (ω ′ ε ) ≥ ω′ ε f ε (θ λ j ′ ε , θ λ g ′ ε ) dλ -o(|ω ′ ε |),
where ω′ ε = {d(x, (ω ′ ε ) c ) ≥ 3}. Combining these facts, we have

F ε (j ′ ε , g ′ ε ) := - ω′ ε f ε (θ λ j ′ ε , θ λ g ′ ε ) dλ ≤ C
and we now apply Theorem 3 to these functionals. We deduce that the measures {Q ε } ε , where Q ε is the image under x → (j ′ ε (x + •), g ′ ε (x + •)) of the uniform normalized probability measure on ω′ ε , converge to a translation invariant probability measure Q on X and lim inf ε→0

F ′ ε (u ′ ε , A ′ ε ) |ω ′ ε | ≥ X lim R→+∞ - U R f (θ λ x) dQ(x).
Moreover, since from (1.14) we have |ω ′ ε | ∼ |ω ′ ε |, we may replace ω′ ε with ω ′ ε in the definition of Q ε and obtain the same limit. But, writing as above χ U R = χ * 1 U R , we have as above from (6.44)

lim R→+∞ - U R f (θ λ x) dλ = lim R→+∞ 1 |U R | χ U R (x) dg(x),
where x = (j, g). Now, if f (x) is finite and x is in the support of Q, then there exists (see Remark

1.6) a sequence {x ε } ε such that (j ′ ε (x ε +•), g ′ ε (x ε +•)) converges to
x in with (6.14) satisfied with U R . It follows from Proposition 6.1 that j ∈ A m λ and that (6.16) is satisfied, thus

lim R→+∞ 1 |U R | χ U R (x) dg(x) = lim R→+∞ lim ε→0 1 |U R | χ U R (x -x ε ) dg ′ ε (x) ≥ lim sup R→∞ W (j, χ U R ) |U R | + m λ γ 2π = W U (j) + m λ γ 2π .
Letting P ε (j) and P (j) denote the marginals of the measures Q ε and Q with respect to the first variable, we immediately deduce that P ε → P and that (6.45) is satisfied. Replacing (6.45) in (5.30) we find (6.46), since |ω

′ ε | = h ex |ω ε,N | = 2πN m ε,N = 2πN m λ + o(N ).
Remark 6.5. As in Remark 2.3 we can also obtain a result of equipartition of energy of F ′ ε on ω ′ ε .

The case of larger applied field

We prove that the conclusions of Proposition 6.3 hold for larger fields as well, that is fields which do not satisfy the assumptions of Proposition 6.1. The technical difficulty is that the applied field is too large to have the energy upper bound that is needed to apply the result of Theorem 6, so we use the strategy of [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] Chapter 8: average over smaller balls where most of the time the local energy is small enough to apply Theorem 6 and the result for small applied fields. Note that in this regime, by item 6 in Proposition 5.6 we have for N

∈ {N - 0 , N + 0 }, G N ε = G N 0 ε + o(h ex ) = 1 2 |Ω|h ex | log ε ′ | + o(h ex ).
Proposition 6.6. Assume that

|log ε| 4 ≪ h ex ≪ 1/ε 2 and that G ε (u ε , A ε ) ≤ hex 2 |Ω|| log ε ′ | + Ch ex .
We also assume that |u ε | ≤ 1 and that A ε is a critical point of G ε (u ε , •). Then, using the same notation as in Propositions 6.1 and 6.3, we have, for any U (6.47)

G ε (u ε , A ε ) ≥ 1 2 |Ω|h ex | log ε ′ | + |Ω|h ex W U (j) dP (j) + γ 2π + o(1) ,
and P -a.e. j ∈ A 1 .

Step 1: Blow-up. The proof follows the ideas in Chapter 8 of [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF]SS5]. We recall the rescaling formula from there. Define x = x/σ and ũε

(x) = u ε (x), Ãε (x) = σA ε (x), Ω = Ω σ , ε = ε σ , hex = σ 2 h ex . Then, denoting B σ x = B(x, σ), we have up to translation G ε (u ε , A ε , B σ x ) = Gε (ũ ε , Ãε , B 1 x),
where G ε (u ε , A ε , B σ x ) denotes the Ginzburg-Landau energy restricted to B σ x , and

Gε (ũ ε , Ãε , B 1 ) = 1 2 B 1 |∇ Ãε ũε | 2 + 1 σ 2 curl Ãε -hex 2 + 1 -|ũ ε | 2 2 2ε 2 . As in [SS4], Chap 8, if |log ε| 4 ≪ h ex ≪ 1/ε 2 we may choose σ ≪ 1 such that (6.48) hex = (log ε) 4 | log ε| ∼ | log ε ′ |, so that σ 2 log 4 1 σ = ε 2 h ex .
Step 2: Fubini. We give a formulation of the energy which follows from Fubini's theorem:

(6.49) G ε (u ε , A ε , Ω) = x∈R 2 G ε (u ε , A ε , B σ x ∩ Ω) |B σ x |
.

Note that if we restrict the integration to the set Ω σ of those x's such that B σ x ⊂ Ω, then we have an inequality.

For x ∈ Ω σ we define P x ε to be the push-forward of the normalized Lebesgue measure on B σ

x by the map x → j ′ ε ( x ℓε + •), where again j

′ ε = (iu ′ ε , ∇ A ′ ε u ′ ε )
, and u ′ ε , A ′ ε are as in (6.10). It is an element of P, the set of probability measures on X = L p loc (R 2 , R 2 ). On P we define the function

f ε (P ) =    G ε (u ε , A ε , B σ x ) |B σ x | - h ex 2 | log ε ′ | if ∃x ∈ Ω σ s.t. P = P x ε +∞ otherwise.
We also define Q ε to be the push-forward under x → P x ε of the normalized Lebesgue measure on Ω σ . It is a probability measure on P. Now (6.49) becomes, after subtracting hex

2 |Ω σ || log ε ′ |, G ε (u ε , A ε , Ω) - h ex 2 |Ω σ || log ε ′ | = |Ω σ | P f ε (P ) dQ ε (P ).
Note that since |Ω \ Ω σ | = O(σ), and since σ| log ε ′ | = o(1) -this easily follows from (6.48) -we deduce from the above that (6.50)

G ε (u ε , A ε , Ω) - h ex 2 |Ω|| log ε ′ | = o(h ex ) + |Ω σ | P f ε (P ) dQ ε (P ).
Step 3: Γ-convergence of 1 hex f ε . Assume that P ε is a probability measure such that f ε (P ε ) ≤ Ch ex and that P ε ∈ P converges to P . Then since f ε (P ε ) < ∞, there exists for each ε some x ε ∈ Ω σ such that P ε = P xε ε and

f ε (P ε ) = G ε (u ε , A ε , B σ xε ) |B σ | - h ex 2 | log ε ′ | = 1 σ 2 Gε (ũ ε , Ãε , B 1 xε ) |B 1 | - σ 2 h ex 2 | log ε ′ | = 1 σ 2 Gε (ũ ε , Ãε , B 1 xε ) |B 1 | - hex 2 | log ε ′ | . (6.51) Since f ε (P ε ) ≤ Ch ex we get Gε (ũ ε , Ãε , B 1 xε ) ≤ C(h ex σ 2 + h ex σ 2 | log ε ′ |) = C hex (1 + | log ε ′ |
) and in view of (6.48), note that ε ′ = ε √ h ex = ε hex , we may apply Proposition 6.3 with ε replaced by ε, h ex by hex , etc. This way we find that, Ñ being given by Proposition 6.3, P is concentrated on A 1 and that

(6.52) Gε (ũ ε , Ãε , B 1 xε ) ≥ 1 2 h ε, Ñ -hex 2 H 1 (B 1 xε ) + π Ñ | log ε ′ | + hex |ω ε| W U (j) d P (j) + γ 2π + o(1),
where h ε, Ñ is the solution of the obstacle problem (1.35), replacing Ω with B 1 xε , h ex with hex and ε with ε ; and µ ε, Ñ = -∆h ε, Ñ + h ε, Ñ , ω ε = Supp(µ ε, Ñ ). We can further check, since hex = | log ε| 4 , that from (5.19), (5.26) in Proposition 5.6, we have, as ε → 0, (6.53)

|ω ε| = |B 1 | + o(1) h ε, Ñ -hex 2 H 1 + π Ñ | log ε ′ | = 1 2 hex |B 1 || log ε ′ | + o( hex ).
In (6.52), P is such that P -a.e. j ∈ A 1 , and is the limit of the probability measures {P ε} ε . Using Remark 6.4, the measure P is the limit of the image of the normalized Lebesgue measure on ω ε by ϕ : x → j(u x , A x ), where

u x (y) = ũε (x + ℓ ε y) = u ε (σx + ℓ ε y), A x (y) = ℓ ε Ãε (x + ℓ ε y) = ℓ ε u ε (σx + ℓ ε y),
while P ε is the image of the normalized Lebesgue measure on B σ xε by the usual blow-up map x → j ′ ε (x + •) which, changing the variable to x, is equal to the image of the normalized Lebesgue measure on B 1 xε by ϕ. From |ω ε| ∼ |B 1 | and ω ε ⊂ B 1 xε , we then deduce that {P ε } ε and {P ε} ε have the same limit, i.e. that P = P . Then (6.52), (6.53) yield

Gε (ũ ε , Ãε , B xε 1 ) ≥ 1 2 hex |B 1 | log 1 ε hex + hex |B 1 | W U (j) dP (j) + γ 2π + o( hex ),
where P = lim ε P ε is concentrated on A 1 . Then, from (6.51), (6.54) lim inf

ε→0 1 h ex f ε (P ε ) ≥ W U (j) dP (j) + γ 2π .
Step 4: {Q ε } ε is tight. A consequence of (6.54) and Proposition 4.1 below is that the function f ε is bounded below independently of ε, thus the results of Section 2 apply. The measures Q ε are regular from their definition hence given δ > 0 there exists compact sets K ε in P such that Q ε (K ε ) > 1 -δ, and since h ex -1 f ε (P ) dQ ε (P ) < C from (6.50) and the energy upper bound, we can also require that h ex -1 f ε < 1/δ on K ε . Now assume P ε ∈ K ε for each ε > 0. Then f ε (P ε )/h ex is bounded independently of ε and therefore from the previous step and after taking a subsequence, P ε → P . This shows that the hypotheses of Lemma 2.1 are satisfied and thus that any subsequence of {Q ε } ε has a convergent subsequence. In addition, we deduce that Q-almost every P satisfies that P -almost every j ∈ A 1 .

Step 5: Conclusion. Combining the convergence of {Q ε } ε and (6.54), we deduce with the help of Lemma 2.2 that (6.55) lim inf

ε→0 1 h ex f ε (P ) dQ ε (P ≥ W U (j) dP (j) dQ(P ) + γ 2π .
It remains to show that (6.56) W U (j) dP (j) dQ(P ) = W U (j) dP (j), where P = lim ε P ε and P ε is the push-forward of the normalized uniform measure on ω ′ ε by the map x → j ′ ε (x + •) where j ′ ε is the current defined from (6.10). But, if ϕ is a continuous and bounded function on X, by definition of Q ε we have

ϕ dP ε = ϕ dP dQ ε (P ) + o(1), since |ω ′ ε | ∼ Ω σ .
Hence, passing to the limit, we find ϕ dP = ϕ dP dQ(P ).

It is straightforward to check that this equality extends to positive measurable functions, and in particular to W , which was proven to be measurable in Proposition 4.1. This proves (6.56).

In addition since Q almost every P satisfies j ∈ A 1 , and since P = P dQ(P ), we also have that P almost every j ∈ A 1 . Renaming P by P , and since |Ω σ | ∼ |Ω| as ε → 0, combining (6.50), (6.55) and (6.56) proves (6.47).

Proof of Theorem 5

We may now combine the results of Propositions 6.3, 6.6, Lemma 5.8 and the upper bound of Theorem 7 below to prove Theorem 5. Assertion 2 of the Theorem is part of Theorem 7 below.

For Assertion 1, in view of Remark 6.4, propositions 6.3 and 6.6 imply, for a suitable choice of N ∈ {N - 0 , N + 0 }, the convergence of P ε to a translation invariant measure P on L p loc such that (1.49) holds.

We next prove (1.50). We have seen in the subsections just above that the upper bound condition implies F

ε (u ε , A 1,ε ) ≤ Ch ex |ω ε,N | or, blowing-up, F ε ′ (u ′ ε , A ′ ε ) ≤ C|ω ′ ε |. We may write µ(u ε , A ε ) -µ ε,N = curl j 1,ε + α + β, where α = curl A 1,ε -µ ε,N , β = µ(u ε , A ε ) -curl A 1,ε -curl j 1,ε .
From (6.12) we have α

2 L 2 ≤ C(F ε ′ + |ω ′ ε |) ≤ Ch ex |ω ε,N |, hence α W -1,p ≤ C √ N .
The same bound holds for β from (5.33). Finally, from (6.12), we have |j

′ | p ≤ C(F ′ ε + |ω ′ ε |) ≤ CN . Rescaling this relation, we get j L p (Ω) ≤ h ex 1 2 -1 p j ′ L p (Ω ′ ε ) ≤ Ch ex 1 2 -1 p N 1 p ≤ CN 1 2
where we have used the fact that N ≤ hex|Ω| 2π ≤ Ch ex . Therefore curl j 1,ε W -1,p ≤ C √ N , which concludes the proof of (1.50).

It remains to prove the statement concerning minimizers

{(u ε , A ε )} ε of G ε . In the case of small applied fields h ex ≤ ε -β , Corollary 7.1 gives G ε (u ε , A ε ) ≤ min N ∈{N - 0 ,N + 0 } G N
ε + CN 0 hence Proposition 6.3 applies. Comparing the lower bound (6.46) to the upper bound (7.2) in Theorem 7, we deduce that N minimizes the right-hand side and that

W U (j) dP (j) ≤ min Am λ W.
Since P is supported on A m λ , we obtain that P -a.e. j minimizes W U over A m λ . Since minimizers of W U are independent of U , we have the result.

In the case of large applied fields

|log ε| 4 ≪ 1 ε 2 , Corollary 7.2 yields min G ε ≤ h ex 2 |Ω|| log ε ′ | + |Ω|h ex min A 1 W + γ 2π + o(1) ,
thus Proposition 6.6 applies and comparing the above to (6.47), we deduce that there is equality and that

W U (j) dP (j) ≤ min A 1 W.
We again deduce that P -almost every j minimizes W over A 1 . From (5.26) in Proposition 5.6, we get that (1.49) holds. Note that Theorem 5 implies Theorem 4 since, from (5.25) in Proposition 5.6, if

h ex = λ|log ε| with λ > λ Ω then G N ε -G N 0 ε = O(1) as ε → 0.
Remark 6.7. If we had chosen from the beginning N = N = 1 2π ν ε (Ω ′ ε ) as indicated in footnote in the proof of Proposition 6.1, the we would obtain the lower bound (1.49) with that N . Combining with the upper bound of Theorem 7 we may deduce that G

N ε = min N ∈N G N ε + o( N ) + o(N 0 ). A careful examination of the variation of G N
ε with N , based on Lemma 5.4 should then allow to obtain that N = N - 0 or N + 0 up to an error which is quantified by the examination of the growth of G N ε = G N 0 ε (this is quite delicate, though). We expect this error to be 0 for small enough applied fields, in particular for h ex < H c 1 + O( |log ε|).

Upper bound

In this section, we use the notation of Section 5, in particular the definitions of h ε,N , ω ε,N ... can be found there. We make here, and here only, the assumption that Ω is convex. This guarantees the smoothness of the solutions to the obstacle problem (A.1), see below. We do not believe this is a serious restriction, but the possible presence of cusps in the coincidence set would certainly add technical difficulties to our construction.

We prove the upper bound matching the lower bound of Theorem 5 (recall the definition of G N ε in (1.41)):

Theorem 7. Assume that Ω is convex and that (1.48) holds. Then for any family of integers {N } depending on ε and satisfying

(7.1) 1 ≪ N ≤ |Ω|h ex 2π
, as ε → 0 the following holds:

1. There exists (u ε , A ε ) such that, as ε → 0,

(7.2) G ε (u ε , A ε ) ≤ G N ε + N 2π m λ min Am λ W + γ + o(1) ,
where λ ∈ [λ Ω , +∞] is the limit of h ex /|log ε| as ε → 0, where A m λ is as in Definition 1.1, and where ε ′ = ε √ h ex .

2. Let 1 < p < 2 be given. For any probability P on L p loc (R 2 , R 2 ) which is invariant under the action of translations and concentrated on A m λ , there exists (u ε , A ε ) such that, letting P ε be the push-forward of the normalized Lebesgue measure on ω ε,N by the map

x → 1 √ hex j(u ε , A ε ) x + •
√ hex , we have as ε → 0, P ε → P weakly and

(7.3) G ε (u ε , A ε ) ≤ G N ε + N 2π m λ W K (j) dP (j) + γ + o(1) .
Corollary 7.1. Under the same assumptions, we have

min G ε ≤ min N ∈{N - 0 ,N + 0 } G N ε + CN 0 ≤ Ch ex | log ε ′ |.
Proof. To obtain an upper bound, we apply the result above with N = N - 0 and N = N + 0 (recall that N 0 is not necessarily an integer) and use (5.25) in Proposition 5.6.

Corollary 7.2. Under the same assumptions, if

|log ε| 4 ≪ h ex ≪ 1 ε 2 , we have min G ε ≤ 1 2 |Ω|h ex | log ε ′ | + h ex |Ω| min A 1 W + γ 2π + o(1) .
Proof. This follows from (7.2) applied with N = N - 0 , (5.26) in Proposition 5.6, |ω ε,N | ≤ |Ω| and λ = +∞.

We now prove Theorem 7.

Properties of ω ε,N

The convexity of Ω guarantees that the coincidence sets {ω m } m for the minimizers of (A.1) are convex (Friedman-Phillips, [FP], see also [Ka], [DM]). Then the density criterion of Caffarelli [Caf] and regularity improvement of Kinderlehrer-Nirenberg [KN] and Isakov [Is] imply that it is in fact analytic for any m. We state a density estimate which is uniform with respect to m ∈ (h 0 , 1]. This is the only place where we use the assumption that Ω is convex.

Lemma 7.3. Assume Ω is convex (so that (1.31) is satisfied), and let L m be as in Proposition A.1. Then there exists α > 0 and r 0 > 0 such that for any m ∈ (h 0 , 1], any r < r 0 , and any

x ∈ ω m |ω m ∩ B(x, L m r)| |B(x, L m r)| ≥ α.
:= -∆h ε,N + h ε,N = m ε,N h ex 1 ω ε,N where m ε,N satisfies (5.2) and m ε,N h ex |ω ε,N | = 2πN . Let ℓ ′ ε = 1 m ε,N h ex .
We have (ℓ ′ ε ) -2 |ω ε,N | ∈ 2πN. Rescaling the previous lemma we find Corollary 7.4. There exists α > 0 such that for any R > 0, any ε small enough depending on R, and x ∈ ω ε,N we have |ω ε,N ∩ B(x, Rℓ ′ ε )| ≥ α|B(x, Rℓ ′ ε )|. Proof. From Proposition 5.6 we have ℓ ′ ε ≪ L ε , where L ε is the value of L m corresponding to m = m ε,N . Thus, if ε is small enough, we have Rℓ ′ ε ≤ r 0 L ε , and the previous lemma applies.

Definition of the test current

The construction follows similar lines as [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF], Chapters 7 and 10, but the estimates must be more precise as only an error of o(1) per vortex is allowed. From now on we assume (1.31), (1.48) and (7.1). We write for simplicity ω ε instead of ω ε,N and m ε instead of m ε,N .

Let R ∈ 4πN be given. We tile R 2 in the obvious way by a collection {K i } i of squares of sidelength 2Rℓ ′ ε . We let

I = {i, K i ⊂ ω ε , dist (K i , ∂ω ε ) ≥ ℓ ′ ε }, ωε = ∪ i∈I K i ωε = ω ε \ ωε .
To prove the first item of the theorem, we apply Corollary 4.4 to a minimizer of W to find j R in K R such that j R • τ on ∂K R where K R = [-R, R] 2 . To prove the second item, let P concentrated on A m λ be given, and let us define P to be the push-forward of P under the rescaling j → 1 √ m λ j

• √ m λ . Then P is concentrated on A 1 and from (1.9), we have (7.4) W K (j) d P (j) = 1 m λ W K (j) dP (j) + 1 4 log m λ .

We then apply Corollary 4.5 to the probability P , it gives again a j R in K R such that j R •τ = 0 on ∂K R . We continue the construction in the same way in either of these two cases.

We next extend j R by periodicity to R 2 and, denoting by c 0 the center of K i 0 , where i 0 ∈ I is arbitrary, we let

ε (x) =    1 ℓ ′ ε j R x -c 0 ℓ ′ ε in ωε 0 in R 2 \ ωε .
In particular, letting Λ = (c 0 + ℓ ′ ε Λ R ) ∩ ωε , where Λ R denotes the support of ν R , and since ε • τ = 0 on ∂ ωε , we have

curl ε = 2π p∈ e Λ δ p -m ε h ex 1 ωε , in R 2 .
We define a current ε as follows. First we note that since

|K i |, |ω ε | ∈ 2πℓ ′ ε 2 N, we have |ω ε | ∈ 2πℓ ′ ε 2 N.
Then, using Corollary 7.4 we may -we omit the cumbersome details -find disjoint measurable sets C 1 , . . . , C n and y i ∈ C i such that, for some c, C > 0 independent of R, ε, (7.5)

1 ωε = i 1 C i , |C i | = 2πℓ ′ ε 2 , B(y i , cℓ ′ ε ) ⊂ C i ⊂ B(y i , Cℓ ′ ε ).
We let j i = -∇ ⊥ f i , where Finally we let j ε = ε + ε , Λ = Λ ∪ Λ. We have Proposition 7.5. The current j ε satisfies

(7.7) curl j ε = 2π p∈Λ δ p -h ex m ε 1 ωε in R 2 j ε = 0 on R 2 \ Ω. Moreover (7.8) lim sup η→0 1 m ε h ex |ω ε | 1 2 Ω\∪ p∈Λ B(p,ηℓ ′ ε ) |j ε | 2 + π#Λ log η ≤ W (j R , 1 K R ) R 2 + o ε (1),
where lim ε→0 o ε (1) = 0. Finally, there exists η 0 > 0 such that for any ε small enough, any p ∈ Λ and any q ∈ [1, +∞) we have (7.9) j ε -∇ ⊥ log | • -p| L q (B(p,η 0 ℓ ′ ε )) ≤ C q ℓ ′ ε 2 q -1 .

Proof. The fact that curl j ε = 2π p∈Λ δ p -h ex m ε is obvious, and j ε = 0 on Ω c follows from the definition of j ε and the fact that d(ω ε , Ω c ) ≥ Cℓ ′ ε if ε is small enough. This is a consequence of the fact that on ω ε we have h ε,N = m ε,N h ex while on ∂Ω we have h ε,N = h ex . The difference is

∆ = h ex (m 0,ε -m ε,N + 1 -m 0,ε ) = c ε,N + 1 2 | log ε ′ | ≈ 1 2 | log ε ′ |,
using (5.12). It follows using (5.20) that d(ω ε , Ω c ) ≥ | log ε ′ |/h ex ≫ ℓ ′ ε . We estimate ε . From (7.6), f i (y) = -log |y -y i | + g i ((y -y i )/ℓ ′ ε ), where g i solves ∆g i (x) = 1 C i (y i + ℓ ′ ε x) in B(0, C) and ∂ ν g i = 0 on ∂B(0, C). Since 1 C i ∈ L ∞ , elliptic regularity implies that ∇g i L q ≤ C q for every q ∈ [1, +∞). We easily deduce that (7.10)

j i -∇ ⊥ log | • -y i | L q (B(y i ,Cℓ ′ ε )) ≤ C q ℓ ′ ε 2 q -1 .
Since j i = 0 outside B(y i , Cℓ ′ ε ) we deduce (7.11) j i L q (R 2 \B(y i ,cℓ ′ ε )) ≤ C q ℓ ′ ε 2 q -1 1 + c 2-q , (7.12)

j i L 2 (R 2 \B(y i ,cℓ ′ ε )) ≤ C 1 + log 1 c .
Then we compute estimates for ε . Since j R is defined independently of ε, there exists η 0 > 0 (depending on R) which bounds from below the distances between the points in Λ R and between Λ R and ∂K R . Since div j R = 0 in K R we have that j R -∇ ⊥ log |•-y| L q (B(y,η 0 )) ≤ C q for any y ∈ Λ R and moreover j R ∈ C ∞ loc (K R \ Λ R ). It follows that ∀y ∈ Λ and ∀q ∈ [1, +∞) we have (7.13) ε -∇ ⊥ log | • -y L q (B(y,η 0 ℓ ′ ε )) ≤ C q ℓ ′ ε 2 q -1 .

Moreover, since j R is uniformly locally bounded in L q ′ , for every q ′ ∈ [1, 2), we have for every x ∈ R 2 and every M > 0

(7.14) ε L q ′ (B(x,M ℓ ′ ε )) ≤ C q ′ M 2 q ′ ℓ ′ ε 2 q ′ -1 .
We are ready to derive estimates for j ε = ε + n i=1 j i . First we note that from (7.5) we have that |y i -y j | and d(y i , ωε ) are bounded below by cℓ ′ ε , if i = j, therefore since Suppj i ⊂ B(y i , Cℓ ′ ε ) and Supp ε ⊂ ωε the overlap number of the supports of ε and the j i 's is bounded by a constant C independent of R, ε. Moreover ωε is included in the complement of ∪ i B(y i , cℓ ′ ε ). We have R 2 ε • ε = i R 2 j i • ε , but the number of i's for which the integrals above are nonzero is bounded by

C |Suppε∩Suppε| ℓ ′ ε 2
since Supp ε ∩ Supp ε ⊂ {d(x, ω c ε ) ≤ Cℓ ′ ε } and using Proposition 5.6, item 4. Applying Hölder's inequality to each of these nonzero integrals, and using (7.11), (7.14), we deduce (7.15) and from (7.10) applied with q = 2 and (7.12), we deduce

R 2 ε • ε ≤ C |Supp ε ∩ Supp ε | ℓ ′ ε 2 × C q ℓ ′ ε 2 q -1 × C q ′ ℓ ′ ε 2 q ′ -1 ≤ Cm ε h ex o ε (
(7.16) 1 2 R 2 \∪ p∈ b Λ B(p,ηℓ ′ ε ) | ε | 2 + π# Λ log η ≤ C# Λ ≤ Cm ε h ex |ω ε | = o ε (1)h ex |ω ε |.
Also, since by definition

W (j R , 1 K R ) = lim η→0 1 2 K R \∪ p∈Λ R B(p,η) |j R | 2 + π#(Λ R ∩ K R ) log η ,
we have, multiplying by the number of squares in ωε , which is

|ω ε |ℓ ′ ε -2 /R 2 = h ex m ε |ω ε |/R 2 , that (7.17) lim η→0 1 m ε h ex |ω ε | 1 2 R 2 \∪ p∈ e Λ B(p,ηℓ ′ ε ) | ε | 2 + π# Λ log η = W (j R , 1 K R ) R 2 ,
and the limit is uniform in ε since, despite the notation, the left-hand side only depends on R. From (7.15), (7.16) and (7.17) we deduce lim sup we obtain (7.8). Then (7.9) follows from (7.10), (7.13).

η→0 1 m ε h ex |ω ε | 1 2 R 2 \∪ p∈Λ B(p,ηℓ ′ ε ) |j ε | 2 + π#Λ log η ≤ W (j R , 1 K R ) R 2 |ω ε | |ω ε | + o ε (

Definition of the test-configuration

We next find A 1,ε such that curl A 1,ε = m ε h ex 1 ωε = µ ε,N , and set

A ε = A 1,ε + ∇ ⊥ h ε,N .
To define u ε , we start by defining its phase ϕ ε by requiring (7.18) ∇ϕ ε = A 1,ε + j ε .

Indeed, denoting by Θ the phase of p∈Λ z-p |z-p| , we have by (7.7) curl (A 1,ε + j ε -∇Θ) = µ ε,N + 2π p∈Λ δ p -m ε h ex 1 ωε -2π p∈Λ δ p = 0, therefore A 1,ε + j ε -∇Θ is the gradient of a function ψ, we may then let ϕ ε = Θ + ψ, this function is well-defined modulo 2π in Ω\Λ and satisfies (7.18). Hence e iϕε is well-defined in Ω\Λ and ∇ϕ ε = A 1,ε + j ε . Fixing M > 1, we then define x) in Ω\ ∪ p∈Λ B(p, M ε)

u ε (x) = e iϕε(
u ε (x) = 1 f (M ) f |x-p| ε e iϕ(x) in B(p, M ε),
where f is the modulus of the unique radial degree-one vortex u 0 (r, θ) = f (r)e iθ (see [BBH, Mi, HH]). f is increasing from 0 to M , so that in particular we deduce |u ε | ≤ 1 everywhere. This construction is possible since, for fixed R, the distances between the points in Λ are bounded below by η 0 ℓ ′ ε (for some η 0 possibly smaller than the one used before) and ℓ ′ ε = 1 √ mεhex ≫ ε since we assume h ex ≪ 1 ε 2 . The test-configuration (u ε , A ε ) is now defined and there remains to evaluate its energy and show that it satisfies (7.2), respectively (7.3).

7.4 Splitting of the energy of (u ε , A ε )

According to Proposition 5.3, we have the relation

G ε (u ε , A ε ) = G N ε + F ε (u ε , A 1,ε ) - Ω (1 -|u ε | 2 )|∇h ε,N | 2 ,
where

F ε (u ε , A 1,ε ) = 1 2 Ω |∇ A 1,ε u| 2 + (curl A 1,ε -µ ε,N ) 2 + (1 -|u ε | 2 ) 2 2ε 2 + Ω (h ε,N -h ex -c ε,N )µ(u ε , A 1,ε ) + c ε,N Ω (µ(u ε , A 1,ε ) -µ ε,N ).
First we observe that Ω (1 -|u ε | 2 )|∇h ε,N | 2 = 0 since ∇h ε,N = 0 in ω ε and |u ε | = 1 outside ∪ p∈Λ B(p, M ε), which is included in ω ε if ε is small enough. Thus G ε (u ε , A ε ) = G N ε + F ε (u ε , A 1,ε ). There remains to evaluate F ε (u ε , A 1,ε ). By definition

µ(u ε , A 1,ε ) = curl (iu ε , ∇ A 1,ε u ε ) + curl A 1,ε = curl (|u ε | 2 (∇ϕ ε -A 1,ε )) + µ ε,N = curl (|u ε | 2 j ε ) + µ ε,N .
Since j ε = 0 on Ω c , we have Ω µ(u ε , A 1,ε ) = Ω µ ε,N = 2πN . Moreover, a direct computation shows that µ(u, A) = 0 where |u| ≡ 1 so µ(u ε , A 1,ε ) is supported in ω ε , where h ε,N -h ex -c ε,N = -1 2 | log ε ′ | (see (5.3)), so we deduce On the other hand, by choice of ϕ ε ,

Ω |∇ A 1,ε u ε | 2 = Ω |∇|u ε || 2 + Ω |u ε | 2 |∇ϕ ε -A 1,ε | 2 = Ω |∇|u ε || 2 + Ω |u ε | 2 |j ε | 2 .
Recalling that curl A 1,ε -µ ε,N = 0, we are thus led to

(7.19) G ε (u ε , A ε ) = G N ε -πN | log ε ′ | + 1 2 Ω |u ε | 2 |j ε | 2 + |∇|u ε || 2 + (1 -|u ε | 2 ) 2 2ε 2 .
It remains to estimate the terms on the right-hand side.

Lemma 7.6 (Energy in B(p, M ε)). For every p ∈ Λ, we have

(7.20) 1 2 B(p,M ε) |u ε | 2 |j ε | 2 + |∇|u ε || 2 + (1 -|u ε | 2 ) 2 2ε 2 = π log M + γ + o M (1) + o ε (1)
where o M (1) → 0 as M → ∞ and o ε (1) → 0 as ε → 0, and γ is the constant of (1.52).

Proof. From (1.52), and since u 0 (r, θ) = f (r)e iθ , we have

γ = lim M →+∞ 1 2 M 0 f ′ 2 + f 2 r 2 + (1 -f 2 ) 2 2 2πr dr -π log M .
On the other hand, from (7.9) in Proposition 7.5 and since |u ε | ≤ 1, we have for any p ∈ Λ, by Hölder's inequality

B(p,M ε) |u ε | 2 j ε -∇ ⊥ log | • -p| 2 ≤ |B(p, M ε)| 1-2 q j ε -∇ ⊥ log | • -p| 2 L q (B(p,M ε)) ≤ M ε ℓ ′ ε 2-4 q = o ε (1),
choosing q > 2 and since ℓ ′ ε ≫ ε. Moreover, choosing q > 2 and 1 q + 1 q ′ = 1, we have by Hölder again

B(p,M ε) |u ε | 2 ∇ ⊥ log | • -p| • (j ε -∇ ⊥ log | • -p|) ≤ j ε -∇ ⊥ log | • -p| L q (B(p,M ε)) 1 |x| L q ′ (B(0,ε)) ≤ C(ℓ ′ ε ) 2 q -1 ε 2 q ′ -1 = o ε (1)
where we used (7.9) and the fact that ℓ ′ ε ≫ ε by (1.48). Finally, since

B(p,M ε) |u ε | 2 ∇ ⊥ log | • -p| 2 = B(p,M ε) f 2 (|x|/ε) |x| 2 dx,
we deduce that for each p ∈ Λ,

B(p,M ε) |u ε | 2 |j ε | 2 = B(0,M ε) f 2 (|x|/ε) |x| 2 + o ε (1).
Following [START_REF] Sandier | Vortices in the Magnetic Ginzburg-Landau Model[END_REF] p. 210, we deduce that (7.20) holds.

Next, we consider the energy in the annuli B(p, ℓ ′ ε η) \ B(p, M ε), which are disjoint when η < η 0 .

Lemma 7.7 (Energy in the annuli). For every p ∈ Λ, we have

(7.21) 1 2 B(p,ℓ ′ ε η)\B(p,M ε) |u ε | 2 |j ε | 2 + |∇|u ε || 2 + (1 -|u ε | 2 ) 2 2ε 2 ≤ π log ηℓ ′ ε M ε + Cη.
Proof. Since |u ε | = 1 on the annulus A = B(p, ℓ ′ ε η) \ B(p, M ε) only the first term in (7.21) needs to be bounded. Using (7.9) in Proposition 7.5 we have for q > 2

A |j ε -∇ ⊥ log | • -p|| 2 ≤ |A| 1-2 q j ε -∇ ⊥ log | • -p| 2 L q (A) ≤ C q η 2-4 q .
A similar argument yields, for any q ′ ∈ [1, 2),

A (j ε -∇ ⊥ log | • -p|) • ∇ ⊥ log | • -p| ≤ C q ′ η 2 q ′ -1 .
We deduce, choosing for instance q = 4, q ′ = 1 above, proving (7.21). From (7.19),(7.8),(7.20),(7.21), and letting M → +∞, η → 0, we deduce, since m ε → m λ as ε → 0 and since 2πN = m ε h ex |ω ε | and ℓ ′ ε ε = 1 √ mεε ′ that for any R > 0

1 2 A |j ε | 2 ≤ 1 2 A |∇ ⊥ log | • -p|| 2 + Cη = π log ηℓ ′ ε M ε + Cη,
(7.22) G ε (u ε , A ε ) ≤ G N ε + N π log 1 √ m λ + γ + 2πN W (j R , 1 K R ) |K R | + o ε (N ).
We recall our choice of j R . To prove item 1 of the theorem j R was the result of applying Corollary 4.4 to a minimizer of W , hence from (4.6) was such that lim sup

R→∞ W (j R , 1 K R ) |K R | ≤ min A 1 W,
so that letting R → ∞ in (7.22) we find

G ε (u ε , A ε ) ≤ G N ε + N 2π(min A 1 W - 1 4 log m λ ) + γ + o(1) .
In view of (1.12), this proves (7.2).

To prove item 2 of the theorem, we chose j R given by Corollary 4.5 applied to P , so that, using (7.4), lim sup

R→∞ W (j R , 1 K R ) |K R | ≤ W K (j) d P (j) = 1 m λ W K (j) dP (j) + 1 4 log m λ .
By letting R → ∞ in (7.22), the result (7.3) follows.

To conclude the proof of the theorem, it remains to show that P ε → P , where P ε is the push-forward of the normalized Lebesgue measure on ω ε,N by x → 1 √ hex j(u ε , A ε ) x + • √ hex . Equivalently it suffices to show that Pε → P where Pε is the push-forward of the normalized Lebesgue measure on ω ε,N by x → 1 √ m λ hex j(u ε , A ε )(x +

• √ m λ hex ). Let Φ be a continuous function on L p loc (R 2 , R 2 ). By definition Φ(j) d Pε (j) = -

ω ε,N Φ 1 √ m λ h ex j(u ε , A ε )(x + • √ m λ h ex ) dx.
For any η > 0, we also have

Φ(j) d Pε (j) = - {dist (x,∂ ωε≥η)} Φ 1 √ m λ h ex j(u ε , A ε )(x + • √ m λ h ex ) dx + o η (1)
where o η (1) → 0 as η → 0. On the other hand by definition of (u ε , A ε ),

j(u ε , A ε ) = j ε + (|u ε | 2 -1)j ε -|u ε | 2 ∇ ⊥ h ε,N . But h ε,N is constant in ω ε,N
, and (|u ε | 2 -1)j ε → 0 in L p so we deduce that in {dist (x, ∂ ωε ) ≥ η} we have

1 √ m λ hex j(u ε , A ε )(x+ • √ m λ hex ) → j R in L p loc (R 2 , R 2 ), as ε → 0. It follows that Φ(j) d Pε (j) = - K R Φ(j R (x • )) dx + o ε (1) + o η (1) = Φ(j) dP R (j) + o ε (1) + o η (1),
where we have used the periodicity of j R , and where P R is as in Corollary 4.5. But P R → P as R → ∞ so letting ε → 0, η → 0, and R → +∞, we obtain the desired result.

Our task here is to establish more precisely this behaviour, in particular obtain some uniform estimates of convergence of ω m (after blow-up at a suitable scale) to an ellipse. We recall that we make the assumption (1.31). It is standard (see [ST]) that there exists an ellipse E Q of measure 1 and a nonnegative function U Q defined in R 2 such that (A.4)

∆U Q = ∆Q 2 1 R 2 \E Q , {U Q = 0} = E Q .
One may check that this U Q is unique and that

U Q = Cst + Q 2 - ∆Q 4π log * 1 E Q .
This ellipse will be shown to be the limit of the coincidence sets ω m as m ց h 0 , rescaled at a scale L m which is not completely obvious to guess since it contains a logarithmic factor, more precisely

(A.5) L m ∼ π(m -h 0 ) h 0 | log(m -h 0 )| as m ց h 0 .
Our main result is the following.

Proposition A.1. Let H m as above the minimizer of (A.2). The following holds. 3. For any δ > 0, if 1 -m is small enough then x ∈ Ω \ ω m implies that d(x, ∂Ω) 2 < (1 -m)(2 + δ). In particular there exists a constant C depending only on Ω such that

(A.6) |Ω\ω m | ≤ C √ 1 -m ∇H m L ∞ (Ω) ≤ C √ 1 -m.
4. Assuming (1.31), there is a length L m such that

(A.7) L m 2 | log L m | ∼ 2π(m -h 0 )/h 0 as m → h 0
and such that for any M, δ > 0, if m is sufficiently close to h 0 then .8) where ω Q = x 0 + L m E Q . In particular |ω m | ∼ L m 2 as m → h 0 .

{d(x, ω Q c ) > δL m } ⊂ ω m ⊂ {d(x, ω Q ) < δL m }, H m -m L m 2 ≤ M ⊂ x 0 + L m {U Q ≤ M + δ}, H m -m L m 2 ≥ M ⊂ x 0 + L m {U Q ≥ M -δ}. (A
We claim that if η is well-chosen and 1 -m is small enough, the function h is an interior barrier for H m . Indeed h(x) = f • d hence ∆h = f ′′ (d)|∇d| 2 + f ′ (d)∆d with

f ′ (d) = 2(1 -m) η 2 (d -η)1 d≤η , f ′′ (d) = 2(1 -m) η 2 1 d≤η .
In addition we have |∇d| = 1 and |∆d| ≤ κ Ω , where κ Ω is the maximum of the curvature on ∂Ω. Thus -∆h + h is trivially positive on {d ≥ η} while on the set {d < η} we have

-∆h + h ≥ -2 1 -m η 2 -2 1 -m η κ Ω + m.
Letting η 2 = (2 + δ)(1 -m), the right-hand side is positive for m close enough to 1 (depending on δ). Then h is an interior barrier for H m and we deduce that {d 2 > (2 + δ)(1 -m)} ⊂ ω m . The first result in (A.6), i.e. |Ω\ω m | ≤ C √ 1 -m, follows immediately, for some appropriate C depending on Ω. The second assertion is a consequence of the first one, together with the estimate H m C 1,1 (Ω) ≤ C, with C independent of m (see [BK]). Indeed, either x ∈ ω m and then ∇H m (x) = 0, or x / ∈ ω m and since there exists y ∈ ω m such that |x -y| ≤ C √ 1 -m, the C 1,1 estimate implies that

|∇H m (x)| ≤ |∇H m (y)| + C √ 1 -m = C √ 1 -m.
Proof of 4). Let ω Q = x 0 + L m E Q , for some L m to be specified below, and define h to be the solution of -∆h + h = m1 ω Q in Ω and h = 1 on ∂Ω. Then we may express h as h(x) = m G Ω (x, y)1 ω Q (y) dy + h 0 (x),

where G Ω (•, y) is the solution of -∆G Ω + G Ω = δ y in Ω and G Ω = 0 on ∂Ω. We further split G Ω as G Ω (x, y) = -1 2π log |x -y| + S Ω (x, y),

where S Ω (•, y) solves -∆S Ω + S Ω = (2π) -1 log | • -y| in Ω, thus is C 1 locally in Ω.

Replacing accordingly in the expression of h and writing x = x 0 + L m x ′ , we obtain (using the fact that the volume of E Q is 1)

h(x) = h 0 - m 2π L m 2 log L m + L m 2 mS Ω (x 0 , x 0 ) + L m 2 1 2 Q(x ′ ) - m 2π log * 1 E Q (x ′ ) + R(x),
where R(x) = h 0 (x) -h 0 -1 2 L m 2 Q(x ′ ) + m (S Ω (x, y) -S Ω (x 0 , x 0 )) 1 ω Q (y) dy.

  (3.15) clearly implies (3.14). To prove (3.16), for any p ∈ Λ * we let K p be the Voronoi cell centered at p, i.e. the set of points in R 2 which are closer to p than to any other point of Λ * . Then (3.17) wN (n, Λ) = p∈Λ * \{0} |p|≥N

  Lemma 4.7. Let χ be a positive function compactly supported in an open set U and assume that curl j = ν -m in U := {x | d(x, U ) < 1}

  which in view of (4.21)-(4.22) and (1.5) proves the bound for |ν(U R ) -|U R ||. The bound for ν

Lemma 5. 1 .

 1 For any 0 ≤ N ≤ |Ω|hex 2π there exists a unique m ∈ [1 -1 2λ Ω , 1] (and conversely) such that h ε,N = h ex H m , and m and N are related by 2πN = h ex m|ω m |. Moreover m and |ω m | are continuous increasing functions of N .

Proof.

  We call d(r) the density ratio above. Since ω m is convex, r → d(r) is decreasing. But from Proposition A.1, the diameter of ω m is bounded by CL m and |ω m | ≥ cL 2 m , where c, C > 0 are independent of m ∈ (h 0 , 1]. Therefore d(C) ≥ c/C 2 . Letting α = c/C 2 and r 0 = C proves the lemma. Now assume the hypothesis of Theorem 7 are satisfied. Then Lemma 5.1 applies and µ ε,N

  -∆f i = 2πδ y i -m ε h ex 1 C i in B(y i , Cℓ ′ ε ) ∂ ν f i = 0 on ∂B(y i , Cℓ ′ ε ),and then, letting j i = 0 on R 2 \ B(y i , Cℓ ′ ε ), we let ε = n i=1 j i . We have, letting Λ = {y 1 , . . . , y n }, curl ε = 2π p∈ b Λ δ p -m ε h ex 1 ωε .

  1), Note that still from Proposition 5.6, item 4,|ω ε | = o ε (1)|ω ε | and, since # Λ = ℓ ′ ε -2 (2π) -1 |ω ε |

  1) , and this holds uniformly in ε. Since|ω ε | = o ε (1)|ω ε |, which implies that |ω ε | = (1 -o ε (1))|ω ε |,

Ω

  (h ε,N -h ex -c ε,N )µ(u ε , A 1,ε ) + c ε,N Ω (µ(u ε , A 1,ε ) -µ ε,N ) = -πN | log ε ′ |.

1.

  The coincidence set ω m is empty if m < h 0 and has positive measure if m > h 0 . H m is increasing with m, the coincidence set ω m as well, and m → m|ω m | is a continuous, strictly increasing bijection from [h 0 , 1] to [0, |Ω|].2. If K is any compact subset of (h 0 , 1) then, uniformly with respect to m ∈ K, lim δ→0 |{x | d(x, ∂ω m ) < δ}| |ω m | = 0, lim δ→0 |{m < H m < m + δ}| |ω m | = 0.

  The first term in R(x) is the remainder of the Taylor expansion of order 2 of h 0 at x 0 . It is therefore O(|x-x 0 | 3 ) and its derivatives areO(|x-x 0 | 2 ) since h 0 is analytic. Since |ω Q | = L m 2 and since y ∈ ω Q =⇒ |y -x 0 | ≤ CL m , the second term in R(x) is clearly O(L m 2 |x -x 0 |) and, since S Ω is C 1 , differentiating under the integral sign shows its derivatives are O(L m 2 ).

  ). If |p| > n 2 it is straightforward to bound the contribution of K p , which we denote M p , by C Λ e -|p|/2 /|p| 2 , and from there to deduce

	(3.19)	lim N →+∞	lim n→+∞	p∈Λ * \{0} |p|≥n 2

  where we have written A η = A \ ∪ p∈Λ B(p, η). For any ball B ∈ B 2 -(k+1) , if B intersects the support of χ then B ⊂ U and thus the previous proposition yields

	Bη

Acknowledgements: We would like to thank warmly S.R.S. Varadhan for his suggestion to use the ergodic theorem. Many thanks also to our colleagues A. Venkatesh, C.S. Guntürk, L. Caffarelli, A. Chambert-Loir, N. Fournier, L. Mazet, M. Traizet and E. Lesigne for providing information and/or help on various steps. We also thank the referee, as well as our students and post-docs H. Ben Moussa, D. Goldman and A. Contreras, for their careful and critical reading. This work would also not have been possible without several invitations of E.S to New York University and S.S to Paris XII. We thank these two institutions, and acknowledge the financial support of the National Science Foundation, an EURYI award (SS), and the Institut Universitaire de France (ES).

It follows from Lemma 5.1 that h ε,N is also the solution to an obstacle problem, hence h ε,N ∈ C 1,1 (see [Fr]) and satisfies h ex m ε,N ≤ h ε,N ≤ h ex and µ ε,N := -∆h ε,N + h ε,N = m ε,N h ex 1 ω ε,N , where ω ε,N = {h ε,N = h ex m ε,N }.

Note that µ ε,N (Ω) = 2πN .

We will also let

It is immediate from (1.48), (5.2) that

The minimizer h 0,ε of (1.35) is equal to h ε,N 0 , where N 0 is given by (1.38). Moreover, we recall (see (1.36)) that h 0,ε = h ex -1 2 | log ε ′ | on its coincidence set, hence h ex m ε,N 0 = h ex -1 2 | log ε ′ | and c ε,N 0 = 0.

On the other hand H m is increasing with respect to m, see Proposition A.1, hence if m 1 ≤ m 2 then H m 1 ≤ H m 2 (see Proposition A.1 in the appendix) so

We will be most interested in the cases where N is one of the two integers closest to N 0 .

Energy-splitting

Let h µ , a "splitting function", be any function such that µ := -∆h µ + h µ ∈ L 2 (Ω) and h µ = h ex on ∂Ω. Let (5.5)

Then, Lemma 5.2. For any (u, A) and h µ as above we have

Proof. From (5.5), we have

where we have used the notation j(u, A) = (iu, ∇ A u). Also, since curl

Proof.

We then claim that (5.32)

Indeed we have

Applying the Cauchy-Schwarz inequality, it follows from (5.31) that

(5.32) is proven, since we recall that 2πN = m ε,N h ex |ω ε,N | with (5.2). Then (5.30) follows from (5.7).

We also note some consequences of the second Ginzburg-Landau equation (1.47).

Lemma 5.9. Assume that (u ε , A ε ) satisfies (1.47). Then, letting

Proof. By definition of j 1,ε and A 1,ε we have

). On the other hand, combining (5.32) with |u ε | ≤ 1 we find (5.34)

By direct calculation we have (5.33) follows again from (5.34).

Proof of Theorem 4, lower bound

In this section we state the key result from [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF] which we need in order to apply the framework of Section 2 to the minimization of the Ginzburg-Landau functional as explained in Section 1.8. In paragraphs 2 (resp. paragraph 3) we use it to derive the lower bound part of Theorems 4, 5 in the case of moderate (resp. high) applied fields. In paragraph 4 we prove Theorems 4 and 5 assuming the upper bound of Theorem 7, proven in Section 7.

On the other hand, from item 2 of Theorem 6 we have

Step 3: Inserting the above and (6.37) in (6.34) we obtain (6. 38)

On the other hand, by (6.33) and choice of χ we have

Since |∇ξ| = o(1) we know that {ξ < 1 4C 0 } ⊂ {ξ < 1 4 } for ε small enough and thus

We have obtained (6.39)

For the terms in (6.38) involving g, g + , we use the fact that ξ > 1 8C 0 on the support of χ:

A Additional results on the obstacle problem

Here we gather a few results on the obstacle problem that we need at various places in the paper. Although these results may be known to experts, we have not been able to find them in the literature, so they may be of independent interest. We focus on the particular type of obstacle problem we are concerned with, which is an obstacle problem with constant obstacle. More precisely, letting Ω be any smooth bounded domain in R 2 , for any m ∈ (-∞, 1] we denote by H m the minimizer of (A.1) min

By convex duality and the maximum principle (cf. [Br, BS]) it is equivalent to the obstacle problem

We also define the coincidence set

For general references on obstacle problems we refer for example to [KS]. (A.2) is a standard obstacle problem where the obstacle is constant and equal to m, and we are interested in the properties of the coincidence set ω m as m varies. It is known that

Note that the regularity of ω m for fixed m is well-known (see [Caf, BK] or the survey [Mo]), however this is not sufficient for our purposes, since we need estimates which are uniform in m. More precisely let us define h 0 to be the minimizer of the unconstrained problem, i.e. the solution of

and set

Then the situation is as follows:

where λ Ω is as in Section 1.6.

2. If h 0 < m ≤ 1 then ω m = ∅. Moreover, as m ր 1, ω m → Ω, and as m ց h 0 , ω m reduces to the set of points where h 0 achieves its minimum h 0 . If we assume in addition that h 0 is achieved at a unique point x 0 then as m ց h 0 , ω m is expected to shrink down to x 0 in an ellipse shape.

The main technique is the construction of barriers:

Lemma A.2. Let H m be as above. Let h be a continuous function.

Both assertions are easy and standard applications of the maximum principle.

We now turn to the proof of the proposition. First recall that from a well known result of J. Frehse ([Fr]), H m is C 1,1 and, as a consequence,

, so H m is increasing in m, and also H m ′ → H m uniformly in Ω as m ′ tends to m from above, and then in the distributions sense too. Hence, using

which is what we want since from 1) we have that inf m∈K |ω m | > 0.

Similarly, we have

Differentiating twice allows to bound its second derivatives by D 2 S Ω L q 1 ω Q L q ′ and using the equation satisfied by S Ω we have that D 2 S Ω L q for every q < +∞, while

Thus the second derivatives of the second term are bounded by C p L m p , for every p < 2. To sum up, for any p < 2 we have (A.9)

Then we note that 1 2 Q -

) and -∆h 0 + h 0 = 0 we have 1 2 ∆Q = ∆h 0 (x 0 ) = h 0 (x 0 ) = h 0 , and the claim follows from (A.4). Thus, if we choose for L m a solution of the following equation

we have

where

Since |h 0 -m| ≤ CL m 2 log L m , we easily deduce that R ′ satisfies the same properties as R, i.e. (A.9). Returning to (A.10), since the term L m 2 log L m dominates, we deduce that L m 2 | log L m | ∼ 2π(m -h 0 )/h 0 as m → h 0 and that (A.7) and (A.5) holds. From (A.11) and (A.9) for R ′ , we deduce that for any M, δ > 0 and if m -h 0 is small enough, then (A.12)

Indeed (A.9), (A.11) imply local uniform convergence of h(x 0 +Lmx ′ )-m

Lm 2

to U Q as m -h 0 decreases to 0, which implies that L m decreases to 0 by (A.5). Thus it suffices to check that h(x 0 + L m x ′ ) ≤ m + M L m 2 implies a uniform bound for x ′ . This is the case because the equation satisfied by

)) and we deduce from (A.11) a bound for U Q (x ′ ), hence for x ′ . Note in particular that the minimum of h(x 0 + L m x ′ ) is achieved in a fixed compact set of R 2 , and then (A.9), (A.11) yield (A.13) h := min

Next, we use h to construct an exterior and an interior barrier for H m . The exterior barrier is defined by h out = max(h-δL m 2 , m). Where h out = m it is obvious that -∆h out +h out = m. Then by definition h out (x) = m implies that h(x) -δL m 2 > m and then from (A.11)-(A.9) if m -h 0 is small enough, we must have

Therefore -∆h out + h out ≤ m1 {hout=m} . The other properties of exterior barriers are trivially verified by h out thus h out ≤ H m and then, using (A.12),

For the interior barrier, let χ : R + → R + be smooth and such that χ = 1 in a neighborhood of 0 and χ |[1,+∞) = 0. Then let h = min(m, min Ω h) and

We have -∆h int + h int = -∆h + m + h -h = m -h ≥ 0 on ω Q c and h int = 1 + (m -h) ≥ 1 on ∂Ω. Moreover, h int ≥ m. It remains to check that -∆h int + h int ≥ 0 in ω Q .

Since U Q = 0 on E Q and using (A.11),(A.9), we have on ω Q

It follows that if m -h 0 is small enough depending on δ, then -∆h int + h int ≥ m/2 in ω Q , finishing the proof that h int is an interior barrier for H m . Then h int ≥ H m in Ω and it follows easily using (A.12) that The relation {h int ≥ m + M L m 2 } = {h ≥ m + M L m 2 } follows from the fact that h int = h outside ω Q and that if m -h 0 is small, then both sets are disjoint from ω Q : Indeed if x ∈ ω Q then from (A.9), (A.11) we have h(x) -m = O(L m 3 ) and h int (x) -m = O(L m 3 ) from (A.13). Then, (A.8) follows from (A.14), (A.15). The fact that |ω m | ∼ L m 2 as m → h 0 is an easy consequence.