Keywords: acquisition précoce du langage, linguistique informatique, phonologie, variation allophonique

Note sur la génération de règles allophoniques

Résumé : Les algorithmes modélisant l'acquisition précoce des règles allophoniques ont besoin de grammaires allophoniques de référence pour évaluer leurs résultats. Cependant, aucune grammaire à large couverture n'a encore été décrite et, surtout, la complexité allophonique est un paramètre qu'il est souhaitable de contrôler. Pour pallier ces problèmes, il est d'usage d'appliquer des grammaires de règles allophoniques artificielles à des corpus de parole adressée aux enfants transcrits phonémiquement. Nous présentons un nouvel algorithme permettant de générer de telles règles. La plausibilité linguistique des règles est garantie par l'utilisation d'une description en traits distinctifs des phonèmes. En contrôlant la taille de la grammaire allophonique, nous sommes capables de de générer des transcriptions présentant des granularités phonétiques variées.

Introduction

Algorithms modeling the acquisition of allophonic rules by infants need a 'standard' allophonic grammar against which to evaluate their results. Even if such grammars may be compiled from the linguistic literature for some well-described languages, we do not know how many phonetic and phonological variants infants actually process when they learn the phonemic categories of their native language. Therefore, to evaluate the algorithms' robustness without a precise idea of the true complexity of the task, we will want to vary the number of phoneme variants as a parameter. As no database of infant-directed speech 1 containing rich phonetic transcriptions or high-quality aligned audio from which such rules could be derived has been released yet, a common workaround in the absence of such resources has been to apply grammars of (attested or artificial) allophonic rules to phonemically-transcribed corpora of infant-directed speech [START_REF] Golinkoff | Approche computationnelle de l'acquisition précoce des phonèmes[END_REF][START_REF] Le Calvez | Bottom-up learning of phonemes: A computational study[END_REF]9,[START_REF] Peperkamp | The acquisition of allophonic rules: statistical learning with linguistic constraints[END_REF]. The automatic application of the rules thus systematizes contextual variation between adjacent segments. Furthermore, controlling the number of rules in the grammar allows for the creation of a continuum of possible inputs, ranging from coarse-grained to fine-grained phonetic transcriptions.

Peperkamp, Le Calvez et al. [START_REF] Golinkoff | Approche computationnelle de l'acquisition précoce des phonèmes[END_REF][START_REF] Le Calvez | Bottom-up learning of phonemes: A computational study[END_REF][START_REF] Peperkamp | The acquisition of allophonic rules: statistical learning with linguistic constraints[END_REF] applied sets of hand-written allophonic rules, e.g. the palatalization rule of the velar consonants /k, g/ before the front vowels and glides /i, y, e, E, ø, oe, j, 4, Ẽ/ in French. This method has an obvious advantage as rules are not only linguistically plausible, but also attested in the language at hand. However, the small allophonic grammars used in these studies introduced a low number of contextual variants: Le Calvez et al. [START_REF] Le Calvez | Bottom-up learning of phonemes: A computational study[END_REF], for example, implemented only 11 allophonic rules. Martin et al. [9] recently circumvented this problem using a simple language-independent algorithm where allophones are nothing but numbered versions of the target phoneme, e.g. using [p 1 , p 2 , p 3 ] as contextual realizations of /p/. Concretely, to generate n allophones of a given phoneme, the set of all its possible contexts (i.e. the phonemic inventory and the utterance boundary) is split into n partitions, and each partition is then randomly tied to one numbered allophone. However, as allophones are conditioned partitioning the set of all possible contexts, and not the set of all attested contexts, this algorithm may generate rules which cannot apply because of phonotactic constraints. If no member of a partition is attested as a context of the target phoneme, e.g. a rule describing the realization of /4/ before /k/ in French, then the allophone will never be realized. The actual granularity of the emulated phonetic transcription may thus be far less than expected.

We present a new algorithm for generating allophonic rules, combining the linguistic plausibility of Peperkamp et al.'s rules and Martin et al.'s ability to emulate rich transcriptions. We first describe the algorithm, and then present experimental results showing we can outperform Martin et al.'s algorithm in increasing the granularity of the phonetic transcriptions we emulate. 1 Infant-directed speech differs from adult-directed speech, and there is evidence that it has special properties that facilitate learning [START_REF] Fernald | The perceptual and affective salience of mothers' speech to infants[END_REF]. Though phonetically-transcribed databases of adult-directed speech exist, they are useless for our purposes. See, however, work by Dautriche using adult-directed speech, where allophones are defined in terms of acoustic similarity [START_REF] Dautriche | Modélisation des processus d'acquisition du langage par des méthodes statistiques[END_REF].

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2 Feature-based generation of allophonic rules

The major contribution is to enforce the linguistic plausibility of the rules we generate. The plausibility of an artificial allophonic rule depends on the relation between the target phoneme, the application context and the allophone. As the vast majority of allophonic rules are assimilatory, we want to guarantee that the allophone shares some properties with both the target phoneme and the context. For example, a rule characterizing voicing assimilation describes a common phenomenon, even if it is not attested in the language at hand. By contrast, a rule where the allophone has nothing in common with neither the target phoneme nor the application context, such as a → 4 / k, is unlikely.

The general idea is to create assimilatory rules whose application contexts span similar contexts of the target phoneme. To do so, we represent the internal structure of the phonemes in terms of distinctive features: similarity between contexts can thus be either articulatory or acoustic, depending on the features. In order to maximize the number of allophones, each attested context of the target phoneme must fall in the scope of exactly one rule. As a consequence, if the desired number of allophones is inferior to the number of contexts, the latter must be aggregated into as many exclusive clusters as desired allophones. We represent a context cluster by an under-specified phonological matrix whose features are the ones shared by the members of that cluster and, to make the rules as plausible as possible, maximize phonological similarity between members.

In line with previously reported experiments [START_REF] Golinkoff | Approche computationnelle de l'acquisition précoce des phonèmes[END_REF][START_REF] Le Calvez | Bottom-up learning of phonemes: A computational study[END_REF]9,[START_REF] Peperkamp | The acquisition of allophonic rules: statistical learning with linguistic constraints[END_REF], we make two simplifying assumptions about the nature of the allophonic rules we generate. First, all rules are of the type p → a / c where a phoneme p is realized as its allophone a before context c. Second, we ensured that no two allophonic rules introduced the same allophone (as in English flapping, where both /t/ and /d/ have an allophone [R]) using parent annotation: each phone is marked by the phoneme it is derived from (e.g. [R] /t/ and [R] /d/ ).

Algorithm

The description of the algorithm relies on a few operations with phonological attribute-value matrices:

• a matrix m is an extension of another matrix m ′ if and only if all attributes in m ′ are specified in m with equal values;

• the fusion of two matrices m and m ′ , m • m ′ , is defined as follows: if an attribute is specified in only one of the two matrices, then it is specified in the resulting matrix with the same value; else, if it is specified in both, then it is specified in the resulting matrix with the value it has in m';

• the fission of a matrix m by an attribute a, m⋆a, yields two new matrices combining the information in m and one of the values a can take, i.e. m ⋆ a = {m • [-a], m • [+a]} in the case of traditional binary attributes.

Starting from an empty matrix, of which any phoneme is an extension, the algorithm successively fission matrices, incorporating more and more features. Each of the final matrices is the application context of an allophonic rule of the type p → a / c. Such rules are assimilatory because we define the allophone

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as the fusion of the target phoneme and its context. Let p be a phoneme and C p the set of its attested contexts, the procedure to generate n ≤ |C p | rules can be detailed as follows:

1. initialize the set of context clusters C with an empty matrix, and a buffer B as an empty set;

2. if B is empty, then:

(a) pick at random and remove a matrix m from the subset of the smallest matrices in C;

(b) let C ′ p be the subset of C p whose members are extensions of m, and a one of the attributes yet not specified in m whose values partition C ′ p in subsets as evenly-sized as possible: fill B with m ⋆ a; 3. pick at random and remove a matrix from B, and add it to C; 4. repeat from step 2 while |C| < n;

5. for each context c ∈ C, create a new rule p → p • c / c.
As shown in Figure 1, the successive states of the set of contexts C can be represented as a tree where the leaves are the actual members of the set, and the history of the fissions can be recollected by a walk up to the root.

-nasal -nasal -high -nasal +high +nasal +nasal -voiced   +nasal -voiced -low     +nasal -voiced +low   +nasal +voiced

Example outputs

Examples of artificial allophonic variation, using Dell's articulatory description of French phonemes [START_REF] Dell | Les règles et les sons[END_REF], are presented in Figures 2 and3 where the original phoneme is at the top of the matrix and feature values modified by rule application are circled.

                        k - + - ⊕ + - + - - - - - ⊕                                                 a - + + + - - - - + - + + +                                                 n + - - + + + - + - - + - +                                                 a - + + + - - - - + - + + +                                                 r - + - + + - - - + - + - ⊖                                                 f + - - + + - - - + - - - -                                                 u - + - + - - + - + + + + +                         Figure 2:
                        i - - - ⊖ ⊕ - + - + - + + -                                                 k - + - - + - + - - ⊕ - ⊕ -                                                 u - + - + ⊕ - + - + + + + +                                                 r - + ⊕ + ⊖ - - - + ⊖ + - +                                                 a ⊕ + + + ⊕ - - - ⊖ - + + ⊖                                                 p + - - - + - - - ⊕ - ⊕ - -                                                 r - ⊖ ⊕ + ⊖ - - - + - + - +                                                 E + - + + ⊕ ⊕ - - + - ⊕ + +                                                 l + - ⊕ + ⊖ + - - ⊖ - + - +                                                 @ ⊖ + + ⊖ - - - - - - - + ⊖                                                 k - + - - + - + - - ⊕ - ⊕ -                                                 o ⊖ + - + ⊕ ⊕ ⊕ - + + - + ⊖                                                 S - ⊕ - + + + + ⊕ + ⊕ - ⊕ -                                                 Õ - + + + - - - + + + + + +                        
Figure 3: Artificial allophonic variation of the French utterance /ikuraprEl@koSÕ/ (il court après le cochon, 'he runs after the pig').

Evaluation

The amount of contextual variation the application of an allophonic grammar introduces in a corpus can be quantified by the average number of allophones per phoneme, i.e. the ratio between the number of attested phones in the now phonetically-transcribed corpus and the number of phonemes in the language. We refer to this quantity as the allophonic complexity of the corpus.

Method

We compare our algorithm to a variant of Martin et al.'s where numbered versions of the target phoneme are randomly tied to partitions of the set of its attested contexts. While preventing the generation of allophonic rules that cannot apply, this modification does not alter the core idea behind their algorithm. Moreover, and for both algorithms, we do not consider the utterance boundary

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as a possible application context as it can not be suitably described in terms of features, would they be articulatory or acoustic. We use corpora of three typologically different languages: English, French and Japanese. All were extracted from transcribed adult-infant verbal interactions collected in the CHILDES database [START_REF] Macwhinney | The CHILDES Project: Tools for Analyzing Talk[END_REF] and were derived for the purpose of previously published experiments [START_REF] Brent | Distributional regularity and phonotactic constraints are useful for segmentation[END_REF][START_REF] Golinkoff | Approche computationnelle de l'acquisition précoce des phonèmes[END_REF][START_REF] Le Calvez | Bottom-up learning of phonemes: A computational study[END_REF][START_REF] Peperkamp | The acquisition of allophonic rules: statistical learning with linguistic constraints[END_REF]. The English corpus is a subset of the Bernstein-Ratner corpus that was transcribed using an inventory of 50 phonemes [START_REF] Brent | Distributional regularity and phonotactic constraints are useful for segmentation[END_REF]. To make the French corpus, the Champaud, Leveillé and Rondal corpora were automatically transcribed and then manually corrected to match a set of 35 phonemes [START_REF] Golinkoff | Approche computationnelle de l'acquisition précoce des phonèmes[END_REF]. The Japanese sample was derived by automatically transcribing the Ishii and Noji corpora from rōmaji, using an inventory of 49 phonemes [START_REF] Golinkoff | Approche computationnelle de l'acquisition précoce des phonèmes[END_REF]. All transcription choices made by the authors were respected.

Both algorithms have a single parameter to vary: n, the desired allophonic complexity. For each language, starting from the phonemically-transcribed corpus, we generate corpora of increasing allophonic complexity, testing the following values of n: from 1 to 5, with increments of 1, and from 5 to the numbers of phonemes (included) in the language at hand, with increments of 5. Then, for each desired complexity, we compare the attested allophonic complexity in each algorithm's output. Due to random picks, both algorithms are non-deterministic processes: all scores reported below are the average of the values obtained over three distinct runs for each model, corpus and desired number of allophones.

Results and discussion

Attested allophonic complexities for both algorithms and all three languages are presented in Figure 4 as functions of the desired complexities. The main observation is that the nature of the rules -i.e., whether contexts are aggregated at random or based on a linguistic description of the phonemes-does not influence the shape of the curves but their growth rate and, as a consequence, the maximal allophonic complexities that can be introduced. It is also worth noting that introducing linguistic constraints in the generation of artificial allophonic rules does not reduce the maximal allophonic complexity, quite to the contrary. Indeed, our feature-based algorithm outperforms Martin et al.'s in all three languages: on average, we were able to introduce up to 23 contextual variants per phoneme in the English corpus (vs. 18 with random ties), 22 (vs. 17) in the French one and 11 (vs. 9) in the Japanese sample.

Both algorithms generate rules with monolateral application contexts and can therefore theoretically produce as many rules as the square of the number of phonemes, describing in this way the realization of each phoneme before any phoneme. Obviously, such allophonic complexities cannot be reached as not all phonemes occur in all contexts: while some are subject to phonotactic sequencing constraints (e.g. /4/ in French, which only occurs before a vowel), others simply occur very infrequently (e.g. /Z/ in English). Similarly, lower values for Japanese are not especially surprising as strict phonotactics constrain the maximal syllabic template to CVC, e.g. /hon/. By contrast, because consonant clusters are legal in these languages, syllables such as /strikt/ (CCCVCC) and /dEkstr/ (CVCCCC) can be observed in French and, equivalently, /stôIkt/ (CCCVCC) and /twElfTs/ (CCVCCCC) in English. In other words, whereas the possible contexts of a Japanese consonant are almost exclusively vowels, the contexts of most English and French consonants include both vowels and consonants. As a consequence, models of allophonic variation that only consider adjacent segments as contexts cannot create as many rules for languages with strict phonotactic constraints, such as Japanese, as they do with English or French.

Conclusion

We described a new algorithm to generate artificial assimilatory allophonic rules which, when applied to a phonemically-transcribed corpus, can emulate phonetic transcriptions. Moreover, controlling the size of the output allophonic grammar, we were able to generate transcriptions of various phonetic granularities. The only a priori linguistic knowledge the algorithm requires is a description of the phonemic inventory in terms of distinctive features. Most importantly, this algorithm satisfies two important properties: a significant number of contextual variants can be introduced for each phoneme and, though artificial, the rules are linguistically admissible.

Although we are looking forward to the setting up of phonetically-transcribed databases of infant-directed speech, this algorithm could be improved in various ways to better emulate phonetic transcriptions, for example by controlling the linguistic well-formedness of the phonological matrices during successive fissions. Furthermore, modeling 'sandwich' rules, of the type p → a / c 1 c 2 , with bilateral contexts would increase the number of attested contexts for each phoneme and, as a consequence, help to reach greater allophonic complexities. Most of all, considering bilateral contexts would improve the plausibility of the output rules not only from a linguistic point of view, but also with respect to speech processing studies [START_REF] Dautriche | Modélisation des processus d'acquisition du langage par des méthodes statistiques[END_REF][START_REF] Makhoul | State of the art in continuous speech recognition[END_REF] where various acoustic models represent contextual realizations as triphones.

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Figure 1 :

 1 Figure 1: Tree representation of application contexts generated by successive fissions of under-specified phonological matrices.

Figure 4 :

 4 Figure 4: Attested allophonic complexity as a function of the desired complexity.


Dos Mickaël

Santos

Study of a 3D-Ginzburg-Landau functional with a discontinuous pinning term

Keywords: Ginzburg-Landau energy, vorticity defects, pinning, energy minimization 2000 MSC: 35J55, 35J50, 35B40

In a convex domain Ω ⊂ R 3 , we consider the minimization of a 3D-Ginzburg-Landau type energy

) 2 with a discontinuous pinning term a among H 1 (Ω, C)-maps subject to a Dirichlet boundary condition g ∈ H 1/2 (∂Ω, S 1 ). The pinning term a : R 3 → R * + takes a constant value b ∈ (0, 1) in ω, an inner strictly convex subdomain of Ω, and 1 outside ω. We prove energy estimates with various error terms depending on assumptions on Ω, ω and g. In some special cases, we identify the vorticity defects via the concentration of the energy. Under hypotheses on the singularities of g (the singularities are polarized and quantified by their degrees which are ±1), vorticity defects are geodesics (computed w.r.t. a geodesic metric d a 2 depending only on a) joining two paired singularities of g p i &n σ(i) where σ is a minimal connection (computed w.r.t. a metric d a 2 ) of the singularities of g and p 1 , ..., p k are the positive (resp. n 1 , ..., n k are the negative) singularities.

Introduction

In a convex domain Ω ⊂ R 3 , we consider the minimization of a 3D-Ginzburg-Landau type energy with a discontinuous pinning term among H 1 (Ω, C)-maps subject to a Dirichlet boundary condition g ∈ H 1/2 (∂Ω, S 1 ). The pinning term a : R 3 → R * + takes a constant value b ∈ (0, 1) in ω, an inner strictly convex subdomain of Ω, and 1 outside ω. The strict convexity of ω is not necessary but it allows to make a simpler description of the techniques used in this article.

Our Ginzburg-Landau type energy is

E ε (u) = 1 2 Ω |∇u(x)| 2 + 1 2ε 2 a(x) 2 -|u(x)| 2 2 dx. (1)
In [START_REF] Bethuel | Small energy solutions to the Ginzburg-Landau equation[END_REF], u ∈ H 1 g := {u ∈ H 1 (Ω, C) | tr ∂Ω u = g}. We are interested in studying the vorticity defects of minimizers of E ε in H 1 g via energetic estimates. In this article, letting u ε be such a minimizer, we aim in describing the set {|u ε | ≤ b/2} (this set is the vorticity defects). In the asymptotic ε → 0 we expect that, at least for special g's, the set {|u ε | ≤ b/2} takes the form of a union of thin wires whose endpoints are in ∂Ω; under this form the vorticity defects are called vorticity lines. We also expect that a concentration of the energy occurs around this set.

Because the pinning term is discontinuous, an energetical noise appears in a small neighborhood of the discontinuity set of a (this set is ∂ω).

In order to study the minimization problem of E ε in H 1 g we first consider an auxiliary minimization problem. Following [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF], we let U ε be the unique minimizer of E ε in H 1 1 := {u ∈ H 1 (Ω, C) | tr ∂Ω u ≡ 1}. The solution U ε plays an important role in the study. It allows to consider a decoupling of E ε (see Section 2). If v ∈ H 1 (Ω, C) and |v| ≡ 1 on ∂Ω, then [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF]

E ε (U ε v) = E ε (U ε ) + F ε (v), where F ε (v) = 1 2 Ω U 2 ε |∇v| 2 + U 4 ε 2ε 2 (1 -|v| 2 ) 2 .
Consequently the study of minimizers of E ε in H 1 g is related to the study of minimizers of F ε in H 1 g . Our techniques are directly inspired from those initially developed by Sandier in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF] (whose purpose was to give, in some special situations, a simple proof of the 3D analysis of the Ginzburg-Landau equation, by Lin and Rivière [START_REF] Lin | Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents[END_REF]), and by their adaptations in [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF].

We prove energy estimates with various error terms depending on our assumptions on Ω and g (see Theorems 2,3 & 4 in Section 4). In some special cases, we identify the vorticity lines via the concentration of the energy. At the end of this section, we will present a strategy which could lead to the localization of the vorticity lines.

The results we present are a first step towards a more precise description of the vorticity defects and of the asymptotic of minimizers.

Before stating our own results, we start by recalling the asymptotic expansion of the energy in the standard 3D-Ginzburg-Landau model (when a ≡ 1).

For g ∈ H 1/2 (∂Ω, S 1 ), if we let

E 0 ε (u) = 1 2 Ω |∇u| 2 + 1 2ε 2 (1 -|u| 2 ) 2 ,
then we have inf

H 1 g E 0 ε = C(g)| ln ε| + o(| ln ε|). (2)
Moreover, C(g) π is given by the length of a minimal connection connecting the singularities of g (in the spirit of Brezis, Coron, Lieb [START_REF] Brezis | Harmonic maps with defects[END_REF]). (See [START_REF] Lin | Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents[END_REF], [START_REF] Lin | A quantization property for static Ginzburg-Landau vortices[END_REF], [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF] and [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF]). For special g's and for a convex domain Ω, (2) was obtained by Lin and Rivière [START_REF] Lin | Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents[END_REF] (see also [START_REF] Lin | A quantization property for static Ginzburg-Landau vortices[END_REF]) and Sandier [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF]. The case of a general data g ∈ H 1/2 (∂Ω, S 1 ) and a simply connected Ω is due to Bourgain, Brezis and Mironescu [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF].

The above articles are the main references in this work. One of our main results is the analog of (2) for the minimization of F ε (Theorem 2). This result is first proved when g is in a dense set H ⊂ H 1/2 (∂Ω, S 1 ) and then extended by density. The upper bound is obtained directly using the techniques developed in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF] and [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF]. The lower bound needs an adaptation in the argument of Sandier [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF]. The main ingredient used to obtain a lower bound in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF] is the existence of a "structure function" adapted to the singularities of g (see Section 7). In the spirit of [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF], we prove, under suitable assumptions on Ω, ω and g, the existence of structure functions adapted to our situation (see Propositions 4,[START_REF] Demazure | Geometry of solutions to nonlinear problems[END_REF]7 and 10).

In our situation (a = b in ω and a = 1 in R 3 \ω), we have an analog of [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF] for inf H 1 g F ε replacing C(g) by C(g, a). When g admits a finite number of singularities, the constant C(g, a)/π is the length of a minimal connection between the singularities of g (see Section 3 for precise definitions). This minimal connection is computed w.r.t. a metric d a 2 depending only on a (see [START_REF] Liu | Vortex-filaments for inhomogeneous superconductors in three dimensions[END_REF]). (This generalizes the case of the standard potential (1 -|u| 2 ) 2 , where the distance is the euclidean one.)

When g has a finite number of singularities, one may prove a concentration of the energy along the vorticity lines (See Theorems 3 and 4). As in [START_REF] Lin | Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents[END_REF] and [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF], we obtain, after normalization, that the energy of minimizer is uniform along the vorticity lines (See Theorem 3). These vorticity lines are identified: they are geodesic segments associated to d a 2 .

In order to sum up our main results we state a theorem in a simplified form. This theorem is a direct consequence of Theorems 2, 3 and 4 stated in Section 4.

Theorem 1. Let g ∈ H 1/2 (∂Ω, S 1 ) then we have

• inf H 1 g E ε = E ε (U ε ) + C(g, a)| ln ε| + o(| ln ε|) where E ε (U ε ) ∼ ε -1
and C(g, a) depends only on the singularities of g and on a (it is the length of a minimal connection of the singularities of g computed w.r.t. the distance d a 2 ).

• Let g be a prepared boundary condition with a finite number of singularities of degree ±1 (g ∈ H, H defined in [START_REF] Chillingworth | Ginzburg-Landau model with small pinning domains[END_REF]) and let p 1 , ..., p k (resp. n 1 , ..., n k ) be the positive (resp. negative) singularities of g. Let Γ be the geodesic link of the singularities (we assume that Γ is unique), i.e., Γ is a union of k geodesic curves joining p i with n σ(i) where σ is a permutation of {1, ..., k} s.t. the total d a 2 -length of the curves is minimal.

Letting v ε be a minimizer of

F ε in H 1 g (i.e. U ε v ε minimizes E ε ) we have U 2 ε 2 |∇v ε | 2 + U 4 ε 4ε 2 (1 -|v ε | 2 ) 2 | ln ε| H 3 weakly converges in Ω in the sense of the measures to πa 2 H 1 |Γ .
Here H 3 is the 3-dimensional Hausdorff measure and H 1 |Γ is the one dimensional Hausdorff measure on Γ The goal of this work is to explain how the vorticity lines are modified under the effect of a pinning term. Although from the theorems below we have an idea about the form of the vorticity defects, we do not identify exactly the set {|u ε | ≤ b/2} (this set is the vorticity defects). In the study of Ginzburg-Landau type energies, it is standard to detect {|u ε | ≤ b/2} in a first step via a concentration of the energy. Once this done, in a second step, coupling energy estimates with an η-ellipticity result (see below) we get that the set where we have a concentration of the energy corresponds to the vorticity defects. In order to have a complete and rigorous description of the vorticity defects, we need an η-ellipticity results in the spirit of [START_REF] Bethuel | Small energy solutions to the Ginzburg-Landau equation[END_REF] for the minimizers of F ε . Namely: fix r > 0 then for small ε and v a minimizer of F ε if, in a ball B(x, r), the quantity

F ε (v, B(x, r)) | ln ε| is small, then |v(x)| ≃ 1.
It seems that an η-ellipticity result cannot be obtained by the standard method, which relies on a monotonicity formula obtained from a Pohozaev identity. The oscillating behavior of U ε yields impossible the direct application of monotonicity formulas. When U ε does not oscillate, it is possible to derive η-ellipticity (see e.g. [START_REF] Liu | Vortex-filaments for inhomogeneous superconductors in three dimensions[END_REF]). In our case, η-ellipticity would require a uniform control on the Lipschitz norm of U ε ; this does not hold in our situation.

Note that, the main result in [START_REF] Bethuel | Small energy solutions to the Ginzburg-Landau equation[END_REF] (an η-ellipticity result for critical points of E 0 ε ) may be applied in a ball B(x, r) s.t. B(x, r) ⊂ Ω \ ∂ω. Indeed, let u ε = U ε v ε be a minimizer of E ε in H 1 g and fix a ball B(x, r) ⊂ Ω \ ∂ω.

We let for z ∈ B(0, a(x)r) ũε (z

) = u ε x + z a(x) /a(x) which solves -∆u = ε -2 u(1 -|u| 2 ) in B(0, a(x)r). Note that if x / ∈ ω then ũ(z) = u(x + z). Because B(x, r) ⊂ Ω \ ∂ω, we have E ε (U ε , B(x, r)) → 0 and thus F ε (v ε , B(x, r)) | ln ε| is small ⇔ E ε (u ε , B(x, r)) | ln ε| is small ⇔ E 0 ε (ũ ε , B(0, a(x)r)) | ln ε| is small.
We may apply the main result of [START_REF] Bethuel | Small energy solutions to the Ginzburg-Landau equation[END_REF] with ũε in B(0, a(x)r) and, because |ũ ε (0

)| = |u ε (x)|/a(x),
our results of concentration of the energy for v ε imply that outside the set of concentration of the energy and "far from" ∂ω we have |u ε | > b/2. But this is not totally satisfying because, by a technical obstruction, we cannot cross over ∂ω. This paper is divided as follows:

• We first present the decoupling of Lassoued-Mironescu and fundamental properties of the special solution U ε (Section 2).

• In Section 3, we define and describe the main geometrical objects in the study (minimal connection/length, geodesic curve/link ...). Once this done, we state the main results in Section 4.

• The proofs of the main results are sketched in Section 5. In particular we explain how we may use [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF] and [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF] and we underline the required adaptations.

• The heart of the argument is based on energetic estimates. In Section 6 we proof upper bounds according to various assumptions (which give various error terms) and in Section 8 we obtain lower bounds. Section 7 is dedicated to the key tool used in Section 8.

• Section 9 is devoted to the last argument in the proof of Theorem 2.

The decoupling of Lassoued-Mironescu

Let ω ⊂ Ω ⊂ R 3 be two smooth bounded open sets s.t. Ω is convex and ω is strictly convex. For b ∈ (0, 1) we define

a : R 3 → {b, 1} x → b if x ∈ ω 1 otherwise . ( 3
)
We denote by E ε the Ginzburg-Landau functional with a as pinning term, namely

E ε (u) = 1 2 Ω |∇u(x)| 2 + 1 2ε 2 a(x) 2 -|u(x)| 2 2 dx.
For ε > 0, we let (see [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF]) U ε be the unique global minimizer of E ε in

H 1 1 := {u ∈ H 1 (Ω, C) | tr ∂Ω u ≡ 1}.
In the following, we will denote also

U ε ∈ H 1 loc (R 3 , C) the extension by 1 of the unique global minimizer of E ε in H 1 1 . Proposition 1. Let U ε be a minimizer of E ε in H 1 1
, then U ε is unique and the following assertions are true

1. U ε : R 3 → [b, 1] (from [8]), 2. -∆U ε = 1 ε 2 U ε (a 2 -U 2 ε ) in Ω, 3. E ε (U ε ) ∼ ε→0 1 ε 2 Ω (a 2 -U 2 ε ) 2 ∼ ε→0 1 ε (same argument as in [8]),
4. There are C, γ > 0 s.t. for x ∈ Ω we have (same proof as in [START_REF] Chillingworth | Ginzburg-Landau model with small pinning domains[END_REF] Proposition 2) [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF]) with

|U ε (x) -a(x)| ≤ Ce -γdist(x,∂ω)/ε , (4)
5. If v ∈ H 1 (Ω, C) is s.t. |tr ∂Ω v| = 1 then E ε (U ε v) = E ε (U ε ) + F ε (v) (same proof as
F ε (v) = 1 2 Ω U 2 ε |∇v| 2 + U 4 ε 2ε 2 (1 -|v| 2 ) 2 , (5)
6. If v minimizes F ε in H 1 g := {v ∈ H 1 (Ω, C) | tr ∂Ω v = g}, then |v| ≤ 1
in Ω (same proof as [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF]).

Assertion 5 of Proposition 1 expresses that minimization of E ε is related with those of F ε . This reformulation is standard in the context of pinned Ginzburg-Landau type energies. It allows to make a cleaning of the energy in order to detect concentration of energy.

To understand this cleaning and our main results (Theorems 2-4) we may see the minimization of E ε w.r.t. the boundary condition g ≡ 1 as the cheapest minimization problem in terms of the energy. Indeed, for u ∈ H

1 (Ω, C) s.t. |tr ∂Ω u| ≡ 1, we have E ε (u) ≥ E ε (|u|) ≥ E ε (U ε ) ∼ ε→0 1 ε .
Note that with the special boundary condition g ≡ 1 and in the study of a simplified Ginzburg-Landau energy without pinning term E 0 ε (obtained from E ε by taking a ≡ 1) we have that U 0 ε , the global minimizer of E 0 ε in H 1 1 , is constant equal to 1; and therefore E 0 ε (U 0 ε ) = 0. In the presence of a pinning term a (given by ( 3)), the special solution U ε carries a concentration of energy along the singular surface ∂ω. More precisely we may prove a refined version of Proposition 1.3: for η > 0 we have

E ε (U ε ) ∼ ε→0 E ε (U ε , {x ∈ Ω | dist(x, ∂ω) < η}) ∼ ε→0 1 ε and E ε (U ε , {x ∈ Ω | dist(x, ∂ω) > η}) → 0 when ε → 0. Here for V an open subset of Ω we denoted E ε (U ε , V ) = 1 2 V |∇U ε | 2 + 1 2ε 2 (a 2 -U 2 ε ) 2 .
One of the advantage of the Lassoued-Mironescu's decoupling is that, in the study of

E ε (•) = F ε ( • Uε ) + E ε (U ε )
, the singular part of E ε is entirely carried by the fixed number E ε (U ε ).

Our best results express this fact: for g ∈ H 1/2 (∂Ω, S 1 )

• The minimal energy takes a standard form:

inf

H 1 g E ε = E ε (U ε ) + inf H 1 g F ε = E ε (U ε ) + C(g, a)| ln ε| + o(| ln ε|) (see Theorems 2&4) where E ε (U ε ) ∼ ε -1 ;
• Under some hypotheses the concentration of the energy of F ε is uniform along curves (see Theorem 3). These curves are called vorticity lines.

To end this section we may underline the interpretation of Lassoued-Mironescu's decoupling

E ε (•) = F ε ( • Uε ) + E ε (U ε )
in terms of concentration of energy:

• At the first order (ε -1 ), the interface ∂ω corresponds to an "artificial" singular surface whose energetic cost is contained in E ε (U ε ),

• At the second order (| ln ε|), minimizers for F ε (•/U ε ) concentrate their energy along the vorticity lines. These curves are described in Section 3.2, they can be interpreted as "line segments" bent by the interface ∂ω.

Minimal connections, geodesic links

In this section we define the main geometrical objects needed in the description of the vorticity lines.

In subsection 3.1 we present a standard method which described the singularities of a map g ∈ H 1/2 (∂Ω, S 1 ): to a map g ∈ H 1/2 (∂Ω, S 1 ) we associate a distribution T g . In Proposition 2 we present the main properties of T g used in this article. We define also an important dense set of H 1/2 (∂Ω, S 1 ).

In subsection 3.2 we state some definitions and we describe the metric d a 2 which plays an important role in this study.

3.1 Length of a minimal connection of a map g ∈ H 1/2 (∂Ω, S 1 )

For g ∈ H 1/2 (∂Ω, S 1 ), following [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF], one may associate to g a continuous linear form

T g : (Lip(∂Ω, R), • Lip ) → R.
Here Lip(∂Ω, R) is the space of Lipschitz functions equipped with the standard norm

ϕ Lip = ϕ L ∞ + sup x,y∈∂Ω x =y |ϕ(x) -ϕ(y)| |x -y| with |x -y| = d eucl (x, y)
is the euclidean distance in R 3 between x and y.

For g ∈ H 1/2 (∂Ω, S 1 ), the map T g : Lip(∂Ω, R) → R is defined by the following way: let ϕ ∈ Lip(∂Ω, R)

• fix u ∈ H 1 g and consider H = 2(∂ 2 u ∧ ∂ 3 u , ∂ 3 u ∧ ∂ 1 u , ∂ 1 u ∧ ∂ 2 u); • fix φ ∈ Lip(Ω, R) s.t. φ = ϕ on ∂Ω;
then Ω H • ∇φ is independent of the choice of u and φ.

Therefore we may define the continuous linear form

T g : Lip(∂Ω, R) → R ϕ → Ω H • ∇φ . Notation 1.
Here "∧" stands for the "vectorial product" in C:

(x 1 + ıy 1 ) ∧ (x 2 + ıy 2 ) = x 1 y 2 -x 2 y 1 , x 1 , x 2 , y 1 , y 2 ∈ R.
Following [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF], we denote, for g ∈ H 1/2 (∂Ω, S 1 ) and d an equivalent distance with d eucl on ∂Ω,

L(g, d) := 1 2π sup {T g (ϕ) | |ϕ| d ≤ 1 } = 1 2π max {T g (ϕ) | |ϕ| d ≤ 1 } (6)
with

|ϕ| d := sup x =y x,y∈∂Ω |ϕ(x) -ϕ(y)| d(x, y) .
Note that, since T g : (Lip(∂Ω, R), • Lip ) → R is continuous and d, d eucl are equivalent on ∂Ω, then L(g, d) is finite.

In the spirit of [START_REF] Lin | Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents[END_REF], [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF] and [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF] we deal with prepared boundary conditions g's. In this article we use the dense subset H ⊂ H 1/2 (∂Ω, S 1 )

H =          g ∈ 1≤p<2 W 1,p (∂Ω, S 1 ) g is smooth outside a finite set C, ∀M ∈ C we have for x close to M : |∇g(x)| ≤ C/|x -M |, ∃ R M ∈ O(3) s.t. g(x) -R M x-M |x-M| ≤ C|x -M |.          . (7)
Here we considered

S 1 ≃ S 1 × {0} ⊂ S 2 .
One may define deg(u, M ), the topological degree of u with respect to

M : if R M ∈ O(3) + then deg(u, M ) = 1 otherwise deg(u, M ) = -1.
In order to justify the term of "degree", assume that in a neighborhood of M ∈ C, ∂Ω is flat. Then, for r > 0 sufficiently small, C = ∂B(M, r) ∩ ∂Ω is a circle centered in M . This circle has a natural orientation induced by B(M, r) ∩ Ω. Thus g |C ∈ C ∞ (C, S 1 ) admits a well defined topological degree (see e.g. [START_REF] Brezis | New questions related to the topological degree[END_REF]), and this degree does not depend on small r.

We may partition the set C into two sets: the positive singularities and the negative singularities. We consider

P = {M ∈ C | deg(u, M ) = 1} and N = {M ∈ C | deg(u, M ) = -1}.
One may also consider for g ∈ H the degree of g with respect to ∂U for U a non empty smooth open set of ∂Ω s.t. ∂U does not contain any singularities of g. This degree is defined as

deg(g, ∂U ) = Card({p ∈ P | p ∈ U }) -Card({n ∈ N | n ∈ U }). (8)
From [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF], we have the following

Proposition 2. Let g, h ∈ H 1/2 (∂Ω, S 1
), then we have

1. T gh = T g + T h and T g = -T g (Lemma 9 [2]), 2. |(T g -T h )(ϕ)| ≤ C|g -h| H 1/2 (|g| H 1/2 + |h| H 1/2 )|ϕ| d eucl , ϕ ∈ Lip(∂Ω, R) (Lemma 9 [2]), 3. H is dense in H 1/2 (∂Ω, S 1 ) (Lemma B.1 [2]), 4. if u ∈ H, then Card(P) = Card(N) and T g = 2π p∈P δ p -2π n∈N δ n (Lemma 2 [2]), 5. if u ∈ H, then L(g, d) = min σ∈S k i d(p i , n σ(i) )
where d is a distance equivalent with d eucl on ∂Ω (Theorem 1 [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF]). Here S k is the set of the permutations of {1, ..., k}.

Minimal connections, minimal length, geodesic links

In the last assertion of Proposition 2, we used the notion of length of a minimal connection w.r.t. a distance d.

Namely, consider d a distance on C = P ∪ N, P, N ⊂ R 3 two sets of k distinct points s.t. P ∩ N = ∅, P = {p 1 , ..., p k } and N = {n 1 , ..., n k }. Definition 1. We denote by L(C, d) the length of a minimal connection of C in (C, d), i.e.,

L(C, d) = min σ∈S k k i=1 d(p i , n σ(i) ). (9)
In [START_REF] Brezis | Harmonic maps with defects[END_REF] (Lemma 4.2), the authors proved that

L(C, d) = max k i=1 {ϕ(p i ) -ϕ(n i )} ϕ : C → R, |ϕ| C d ≤ 1 (10)
with

|ϕ| C d = sup x =y x,y∈C |ϕ(x) -ϕ(y)| d(x, y) . Definition 2. A permutation σ s.t. i d(p i , n σ(i) ) = L(P ∪ N, d) is called a minimal connection of (P ∪ N, d).
In the following we will consider a special form of distance d on ∂Ω: a geodesic distance in Ω equipped with a metric that we will describe below.

Let us first introduce some notations. Let f : R 3 → [b 2 , 1] be a Borel function and let Γ ⊂ Ω be Lipschitz curve. We denote by long f (Γ) the length of Γ in the metric f h (here h is the euclidean metric in R 3 ), i.e.,

long f (Γ) := 1 0 f [γ(s)] |γ ′ (s)|ds, γ : [0, 1] → Γ is an admissible parametrization of Γ.
In this paper, when we consider a curve (or arc) Γ, it will be implicitly that it is a Lipschitz one.

We define d f as the geodesic distance in f h (h is the euclidean metric in R 3 ). Thus, for x, y ∈ R 3 , x = y, we have

d f (x, y) = inf Γ Lipschitz arc with endpoints x, y long f (Γ). (11)
In the special case f = a 2 , one may easily prove the following proposition Proposition 3. Let x, y ∈ R 3 , x = y. The following assertions are true 1. In [START_REF] Liu | Vortex-filaments for inhomogeneous superconductors in three dimensions[END_REF] the infimum is attained.

We denote by Γ 0 a minimal curve in [START_REF] Liu | Vortex-filaments for inhomogeneous superconductors in three dimensions[END_REF].

2. If x, y ∈ Ω then a geodesic Γ 0 is included in Ω.
3. A geodesic Γ 0 is a union of at most three line segments:

Γ 0 = ∪ l i=1 S i 4. These line segments are such that a. if x, y ∈ ω then l = 1, b. if l = 2 then S 1 ∩ S 2 ⊂ ∂ω, c. if l = 3 then x, y ∈ R 3 \ ω and S 2 is a chord of ω, d. if [x, y] ∩ ω = {z} then l ∈ {2, 3}. 5. For M ∈ R 3 we have ψ M = d a 2 (•, M ) ∈ W 1,∞ loc (R 3 , R) and |∇ψ M | = a 2 . Definition 3. In the case d = d a 2 and C = P ∪ N ⊂ ∂Ω, we say that ∪ i Γ i is a geodesic link when σ is a minimal connection in (C, d a 2
) and Γ i is a geodesic joining p i to n σ(i) .

In Figure 1, we have represented a geodesic link for k = 2 and a certain b ∈ (0, 1). Remark 1. We consider the situation d = d a 2 and C = P ∪ N ⊂ ∂Ω.

1. If the minimal connection is unique, then, letting ∪ i Γ i be a geodesic link, we have Γ i ∩ Γ j = ∅ for i = j. In particular, if the geodesic link ∪ i Γ i is unique then Γ i ∩ Γ j = ∅ for i = j.

If a geodesic link

∪ i Γ i is s.t. for all i Γ i is a line segment, then we have Γ i ∩ Γ j = ∅ for i = j.

The main results

We now state our main results. The first result (Theorem 2) is an energetic estimation of inf H 1 g F ε in the spirit of [START_REF] Lin | Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents[END_REF], [START_REF] Lin | A quantization property for static Ginzburg-Landau vortices[END_REF], [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF] and [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF]. This result is stated for the most general boundary condition: g ∈ H 1/2 (∂Ω, S 1 ).

The second theorem (Theorem 3) is valid for the prepared boundary condition: g ∈ H. This result expresses that for solutions of inf H 1 g F ε , a concentration of the leading part of the energy occurs in a small neighborhood of the geodesic link of C in (R 3 , d a 2 ).

The last result (Theorem 4) gives a more precise result than Theorems 2&3 (a bounded error term instead of a o(| ln ε|)-error term) under strong symmetric hypothesis.

Theorem 2. Let g ∈ H 1/2 (∂Ω, S 1 ). Then we have inf Let v ε be a minimizer of F ε in H 1 g . Then the normalized energy density

H 1 g F ε = πL(g, d a 2 )| ln ε| + o(| ln ε|).
µ ε = U 2 ε 2 |∇v ε | 2 + U 4 ε 4ε 2 (1 -|v ε | 2 ) 2 | ln ε| H 3 weakly converges in Ω in the sense of the measures to πa 2 H 1 |∪iΓi .
Here H 3 is the 3-dimensional Hausdorff measure and

H 1 |∪iΓi is the one dimensional Hausdorff measure on ∪ i Γ i .
In other words

∀φ ∈ C 0 (Ω, R) ∩ L ∞ (Ω, R) we have Ω φ dµ ε → π ∪iΓi φa 2 dH 1 .
Note that this result gives a (uniform) energy concentration property of the minimizers along the geodesic link. Namely, for all compact K s.

t. K ∩ ∪ i Γ i = ∅, we have F ε (v ε , K) = o(| ln ε|).
In order to obtain a more precise statement we assume that we are in a very symmetrical case: Ω = B(0, 1) and ω = B(0, r 0 ), r 0 ∈ (0, 1), g ∈ H is s.t. C = {p, n} with p = -n. Under these hypotheses we have the following theorem.

Theorem 4. The following estimate holds inf

H 1 g F ε = πd a 2 (p, n)| ln ε| + O(1).
Moreover, for all η > 0, there is

C η > 0 s.t. denoting V η = {x ∈ Ω | dist(x, [p, n]) ≥ η} and v ε a minimizer of F ε in H 1 g , we have F ε (v ε , V η ) ≤ C η .

Outline of the proofs

The proofs of the above theorems strongly rely on the techniques developed in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF]. The proofs of Theorems 2, 3 and 4 consist essentially into two parts devoted to obtaining respectively lower and upper bounds.

The upper bound is obtained by the construction of a test function. The test function was obtained by Sandier in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF] in the situation where Ω is a strictly convex domain and there is a geodesic link in (C, d a 2 ) which is a union of line segments. In this special case, one may obtain (see Section 6.1): inf

H 1 g F ε ≤ πL(g, d a 2 )| ln ε| + O(1). (12)
For the general case, when the geodesic links are not unions of line segments, in [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF], Bourgain, Brezis and Mironescu adapted the construction of Sandier. In our case this leads to the bound:

inf H 1 g F ε ≤ πL(g, d a 2 )| ln ε| + o(| ln ε|). (13)
(See Section 6.2.)

The lower bounds are obtained as in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF]. The key ingredient is the construction of a "structure function" ξ : R 3 → R (see Section 7 for a precise definition). Due to the fact that for

M ∈ R 3 , x → ψ M (x) = d a 2 (x, M ) is not C 1 (its gradient is not continuous on ∂ω since |∇ψ M | = a 2 in R 3 \ ∂ω),
we cannot obtain ξ with exactly the same properties as in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF] (see Proposition 4). The consequence of this lack of smoothness of the distance function implies that our best lower bound is inf

H 1 g F ε ≥ πL(g, d a 2 )| ln ε| -o(| ln ε|). (14)
However, under strong symmetry hypotheses, namely, Ω = B(0, 1), ω = B(0, r 0 ) and C = {p, n = -p}, the structure function ξ enjoys additional properties (see Proposition 10). In this symmetric case, one may obtain the sharper bound inf

H 1 g F ε ≥ πL(g, d a 2 )| ln ε| -O(1). ( 15
)
The estimate on inf H 1 g F ε in Theorem 2 (resp. Theorem 4) is a direct consequence of (13), (14) (resp. ( 12) and ( 15)) and of the density of H in H 1/2 (∂Ω, S 1 ) (see Section 9) .

Theorem 3 is proved along the main lines in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF]. Roughly speaking, under the hypotheses of Theorem 3, for all x ∈ Ω, there is ρ x > 0 s.t. for K = B(x, ρ x ), one may consider a structure function ξ adapted to C which is constant in K (see Section 7.3). Arguing as in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF], if K does not intersect the geodesic link, then we obtain that in K, a minimizer of F ε has its energy of order o(| ln ε|) (see (35)). Thus µ, the weak limit of

µ ε = { U 2 ε 2 |∇v ε | 2 + U 4 ε 4ε 2 (1 -|v ε | 2 ) 2 }H 3 /| ln
ε| (which exists up to subsequence), is supported in Ω \ K. Therefore, one may prove that the support of µ is included in the geodesic link.

Otherwise, if x is on the geodesic link, as explained in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF], then we obtain for v ε a minimizer and ρ sufficiently small that (with K = B(x, ρ))

lim sup F ε (v ε , K) | ln ε| ≤ πlong a 2 (K ∩ ∪ i Γ i ).
Theorem 3 is obtained by comparing µ to πa 2 H 1 |∪iΓi .

6 First step in the proof of Theorem 2: an upper bound for inf H 1 g F ε , g ∈ H 6.1 The case C admits a geodesic link in (R 3 , d a 2 ) which is a union of lines

Assume that

• Ω is strictly convex;

• there is Γ = ∪Γ i , a geodesic link of C in (R 3 , d a 2 ), s.t. Γ i is a line segment for all i.
One may assume that the minimal connection associated to Γ is the identity and that Γ i is a geodesic curve between p i and n i .

It is easy to see that

• by strict convexity of Ω we have Γ i ⊂ Ω;

• from Remark 1.2 we have for i = j that Γ i ∩ Γ j = ∅.

In this situation, we may mimic the construction of the test function made in Section 1 of [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF]. The test function is a fixed (independent of ε) S 1 -valued function outside V η , an η-tubular neighborhood of Γ (η small and independent of ε).

Inside each tubular neighborhood V η,i of Γ i , the test function takes (essentially) the form (in an orthonormal basis {p i , (e x , e y , e z )} s.t.

n i = (0, 0, |p i -n i |)) v ε (x, y, z) =                    (x, y) |(x, y)| if η < z < |p i -n i | -η and ε < |(x, y)| < η (x, y) ε if η < z < |p i -n i | -η and |(x, y)| < ε 1 2 Ṽη,i |∇v ε | 2 + 1 2ε 2 (1 -|v ε | 2 ) 2 ≤ 2πη| ln ε| + C(g, η) with Ṽη,i = V η,i ∩ {0 < z < η or |p i -n i | -η < z < |p i -n i |} . ( 16
)
(see [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF] for more details).

From the strict convexity of ω, for all line D ⊂ R 3 , we have

long D ∩ {x ∈ R 3 | dist(x, ∂ω) ≤ √ ε} ≤ C √ ε with C > 0 independent of ε.
Thus one may obtain from Proposition 1.4. that (12) holds.

The general case

One may adapt the above construction to a more general situation: the geodesic links are not unions of line segments and Ω is convex (not strictly convex). Without loss of generality we may assume that σ = Id is a minimal connection.

For the standard Ginzburg-Landau energy, this has been done in [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF]; there, Ω is not supposed convex. Roughly speaking, the argument there consists in replacing in Sandier's proof, line segments by curves.

Their construction begins with

• the modification of Ω (flattening ∂Ω close to the singularities)

• for η > 0, we construct an approximate (smooth) geodesic link

Γ η = ∪ i Γ η i ⊂ Ω s.t. -long a 2 (Γ η ) ≤ long a 2 (Γ) + η,
-Γ is a geodesic link and Γ η i is an approximate geodesic curve between p i and n i , -Γ η i ∩ Γ η j = ∅ for i = j and Γ η i is orthogonal to ∂Ω at p i and n i . In order to be applicable to our situation, this construction requires the additional property about its intersection with a small neighborhood of ∂ω:

H 1 (Γ η ∩ {dist(x, ∂ω) < √ ε}) = O( √ ε).
We can clearly find Γ η satisfying this property. By adapting the construction of v ε in (16), one may construct a test function v η ε having Γ η as set of zeros and satisfying, for each η > 0, the estimate

inf v∈H 1 g F ε (v) | ln ε| ≤ F ε (v η ε ) | ln ε| + o ε (1) = πlong a 2 (Γ η ) + o ε (1) ≤ πlong a 2 (Γ) + η + o ε (1).
In order to obtain this estimate we rely on the formula of v η ε , Proposition 1.4. and the assumption

H 1 (Γ η ∩ {dist(x, ∂ω) < √ ε}) = O( √ ε).
Consequently we deduce that (13) holds.

Structure functions

For g ∈ H, we construct suitable structure functions adapted to the singularities of g. Very roughly speaking, a structure function ξ ∈ Lip(R 3 , R) is a smooth map whose restriction on ∂Ω almost maximizes [START_REF] Demazure | Geometry of solutions to nonlinear problems[END_REF]. More qualitative properties of ξ will be described in Propositions 4, 6, 7 and 10.

Structure functions are used to compute a lower bound of the energy. We may sketch the proof of a (sharp) lower bound of F ε (v ε ):

• We integrate by layers (on hypersurfaces) using the Coarea formula;

• The layers are the level sets of the structure function ξ : R which need a control on the second fundamental form of {ξ = t}. This is an another property of the structure function.

• The "quantity" of layers is related with the length of a minimal connection L(C, d a 2 ). In order to get a sharp lower bound we need that i {ξ(p i )ξ(n i )} L(C, d a 2 ); it is another important property of ξ.

We present below four structure functions corresponding to four different settings.

Second step in the proof of Theorem 2: a structure function

This subsection is devoted to the presentation of a structure function adapted to C and a. This function is given in the following. Proposition 4. For all η > 0, there is

C η > 0, E η ⊂ R, ξ η ∈ C ∞ (R 3 , R) and ε η > 0 s.t. for 0 < ε < ε η , 1. |∇ξ η | ≤ min(a 2 , U 2 ε + ε 4 ) in R 3 , 2. i∈N k {ξ η (p i ) -ξ η (n i )} ≥ L(C, d a 2 ) -η, 3. H 1 (E η ) ≤ η and for all t ∈ R \ E η , {ξ η = t} is a closed hypersurface whose second fundamental form is bounded by C η .
This proposition is proved in Section 7.2 Remark 2. In Proposition 4, the structure function ξ η does not dependent on (small) ε. The structure functions given by Propositions 6 and 7 (below) are also independent of small ε. On the other hand, the structure function given by Proposition 10 (below) depends on ε.

The proof of Proposition 4 7.2.1 Some definitions

Let δ 0 = 10 -2 min{1; dist(∂ω, ∂Ω)}. For 0 < δ ≤ δ 0 , we define ω δ := ω + B(0, δ), α 0 = a 2 and

α δ : R 3 → {1, b 2 } x → b 2 if x ∈ ω δ 1 otherwise .
For x, y ∈ R 3 and 0 ≤ δ < δ ′ ≤ δ 0 , we have

d α δ ′ (x, y) ≤ d α δ (x, y) ≤ d α δ ′ (x, y) + O(|δ ′ -δ|). (17)
The first inequality is a direct consequence of α δ ′ ≤ α δ . We prove the second inequality. Consider x, y ∈ R 3 s.t. d α δ ′ (x, y) < d α δ (x, y). We obtain that if Γ is a geodesic joining x and y in (R 3 , d α δ ′ ), then we have Γ ∩ ∂ω δ ′ = ∅. Note that by Proposition 3, we have

Card(Γ ∩ ∂ω δ ′ ) ∈ {1, 2}. Assume that Γ ∩ ∂ω δ ′ = {x ′ , y ′ } with d α δ ′ (x, x ′ ) < d α δ ′ (x, y ′ ).
The situation where Γ ∩ ∂ω δ ′ = {z} is similar.

Consider x ′′ = Π ω δ (x ′ ) and y ′′ = Π ω δ (y ′ ). Here Π ω δ stands for the orthogonal projection on ω δ . By the definition of x ′′ and y ′′ we have d eucl (x ′ , x ′′ ) = d eucl (y ′ , y ′′ ) = δ ′δ. By Proposition 3, we deduce that d α δ ′ (x ′′ , y ′′ ) = d α δ (x ′′ , y ′′ ).

Since x ′ , y ′ ∈ Γ, we have

d α δ ′ (x, y) = d α δ ′ (x, x ′ ) + d δ ′ (x ′ , y ′ ) + d α δ ′ (y ′ , y) ≥ d α δ ′ (x, x ′ ) + d α δ ′ (x ′′ , y ′′ ) + d α δ ′ (y ′ , y) -2|δ ′ -δ| ≥ d α δ (x, x ′ ) + d α δ (x ′′ , y ′′ ) + d α δ (y ′ , y) -2|δ ′ -δ| ≥ d α δ (x, x ′ ) + d α δ (x ′ , y ′ ) + d α δ (y ′ , y) -4|δ ′ -δ| ≥ d α δ (x, y) -4|δ ′ -δ|
Consequently, (17) holds. Thus, for C = P ∪ N defined above, we obtain that

L(C, d α δ ) = L(C, d α δ ′ ) + O(|δ ′ -δ|). ( 18
)
Remark 3. Using (18), it is easy to check that there is

δ b,ω,C < δ 0 s.t. for δ < δ b,ω,C : If σ is a minimal connection in (C, d α δ ), then σ is a minimal connection in (C, d α0 ).
We now state a preliminary result: Proposition 5. Proposition 4 will be a direct consequence of Proposition 5. Proposition 5. For η > 0 there is δ η > 0 s.t. for δ η > δ > 0 there are C η,δ > 0, E η,δ ⊂ R and

ξ η,δ ∈ C ∞ (R 3 , R) s.t. 1. |∇ξ η,δ | ≤ α δ in R 3 2. i∈N k {ξ η,δ (p i ) -ξ η,δ (n i )} ≥ L(C, d α δ ) -η 3. H 1 (E η,δ ) ≤ η
and for all t ∈ R \ E η,δ , {ξ η,δ = t} is a closed two dimensional surface with its second fundamental form which is bounded by C η,δ .

Proof of Proposition 4. Let η > 0 and fix 0 < δ < δ η/2 (δ η/2 given by Proposition 5) s.t.

L(C, d α δ ) + η 2 ≥ L(C, d a 2 ) (using (18)).
Consider ε η > 0 s.t. for 0 < ε < ε η we have Ce -γδ/ε < ε 4 (C and γ are given by ( 4)).

We take ξ η = ξ η/2,δ obtained from Proposition 5.

Clearly, ξ η satisfies 2. and 3. with

E η = E η/2,δ and C η = C η/2,δ . It is direct to obtain that |∇ξ η | -U 2 ε ≤ α δ -U 2 ε ≤ b 2 -U 2 ε ≤ 0 if dist(x, ω) < δ ε 4
otherwise .

It follows that ξ η satisfies 1 since α δ ≤ a 2 .

The proof of Proposition 5

We construct ξ η,δ in five steps.

Let η > 0 and 0 < δ < δ ′ < δ b,ω,C (here δ b,ω,C is defined in Remark 3). We denote α = α δ and α ′ = α δ ′ . Assume that P = {p 1 , ..., p k } and N = {n 1 , ..., n k } are s.t. σ = Id is a minimal connection in (C, d α ′ ).

Step 1: There is

ξ 0 : C → R s.t. ξ 0 is 1-Lipschitz in (C, d α ′ ) and ξ 0 (p i ) -ξ 0 (n i ) = d α ′ (p i , n i )
This step is a direct consequence of Lemma 4.2 in [START_REF] Brezis | Harmonic maps with defects[END_REF] (see also Lemma 2.2 in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF] or Lemma 2 in [START_REF] Brezis | Liquid crystals and energy estimates for S 2 -valued maps[END_REF]).

Step 2: We extend ξ 0 to R 3 : there is some

ξ 1 ∈ Lip(R 3 , R) s.t. |∇ξ 1 | = α ′ and ξ 1|C ≡ ξ 0
Although the argument is the same as in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF], for the convenience of the reader, we recall the construction. Consider

ξ 1 (x) = max i {ξ 0 (p i ) -d α ′ (x, p i )} , x ∈ R 3 .
Then we have the following.

• If k = 1, ξ 1 (x) = ξ 0 (p 1 ) -d α ′ (x, p 1 ) is s.t. ξ 1 ∈ Lip(R 3 , R) s.t. |∇ξ 1 | = α ′ and ξ 1|C ≡ ξ 0 . We now assume that k ≥ 2.
• ξ 1|C ≡ ξ 0 : let M ∈ C and i be s.t. M ∈ {p i , n i } and j = i, it is clear that

ξ 0 (p i ) -d α ′ (M, p i ) -ξ 0 (p j ) + d α ′ (M, p j ) = ξ 0 (p i ) -ξ 0 (p j ) + d α ′ (p i , p j ) ≥ 0 if M = p i ξ 0 (n i ) -ξ 0 (p j ) + d α ′ (n i , p j ) ≥ 0 if M = n i ;
• |∇ξ 1 | = α ′ : for all i we have (Proposition 3.5)

|∇ [ξ 0 (p i ) -d α ′ (x, p i )]| = |∇d α ′ (x, p i )| = α ′ in L ∞ (R 3 ).
Step 3: We construct a smooth approximation: let 0 < β < δ, there is

ξ 2 ∈ C ∞ (R 3 , R) s.t. |∇ξ 2 | ≤ (1 -β)α and i∈N k {ξ 2 (p i ) -ξ 2 (n i )} ≥ L(C, d α ) -η/2 (19)
Let (ρ t ) δ ′ -δ>t>0 be a classical mollifier, namely

ρ t (x) = t -3 ρ(x/t) with ρ ∈ C ∞ (R 3 , [0, 1]), Supp ρ ⊂ B(0, 1) and R 3 ρ = 1. Consider ξ 2 (x) := (1 -β)ξ 1 * ρ t (x).
Because for x ∈ ω δ we have x + B(0, t) ⊂ ω δ ′ , we have

|∇ξ 2 | ≤ (1 -β)α: |∇ξ 2 (x)| (1 -β) = |[∇ξ 1 * ρ t ](x)| ≤ 1 if x / ∈ ω δ [|∇ξ 1 | * ρ t ](x) = [α ′ * ρ t ](x) = b 2 if x ∈ ω δ .
Moreover Condition (19) is satisfied:

i∈N k {ξ 2 (p i ) -ξ 2 (n i )} ≥ (1 -β) i∈N k {ξ 1 (p i ) -ξ 1 (n i )} -O(δ ′ ) .
Thus letting δ η sufficiently small s.t. for 0 < β < δ < δ ′ < δ η we have

(1 -β) i∈N k {ξ 1 (p i ) -ξ 1 (n i )} -O(δ ′ ) ≥ L(C, d α ′ ) -η/3 ≥ L(C, d α ) -η/2.
Therefore ξ 2 has the desired properties for β < δ < δ η .

Step 4: Let Ω be a smooth and bounded open neighborhood of Ω. We approximate ξ 2 by a Morse function ξ η,δ in C 1 ( Ω). Function ξ η,δ is s.t. we have ξ η,δ ∈ C ∞ (R 3 , R) and

ξ η,δ -ξ 2 L ∞ ( Ω) ≤ η/(4k), |∇ξ η,δ | ≤ α, ξ η,δ is a Morse function, ∃R = R(η, δ) > 0 s.t. in R 3 \ B(0, R), ξ η,δ = |x|/2
Clearly ξ η,δ satisfies 1. and 2. of Proposition 5.

Step 5: We follow [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF]. We construct E η,δ

Let {x 1 , ..., x l } be the set of the critical points of ξ η,δ . Then there is

C = C(η, δ) > 0 s.t.: inf B(0,R)\∪iB(xi,ρ) |∇ξ η,δ | ≥ ρ C
since the critical points are not degenerate and

H 1 [ξ η,δ (∪ i B(x i , ρ))] ≤ Cρ.
We consider ρ > 0 s.t. Cρ ≤ η and set E η,δ = ξ η,δ (∪ i B(x i , ρ)).

For t / ∈ E η,δ , we have

• if x ∈ {ξ η,δ = t} \ B(0, R), then the second fundamental form of {ξ η,δ = t} in x is bounded by C ′ η,δ , • if x ∈ {ξ η,δ = t}∩B(0, R), then the second form is bounded by C ′′ η,δ = C sup B(0,R) |D 2 ξ η,δ | inf B(0,R)\∪iB(xi,ρ) |∇ξ η,δ | .
We find that the second fundamental form is globally bounded by C η,δ = max{C ′ η,δ , C ′′ η,δ }.

Second step in the proof of Theorem 3: two structure functions

Letting

µ ε = U 2 ε 2 |∇v ε | 2 + U 4 ε 4ε 2 (1 -|v ε | 2 ) 2
| ln ε| H 3 , because by (13) µ ε is bounded, we get that, up to pass to a subsequence, µ ε converges in the sense of the measures to µ.

The proof of Theorem 3 consists in the identification of µ. We want to prove that µ = πa 2 H 1 |∪Γi with ∪Γ i is the unique the geodesic link joining the singularities of g. Because by Theorem 2 we have that µ and πa 2 H 1 |∪Γi has the same total mass, it suffices to prove that for all x 0 ∈ Ω there is

r x0 > 0 s.t. µ(B(x 0 , r x0 )) ≤ πa 2 H 1 |∪Γi (B(x 0 , r x0 )
). Therefore we have to obtain lower bound for

F ε (v ε , Ω \ B(x 0 , r x0 ))/| ln ε|.
In order to get a suitable lower bound to prove Theorem 3 we need two kinds of structure function:

• a structure function which is constant in a neighborhood K = B(x 0 , r x0 ) of an arbitrary point x 0 which is not on a geodesic link joining the singularities;

• a structure function which is constant in a neighborhood K = B(x 0 , r x0 ) of an arbitrary point x 0 which is on the geodesic link joining the singularities (assuming the geodesic link is unique).

We briefly sketch the strategy used in Section 8 to get that µ = πa 2 H 1 |∪Γi . Using the upper bound (13), if x 0 is not on a geodesic link joining the singularities, then we get the lower bound (letting

K = B(x 0 , r x0 )) F ε (v ε , Ω \ K) ≥ F ε (v ε , Ω) -o(| ln ε|). ( 20
)
From the estimate (20) we deduce that F ε (v ε , K) = o(| ln ε|). Therefore µ is supported in the union of the geodesic links joining the singularities.

Assuming that the geodesic link is unique (we denote this geodesic link by ∪Γ i ), we get a structure function which allows to obtain for x 0 which is on the geodesic link and small r x0 the estimate

F ε (v ε , K) ≤ 2πa(x 0 ) 2 r x0 | ln ε| + o(| ln ε|) if x 0 / ∈ ∂ω π(1 + b 2 )r x0 | ln ε| + o(| ln ε|) if x 0 ∈ ∂ω . ( 21
)
Therefore, we get that µ ≤ πa 2 H 1 |∪Γi and thus we obtain that µ = πa 2 H 1 |∪Γi . In order to get estimates (20) and ( 21) we need two kinds of structure functions. In the spirit of Proposition 4, to get that the support of µ is included in the geodesic links (Estimate (20)), we have the following structure function (Proposition 6 does not need the hypothesis that there exists a unique geodesic link)

Proposition 6. Let x 0 ∈ R 3 be s.t. x 0 is not on a geodesic link of C in (R 3 , d a 2 ). There is r x0 > 0 s.t. denoting K = B(x 0 , r x0 ), for η > 0 there are ξ η,K ∈ C ∞ (R 3 , R), E η,K ⊂ R, C η,K > 0 and ε η,K > 0 s.t. for 0 < ε < ε η,K , 1. |∇ξ η,K | ≤ min(a 2 , U 2 ε + ε 4 ) in R 3 and ξ η,K is constant in K 2. i∈N k {ξ η,K (p i ) -ξ η,K (n i )} ≥ L(C, d a 2 ) -η 3. H 1 (E η,K ) ≤ η and for all t ∈ R \ E η,K , {ξ η,K = t} is a closed hypersurface whose second fundamental form is bounded by C η,K .
And in order to compare µ with πa 2 H 1 |∪Γi (Estimate ( 21)) we have the following structure function Proposition 7. Assume that there exists a unique geodesic link ∪Γ i joining the singularities of g

in (R 3 , d a 2 ). Let x 0 ∈ ∪Γ i \ C, then there is r x0 > 0 s.t. denoting K = B(x 0 , r x0 ), for η > 0 there are ξ η,K ∈ C ∞ (R 3 , R), E η,K ⊂ R, C η,K > 0 and ε η,K > 0 s.t. for 0 < ε < ε η,K , 1. |∇ξ η,K | ≤ min(a 2 , U 2 ε + ε 4 ) in R 3 and ξ η,K is constant in K 2. i∈N k {ξ η,K (p i ) -ξ η,K (n i )} ≥ L(C, d a 2 ) -a 2 H 1 |∪Γi (K) -η = L(C, d a 2 ) -2a 2 (x 0 )r x0 -η if x / ∈ ∂ω L(C, d a 2 ) -(1 + b 2 )r x0 -η if x ∈ ∂ω 3. H 1 (E η,K ) ≤ η and for all t ∈ R \ E η,K , {ξ η,K = t} is a closed hypersurface whose second fundamental form is bounded by C η,K .
The proofs of Propositions 6 and 7 are given in Section 7.4.

Proof of Propositions 6 and 7

In order to prove Propositions 6 & 7, we follow the proof of Proposition 4 using a special pseudo metric taking care of the compact set K.

Definition and properties of a special pseudometric

Let f : R 3 → [b 2 , 1] be a Borel function and let ∅ = K ⊂ R 3 be a smooth compact set. We define

d K f (x, y) = min {d f (x, y), d f (x, K) + d f (y, K)} . Here d f (x, K) = min y∈K d f (x, y). Then d K f is a pseudometric in R 3 . If, in addition K ∩C = ∅, then d K f is a distance in C. Therefore the minimal connection of C and the length of a minimal connection L(C, d K f ) with respect to d K f make sense. Clearly, if x, y ∈ R 3 , then we have d K f (x, y) = 0 ⇔ x = y or x, y ∈ K. One may easily prove that d K f (x, y) ≤ d f (x, y) ≤ d K f (x, y) + diam(K). (22)
We are interested in the special case K = B(x 0 , r) for some x 0 ∈ Ω and f = α δ with δ ∈ [0, δ 0 ], δ 0 = 10 -2 min{1; dist(∂ω, ∂Ω)}.

Note that we have a similar estimate to (18), namely for 0

≤ δ < δ ′ < δ 0 L(C, d K α δ ) = L(C, d K α δ ′ ) + O(|δ ′ -δ|). (23)
From ( 22), ( 23) and as in Remark 3 we have the following. Remark 4. There is δb,ω,C < 10 -2 •dist(∂ω, ∂Ω) s.t. for x 0 ∈ R 3 , 0 < δ, r < δb,ω,C and K = B(x 0 , r)

s.t. K ∩ C = ∅: If σ is a minimal connection in (C, d K α δ ), then σ is a minimal connection in (C, d α0
). Definition 4. For y / ∈ K and x ∈ R 3 , we say that

• Γ is a K-curve joining x, y if Γ is a finite union of curves included in R 3 \ K (except perhaps their endpoints) s.t.:
-their endpoints are either x or y or an element of ∂K,

-if z ∈ {x, y} \ K, then z is an endpoint of a connected component of Γ. • Γ is a minimal K-curve joining x, y if Γ = ∪ i Γ i is a K-curve joining x, y, where the Γ i 's are disjoint curves and i long a 2 (Γ i ) = d K a 2 (x, y).
We next sum up the main properties of d K a 2 .

Proposition 8. Let x 0 ∈ R 3 , r > 0 and K = B(x 0 , r).

1. If y / ∈ K then for all x ∈ R 3 there is a minimal K-curve joining x, y. Moreover, a minimal K-curve is the union of at most two geodesics in (R 3 , d a 2 ).

2. For x, y ∈ R 3 , x = y and x 0 = x, y we have: i. If x 0 ∈ R 3 \ ∂ω and x 0 is on a geodesic joining x, y in (R 3 , d a 2 ), then there is r x0,x,y > 0 s.t. for all r < r x0,x,y , d K a 2 (x, y) = d a 2 (x, y) -2a 2 (x 0 )r, ii. If x 0 ∈ ∂ω and x 0 is on a geodesic joining x, y in (R 3 , d a 2 ), then there is r x0,x,y > 0 s.t. for all r < r x0,x,y , d K a 2 (x, y) = d a 2 (x, y) -(1 + b 2 )r, iii. If x 0 is not on a geodesic joining x, y in (R 3 , d a 2 ), then there is r x0,x,y > 0 s.t. for all r < r x0,x,y , d K a 2 (x, y) = d a 2 (x, y).

Proposition 8 is proved in the Appendix A. As a direct consequence of Proposition 8 we have the following. Remark 5. Let C ⊂ ∂Ω as above and x 0 ∈ Ω \ C.

1. If for all minimal connection σ of C and for i ∈ {1, ..., k}, we have that x 0 is not on a geodesic joining p i with n σ(i) in (R 3 , d a 2 ), then there is r x0,C > 0 s.t. for all 0 < r < r x0,C , we have for

K = B(x 0 , r) L(C, d K a 2 ) = L(C, d a 2 ). (24)
2. Assume that there exists a unique geodesic link ∪Γ i joining P with N (we may assume that σ = Id is the minimal connection). Therefore from Remark 1.1, for i = jwe have Γ i ∩ Γ j = ∅.

If for some i we have that x 0 which is on the geodesic joining p i with n i , then there is r x0,C > 0 s.t. for all 0 < r < r x0,C , we have for K = B(x 0 , r)

L(C, d K a 2 ) = L(C, d a 2 ) -a 2 H 1 |∪Γi (K) = L(C, d a 2 ) -2a 2 (x 0 )r if x 0 / ∈ ∂ω L(C, d a 2 ) -(1 + b 2 )r if x 0 ∈ ∂ω . (25)

The proofs of Propositions 6 & 7

This pseudo metric allows us to construct an intermediate structure function in the spirit of Proposition 5 which is constant in a "small" compact K. Proposition 9. Let K = B(x 0 , r) be s.t. B(x 0 , 2r) ⊂ R 3 \ C and η > 0.

Then there is δ η,K > 0 s.t. for 0 < δ < δ η,K there are

C η,K,δ > 0, E η,K,δ ⊂ R and ξ η,K,δ ∈ C ∞ (R 3 , R) satisfying 1. |∇ξ η,K,δ | ≤ α δ in R 3 and ξ η,K,δ is constant in K (i.e. |∇ξ η,K,δ | ≤ α δ 1I R 3 \K ), 2. i∈N k {ξ η,K,δ (p i ) -ξ η,K,δ (n i )} ≥ L(C, d K α δ ) -η, 3. H 1 (E η,K,δ ) ≤ η and for t ∈ R \ E η,K,δ , {ξ η,K,δ = t} is a closed hypersurface whose second fundamental form is bounded by C η,K,δ .
Proof of Proposition 6. Assume that σ = Id is a minimal connection in (C, d a 2 ). Let r x0,C > 0 (given by Remark 5) be s.t. for

K = B(x 0 , r x0,C /2), we have L(C, d a 2 ) = L(C, d K a 2 ). We fix r x0 of Proposition 6 s.t.: r x0 ≤ r x0,C /2 and B(x 0 , 2r x0 ) ∩ C = ∅. We let K = B(x 0 , r x0 ).
Now we apply Proposition 9: there is δ η/2,K > 0 s.t. for 0 < δ < δ η/2,K , there are

C η/2,K,δ > 0, E η/2,K,δ ⊂ R and ξ η/2,K,δ ∈ C ∞ (R 3 , R) satisfying the conclusions of Proposition 9.
From ( 23) and ( 24), one may fix 0 < δ < δ η/2,K s.t.

L(C, d a 2 ) -L(C, d K α δ ) < η/2.
We consider ε η,K > 0 s.t. for 0 < ε < ε η,K , we have Ce -γδ/ε < ε 4 (C and γ are given by ( 4)). We obtain the result by taking

C η,K = C η/2,K,δ , E η,K = E η/2,K,δ and ξ η,K = ξ η/2,K,δ .
Proof of Proposition 7. Assume that σ = Id is the minimal connection of C and that x 0 is on the geodesic joining p 1 with n 1 . Let r x0 < min{r x0,p1,n1 , min i=2,...,k r x0,pi,ni } (r x0,pi,ni given by Proposition 8.2) s.t. B(x 0 , 2r x0 ) ⊂ R 3 \ C.

It suffices to apply Proposition 9 as in the proof of Proposition 6 combined with (25).

The proof of Proposition 9

The main point is that we require that ξ η,K,δ is constant in K. All the other requirements are satisfied by the map ξ η,δ constructed in Proposition 5.

For δ < r/2, let K 1 = B(x 0 , r + 2δ) and K 2 = B(x 0 , r + δ). We denote α = α δ and α ′ = α 2δ . We assume that σ = Id is a minimal connection for (C, d K1 α ′ ) and we fix η.

Step 1: As in the proof of Proposition 5, there is a function ξ

0 : C → R 1-Lipschitz in (C, d K1 α ′ ) and s.t. ξ 0 (p i ) -ξ 0 (n i ) = d K1 α ′ (p i , n i ).
Step 2: We extend ξ 0 to a map ξ 1 : R 3 → R, 1-Lipschitz w.r.t. d K1 α ′ and thus constant in K 1 For example, we may take

ξ 1 (x) = max i {ξ 0 (p i ) -d K1 α ′ (x, p i )}.
As in the proof of Proposition 5, ξ 1|C = ξ 0 and

|∇ξ 1 | ≤ α ′ . Moreover, ξ 1 is constant in K 1 . Indeed, for all x ∈ K 1 , we have ξ 0 (x) = max i ξ 0 (p i ) -d K1 α ′ (x, p i ) = max i ξ 0 (p i ) -d α ′ (K 1 , p i ). Step 3: For 0 < β < δ, we approximate ξ 1 by ξ 2 ∈ C ∞ (R 3 , R) satisfying |∇ξ 2 | ≤ (1 -β)α, i∈N k {ξ 2 (p i ) -ξ 2 (n i )} ≥ L(C, d K α ) -η/2
(for δ sufficiently small), and s.t. ξ 2 is constant in K 2 The approximation ξ 2 is obtained (as in Proposition 5) by regularization of (1β)ξ 1 using a mollifier ρ t (t < 10 -2 δ) and by noting that

L(C, d K α ) ≥ L(C, d K α ′ ) ≥ L(C, d K2 α ′ ) ≥ L(C, d K1 α ′ ) ≥ L(C, d K α ′ ) -O(δ) = L(C, d K α ) -O(δ).
We fix δ η,K > 0 s.t. for δ < δ η,K we have

• L(C, d K1 α ′ ) ≥ L(C, d K α ) -η/8 and 2kη 2 δ 2 ≤ 10 -2 η (this condition is used below) • ξ 1 -ξ 2 L ∞ ≤ η/(16k) and δL(C, d a 2 ) ≤ η/4.
Step 4: Let Ω be a neighborhood of Ω. We approximate ξ 2 in C 1 ( Ω) by a Morse function

ξ 3 ∈ C ∞ (R 3 , R) We let ξ 3 ∈ C ∞ (R 3 , R) be s.t. ξ 3 -ξ 2 C 1 ( Ω) < η 2 δ 2 , |∇ξ 3 | ≤ (1 -β/2)α, ξ 3 is a Morse function, ∃R > 0 s.t. in R 3 \ B(0, R), ξ 3 = |x|/2.
Step 5: We modify ξ 3 in order to have ξ η,K,δ ≡ C 0 in K By construction, there is

C 0 ∈ ξ 3 (K) s.t. ξ 3 -C 0 C 1 (K2) < η 2 δ 2 . Noting that dist(∂K 2 , K) = δ, one may construct ξ η,K,δ ∈ C ∞ (R 3 ) s.t. ξ η,K,δ = ξ 3 in R 3 \ K 2 , ξ η,K,δ ≡ C 0 in K, ξ η,K,δ -C 0 L ∞ (K2) < η 2 δ 2 and |∇ξ η,K,δ | ≤ b 2 in K 2 .
Clearly ξ η,K,δ satisfies 1. and 2. in Proposition 9.

Step 6: We construct E η,K,δ For ρ > 0, we consider E 1 η,K,δ = ξ η,K,δ (∪ i B(x i , ρ)) where {x 1 , ..., x l } is the set of the critical points of ξ η,K,δ in B(0, R) \ K 2 .

For the same reasons as in Proposition 5, we have

H 1 (E 1 η,K,δ ) ≤ Cρ. We also define E 2 η,K,δ = ξ η,K,δ (K 2 )
. By construction, we have H 1 (E 2 η,K,δ ) ≤ 2η 2 δ 2 . Thus it suffices to consider ρ sufficiently small in order to have Cρ ≤ 10 -2 η and to set E η,K,δ = E 1 η,K,δ ∪ E 2 η,K,δ .

7.5 Second step in the proof of Theorem 4: a structure function in presence of symmetries

In this section we assume that Ω = B(0, 1) and that ω = B(0, r 0 ), with r 0 ∈ (0, 1). Consider C = {(1, 0, 0), (-1, 0, 0)} = {p, n}, p = (1, 0, 0). It is clear that in this situation, the line segment [p, n] is the unique geodesic between p and n in (R 3 , d a 2 ).

In order to get a very sharp lower bound (matching with the upper bound up to a O(1) term), we need a structure function as in Proposition 6 but with "η = 0". This structure function is given in the following proposition.

Proposition 10. Let M ∈ Ω \ [p, n]. Then there is V, an open neighborhood of M s.t. for ε > 0, there is ξ ε : R 3 → R a Lipschitz function s.t. 1. ξ ε (p) -ξ ε (n) = d U 2 ε (p, n), 2. |∇ξ ε | ≤ U 2 ε , 3. ξ ε ≡ 0 in V, 4. ∀ t ∈ ξ ε (R 3 ) \ {0, ξ ε (p), ξ ε (n)}, {ξ ε = t} is a sphere whose radius is at least 1.
The proof of Proposition 10 is in the next section (Section 7.6).

Proof of Proposition 10

Using the spherical symmetry of Ω, ω and the minimality of U ε , one may easily prove the following proposition.

Proposition 11. The unique minimizer U ε of E ε in H 1 1 , is radially symmetric and non decreasing.

Proposition 10 is a particular case of the following lemma (by taking U = U ε ). Then there are ξ : R 3 → R and B + , B -two distinct open balls, B + , B -are exteriorly tangent and independent of U s.t. Using the symmetry of the situation, the function ξ is represented in the Figure 2.

1. ξ(p) -ξ(n) = d U 2 (p, n), 2. |∇ξ| ≤ U 2 , 3. ξ ≡ 0 in V := R 3 \ (B + ∪ B -),
M 1 M 2 × 2n × 2p n p ξ ≡ 0 ξ ≡ ξ(n) ξ ≡ ξ(p)
The Proof. Let p, n ∈ ∂Ω, p = -n and {0, (e 1 , e 2 , e 3 )} be an orthonormal and a direct coordinate system of R 3 s.t. p = (1, 0, 0) and n = (-1, 0, 0). Let M (x 0 , y 0 , z 0 ) ∈ Ω \ [p, n].

Step 1:

ξ 0 : [-1, 1] → R s.t. ξ 0 (1) -ξ 0 (-1) = d U 2 (p, n), ξ 0 ′ (s) = U 2 (s, 0, 0) and ξ 0 (x 0 ) = 0 It suffices to consider ξ 0 (s) = s x0 U 2 (t, 0, 0) dt.
Step 2: We construct ξ : R 3 → R We denote Σ + r = ∂B((2, 0, 0), r) for r ∈ (1, 2x 0 ) and Σ - r = ∂B((-2, 0, 0), r) for r ∈ (1, 2 + x 0 ).

1) We have

lim inf ε→0 F ε (v ε ) | ln ε| ≥ πL(g, d a 2 ). (26)
2) We denote by < Γ > the union of all geodesic links of C in (R 3 , d a 2 ) and for µ > 0,

K µ := {x ∈ Ω | dist(x, < Γ >) ≥ µ}. Then we have lim inf ε→0 F ε (v ε , Ω \ K µ ) | ln ε| ≥ πL(g, d a 2 ). (27)
3) Assume that there exists a unique geodesic link ∪Γ i joining the singularities of g. Then for

x 0 ∈ ∪ i Γ i \ C there exists r x0 > 0 s.t. letting K = B(x 0 , r x0 ) we have lim inf ε→0 F ε (v ε , Ω \ K) | ln ε| ≥ π L(g, d a 2 ) -a 2 H 1 |∪Γi (K) . (28)
4) Moreover, if we are in the symmetric case of Section 7.5, then we have < Γ >= [p, n] and there is

C µ > 0 s.t. F ε (v ε , Ω \ K µ ) ≥ πd a 2 (p, n)| ln ε| -C µ . (29)
Theorem 2 for g ∈ H, as well as Theorems 3&4, are straightforward consequences of Proposition 13 combined with the upper bounds ( 12), (13).

We prove in detail (26), and we will sketch the proofs of ( 27),(29) which are, as explained in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF], obtained exactly in the same way as (26).

We prove that for all η := η(8k 2 + 3k + 1) > 0, the following holds

lim inf ε→0 F ε (v ε ) | ln ε| ≥ πL(g, d a 2 ) -η. (30)
Let η > 0, ε n ↓ 0, let (v n ) n ⊂ H 1 g be a sequence of minimizers of F εn in H 1 g and let ξ η , C η , E η be given by Proposition 4 (for n sufficiently large).

Let 0 < ρ < 10 -2 min{η, ε η } (ε η defined in Proposition 4) and set

Ω ρ := {x ∈ R 3 | dist(x, Ω) < ρ and dist(x, C) > ρ}.
One may assume that ρ is sufficiently small s.t. in Ω ρ \ Ω, Π ∂Ω , the orthogonal projection on ∂Ω, is well defined and smooth. Then we extend v n (we use the same notation for the extension) by letting

v n : Ω ρ → R 2 , x → v n (x) if x ∈ Ω g(Π ∂Ω (x)) if x ∈ Ω ρ \ Ω .
Since g ∈ H and v n|Ωρ\Ω does not depend on n and takes its values in S 1 , we obtain the existence of C 0 (ρ) depending only on ρ, Ω, g s.t. (for small ρ)

F εn (v n , Ω) ≥ F εn (v n , Ω ρ ) -C 0 (ρ) (31) If we define F = F η,ρ := E η ∪ [ξ η (C) -2ρ, ξ η (C) + 2ρ] (ξ η , E η
given by Proposition 4), then we have

H 1 (F ) ≤ 8kρ + η ≤ (8k + 1)η.
If t ∈ R \ F , we denote Σt := {ξ η = t}. We construct for almost all t ∈ R \ F a smooth closed submanifold Σ t ⊂ Σt .

Note that for t ∈ R \ F , we have dist(t, ξ η (C)) ≥ 2ρ (by definition of F ). Consequently, for t ∈ R \ F , we obtain that Σt ∩ {Ω + B(0, ρ)} = Σt ∩ Ω ρ (because ξ η is 1-Lipschitz).

Since t ∈ R \ F is not a critical value of ξ η , the connected components W 's of Σt = ∂{ξ η ≥ t} = {ξ η = t} have no boundary. If such W intersects Ω ρ , then we distinguish two cases:

a) W ∩ ∂Ω ρ = ∅ b) W ∩ ∂Ω ρ = ∅.
Denote by W a , resp. W b , the set of the connected components satisfying a), resp. b).

If W b = ∅, then we define

Σ t = Σt ∩ Ω ρ = {ξ η = t} ∩ Ω ρ . Thus it remains to construct Σ t when W b = ∅. Consider f : Ω + B(0, ρ) → R 2 x → (ξ η (x), dist[x, ∂(Ω + B(0, ρ))]
) .

Using the Constant Rank Theorem (see Theorem 4.3.2, page 91 in [START_REF] Demazure | Geometry of solutions to nonlinear problems[END_REF]), the set f -1 ({t}×[r, ∞)) (r ∈ (0, ρ/2)) is a manifold with boundary when • t is a regular value of ξ η ,

• (t, r) is a regular value of f . Thus, using Sard's Lemma, for almost all t ∈ R \ F s.t. W b = ∅, there is r = r(t) ∈ (0, ρ/2) s.t.

Σ t = f -1 ({t} × [r, ∞)) ⊂ Σt is a closed submanifold with boundary. ( 32
)
Moreover, we have

∂Σ t ⊂ ∂{Ω + B(0, ρ -r)} ∩ Ω ρ .
We denote by G the set A key argument in this proof is the way to pass from lower bounds on hypersurfaces to a lower bound in Ω. To do this we have the following. With the help of Lemma 2, we have Estimate (26) in Proposition 13 is obtained by letting η → 0 in the above estimate. We now sketch the arguments leading to (27) and (29). The fundamental ingredient is a lower bound for F ε (v ε , Ω \ K µ ). Without loss of generality, by compactness of K µ , we may only consider the situation K = B(x, r x ) for some x which does not belong to a geodesic link between the singularities of g; here, r x > 0 is some small number.

G := {t ∈ R \ F | W b = ∅ or W b = ∅
F εn (v n , Ω ρ ) ≥ (Prop. 4 & (31)) ≥ 1 2 Ω (|∇ξ η | -ε 4 n ) |∇v n | 2 + U 2 εn 2ε 2 n (1 -|v n | 2 ) 2 -C 0 ≥ (13) ≥ R\F 1 2 Σt |∇v n | 2 + b 2 2ε 2 n (1 -|v n | 2
In order to prove (27), we use Proposition 6: we consider ξ η,K s.t. ∇ξ η,K = 0 in K.

Following the same lines of proof of (26), we find that

lim inf n F εn (v n , Ω \ K) | ln ε n | ≥ πL(g, d a 2 ).
Combining this lower bound with (13), we obtain

F εn (v n , K) = o(| ln ε n |). (35)
To get (28), we use Proposition 7 and we get

lim inf n F εn (v n , Ω \ K) | ln ε n | ≥ π L(g, d a 2 ) -a 2 H 1 |∪Γi (K) .
This lower bound is exactly (28) and by combining (28) with (13) yields (21). Therefore, as explained in Section 7.3, from (35) and (21) we prove Theorem 3.

In the symmetric case, using Proposition 10, for x ∈ Ω \ [p, n] we obtain the existence of r x > 0 s.t., with K = B(x, r x ), we have

F εn (v n , Ω \ K) ≥ πd a 2 (p, n)| ln ε| -C K .
This estimate is obtained exactly as (26) because we may have a structure function ξ with "η = 0".

Consequently from the upper bound (12), we deduce

F εn (v n , K) ≤ C ′ K which implies Theorem 4.
9 Extension by density of Theorem 2

From ( 27) and (13), we obtain that Theorem 2 holds for g ∈ H. This section is devoted to the extension of Theorem 2 to the general case g ∈ H 1/2 (∂Ω, S 1 ).

For g ∈ H 1/2 (∂Ω, S 1 ), we denote

f ε,g = min v∈H 1 g F ε (v).
Using same arguments as in [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF], we have Proposition 14.

1. Let δ ∈ (0, 1). There is C(δ) > 0 s.t. for g 1 , g 2 ∈ H 1/2 (∂Ω, S 1 ), we have ((5.1),(5.2) in [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF])

(1 -δ)f ε,g1 -C(δ)f ε,g2 ≤ f ε,g1g2 ≤ (1 + δ)f ε,g1 + C(δ)f ε,g2 . (36
Proof. The estimates (36) & (37) are obtained using exactly the same arguments than in [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF]. Estimate (37) is proved in [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF]. We now focus on (39) which is not exactly proved in [START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF] (similar estimates are obtained for other metrics). We have easily proved by the definition of L(g, d a 2 ) = (2π) The proof of Theorem 2 is complete.

Figure 1 :

 1 Figure 1: Illustration of a geodesic link with k = 2: the boundary of Ω is in wire, the one of ω is black filled, the positive points are white, the negative ones are black and a geodesic link is represented in white. The shaded off portion on the two penetration points gives indications about the 3D-geometry of the geodesic link and of the inclusion. (Courtesy of Alexandre Marotta)

Lemma 1 .

 1 [The dumbbell lemma] Let U : R 3 → [b, 1] be a radially symmetric and non decreasing Borel function. Fix p, n ∈ S 2 , p = -n and let M ∈ Ω \ [p, n].

4 .

 4 M ∈ T with T which is the common tangent plan of B + and B -,5. B + is centered in 2p, B -is centered in 2n,6. denoting B+ (resp. B-) the ball centered in 2p (resp. 2n) with radius 1, ξ is locally constant in B+ ∪ B-, 7. ∀t ∈ ξ(R 3 ) \ {0, ξ(p), ξ(n)}, {ξ = t} is a sphere centered in 2p or 2n whose radius is at least 1.

Figure 2 :

 2 Figure 2: The geometry of the level sets of ξ (intersected with the plane defined by p, n, M )

Lemma 2 .m

 2 The following lower bound holdsR\F d(n, t) dt ≥ L(g, d a 2 )η(8k 2 + 3k + 1). Proof. Let m = inf Ωρ ξ η and for t ∈ R let S(t) = Card {p i ∈ {ξ η ≥ t + ρ}} -Card {n i ∈ {ξ η ≥ t -ρ}} = k i=1 {1I ξη(pi)≥t+ρ -1I ξη (ni)≥t-ρ }. We have R\F d(n, t) dt = R\F Card{p t i ∈ {ξ η ≥ t}} -Card{n t i ∈ {ξ η ≥ t}} [ξ η is 1-Lipschitz] ≥ R\F Card{p i ∈ {ξ η ≥ t + ρ}} -Card{n i ∈ {ξ η ≥ t -ρ}} [S(t) ≤ k & H 1 (F ) ≤ (8k + 1)η] ≥ R S(t)k(8k + 1)η {1I ξη (pi)≥t+ρ -1I ξη(ni)≥t-ρ }k(8k + 1)η ≥ k i=1 {ξ η (p i )ξ η (n i )} -2kρk(8k + 1)η [ρ ≤ η] ≥ L(g, d a 2) -(8k 2 + 3k + 1)η.

F

  ) 2 dt -(C 0 + 1) ≥ (34) ≥ π(ln bρ ε n -C) R\F |d(n, t)| -(C 0 + 1) ≥ (Lemma 2) ≥ π(ln bρ ε n -C) L(g, d a 2 ) -(8k 2 + 3k + 1)η -(C 0 + 1) ≥ π| ln ε n | L(g, d a 2 ) -(8k 2 + 3k + 1)η -C. εn (v n , Ω) | ln ε n | ≥ πL(g, d a 2) -(8k 2 + 3k + 1)η, ∀ η > 0.

  3 → R. Lower bounds on each hypersurfaces are obtained via a standard result on Ginzburg-Landau energy (Proposition 12). Because the use of Coarea formula adds an extra-weight 1 |∇ξ| on the layers, in order to get a lower bound we drop the weights U 2 ε and U 4 ε ≥ b 2 U 2 ε by taking ξ s.t. Therefore, an important property of ξ is that its gradient is controlled by U 2 ε (|∇ξ| U 2 ε ). • In order to treat the integrals on {ξ = t} we use a standard lower bound (Proposition 12)

	U 2 ε |∇ξ|	1.

  Card({n t i ∈ {ξ η ≥ t}}) takes at most 2k values, one may assume that C(deg(v n , ∂Σ t )) is uniformly bounded in n and t. Note that d(n, t) is defined for almost all t ∈ R \ F (for t ∈ G).

	1 2 Σt	|∇v n | 2 +	b 2 2ε 2

and there is r(t) ∈ (0, ρ/2) s.t. (32) holds}. For t ∈ G we have dist(∂Σ t , Ω) ≥ ρ/2. (33) Let x ∈ Σ t be s.t. dist(x, ∂Σ t ) < ρ/2. Using (33), we have x ∈ Ω ρ \ Ω and therefore |v n (x)| = 1. Finally, we are in a position to apply Proposition 12: n (1 -|v n | 2 ) 2 ≥ π|deg(v n , ∂Σ t )| ln bρ ε n -C(deg(v n , ∂Σ t )). (34) For M ∈ C and for t ∈ G we denote M t ∈ ∂(Ω + B(ρr(t)) s.t. Π ∂Ω (M t ) = M . Here we set r(t) = 0 when W b = ∅, i.e., when Σ t = Σt ∩ Ω ρ . It is clear that M t is uniquely defined. Since d(n, t) = deg(v n , ∂Σ t ) = Card({p t i ∈ {ξ η ≥ t}}) -

  ) 2. There is C > 0 depending only on Ω s.t. for g ∈ H 1/2 (∂Ω, S 1 ) we have ((5.4) in[START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF])f ε,g ≤ C|g| 2 H 1/2 (∂Ω) (1 + | ln ε|). (37) 3. If (g n ) n ⊂ H is s.t. g n → g in H 1/2 (∂Ω) then Lemma 17 in [2] applied with u n = g n /g and v = g yields g n g H 1/2 (∂Ω) → 0. (38)4. There is C > 0 depending only on Ω and on a s.t. for g 1 , g 2 ∈ H 1/2 (∂Ω, S 1 ) we have ((2.6) in[START_REF] Bourgain | H 1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg-Landau equation[END_REF])|L(g 1 , d a 2 ) -L(g 2 , d a 2 )| ≤ C|g 1g 2 | H 1/2 (∂Ω) |g 1 | H 1/2 (∂Ω) + |g 2 | H 1/2 (∂Ω) .

  -1 sup T g (ϕ) |ϕ| d a 2 ≤ 1 and Proposition 2.1 that|L(g 1 , d a 2 ) -L(g 2 , d a 2 )| ≤ L(g 1 g 2 , d a 2 ) ≤ L(g 1 g 2 , d eucl ) ≤ [Eq. (2.8) in [2]] ≤ C|g 1g 2 | H 1/2 (∂Ω) |g 1 | H 1/2 (∂Ω) + |g 2 | H 1/2 (∂Ω) . this proposition, Theorem 2 is proved as follows. Let g ∈ H 1/2 (∂Ω, S 1 ). By Proposition 2.3, there is (g n ) n ⊂ H s.t. g n → g in H 1/2 (∂Ω).Let ε ∈ (0, 1) and δ > 0. Then, by (36), we have , da 2 ) + C ′ (δ)|g/g n | H 1/2 . (40) Using (39), we obtain that L(g n , d a 2 ) → L(g, d a 2 ). If, in (40), we first let n → ∞, we use (38) and we next let δ → 0, we obtain that lim ε f ε,g | ln ε| = πL(g, d a 2 ).

	Using (1 -δ)	f ε,gn | ln ε|	-C(δ)	f ε,g/gn | ln ε|	≤	f ε,g | ln ε|	≤ (1 + δ)	f ε,gn | ln ε|	+ C(δ)	f ε,g/gn | ln ε|	.

From (37) and the fact that Theorem 2 holds for g n , we have

(1δ)πL(g n , d a 2 ) -C ′ (δ)|g/g n | H 1/2 ≤ lim inf ε f ε,g | ln ε| ≤ lim sup ε f ε,g | ln ε| ≤ (1 + δ)πL(g n

Acknowledgements. The author is grateful to Professor Petru Mironescu for having introduced him to the subject of this paper and for stimulating conversations. He wants also to thank Professor Pierre Bousquet for helpful comments. The author gratefully acknowledge the comments of the anonymous Referee.

We define ξ : R 3 → R by its level sets:

in B((-2, 0, 0), 1) ξ 0 [START_REF] Bethuel | Small energy solutions to the Ginzburg-Landau equation[END_REF] in B((2, 0, 0), 1) 0 otherwise .

Step 3: ξ satisfies the properties of Lemma 1 Assertion 1. is easily satisfied since ξ(p) = ξ 0 (1), ξ(n) = ξ 0 (-1) and ξ 0 (1)ξ 0 (-1) = d U 2 (p, n).

We take B + = B((2, 0, 0), 2x 0 ) and B -= B((-2, 0, 0), 2 + x 0 ). Clearly Assertions 3., 4., 5., 6. and 7. hold. We check 2.. Since ξ is locally constant in

The key argument is the fact that for

. This is obvious if we draw a picture and may be easily justified. Indeed, if Q 0 is a minimal point, then the line segment [Q, Q 0 ] is orthogonal to ∂B(Q ′ , r). Only two points on ∂B(Q ′ , r) satisfy this condition and one of them is clearly not minimal.

Consequently, by taking Q = 0 and Q ′ ∈ {2p, 2n} we have that min

Note that U is radially symmetric and non decreasing. Since in each connected components of

ξ admits a spherical symmetry, we have

8 Lower bound for inf H 1 g F ε when g ∈ H: the argument of Sandier As briefly explain in Section 7, we compute a lower bound via the Coarea formula: we integrate on the level sets of ξ : R 3 → R (a structure function).

Thus the obtention of a lower bound for F ε (v ε ) is related with lower bounds on hypersurfaces. On hypersurfaces, the main ingredient to get the desired estimate is Proposition 3.5 in [START_REF] Sandier | Ginzburg-Landau minimizers from R n+1 to R n and minimal connections[END_REF]. For the convenience of the reader, we recall this result. Proposition 12. Let Σ be a (smooth) closed and oriented hypersurface in R 3 whose second fundamental form is bounded by K. We denote by d(•, •) the Euclidean distance restricted to Σ.

Then we have the existence of C > 0 depending only on K and deg(v, ∂Σ) s.t.

This section is devoted to the proof of the following proposition.

Proposition 13. Let g ∈ H and let C = P ∪ N be the set of its singularities.

Appendix A Proof of Proposition 8

We prove the first assertion. There are two cases to consider: x ∈ K and x / ∈ K. If x ∈ K and y / ∈ K, then we have the existence of a unique point y 0 ∈ K which minimizes d a 2 (y, z) among the points z ∈ K. Clearly considering Γ a geodesic in (R 3 , d a 2 ) joining y with y 0 , by definition of y 0 , Γ ∩ K = ∅. Thus Γ is a minimal K-curve joining x with y according to the definition given above.

If x, y / ∈ K, then we consider Γ a geodesic joining x, y in (R 3 , d a 2 ) and, for z ∈ {x, y}, let Γ z be a minimal curve in (R 3 , d a 2 ) joining z with K (i.e. joining z with x 0 ).

If

Otherwise, long a 2 (Γ x ) + long a 2 (Γ y ) ≥ d a 2 (x, y). Consequently, denoting Γ a geodesic in (R 3 , d a 2 ) joining x with y, Γ \ K is a K-curve and has a minimal length.

It remains to prove that Γ, a minimal K-curve, is a union of at most two geodesics in (R 3 , d a 2 ). If Γ is connected, then, by the definition of a K-curve, Γ ∩ K = ∅. Thus Γ is a geodesic joining x, y.

Otherwise, assume that Γ is not connected. By the definition of a K-curve and by the minimality of Γ, for z ∈ {x, y}, there are z ′ ∈ ∂K and Γ z a connected component of Γ s.t. z, z ′ are the endpoints of Γ z . Thus, by minimality of Γ, Γ z is a geodesic joining z, z ′ in (R 3 , d a 2 ) and Γ = Γ x ∪ Γ y . Now we prove the second assertion. First, we assume that x 0 / ∈ ∂ω and that x 0 is on a geodesic curve joining x, y in (R 3 , d a 2 ).

Consider r x0,x,y = 10 -2 min {|xx 0 |, |yx 0 |, dist(x 0 , ∂ω)}. Then, for r < r x0,x,y , considering K = B(x 0 , r) and a K-curve Γ \ K where Γ is a geodesic joining x, y in (R 3 , d a 2 ) and containing x 0 , we obtain that

This comes from the fact that Γ ∩ K is a diameter of K and that this diameter is contained in the same connected component of R 3 \ ∂ω as x 0 . To obtain the reverse estimate, it suffices to consider Γ, a minimal K-curve joining x, y. From (A.1), we know that Γ has exactly two connected components: Γ x , Γ y with Γ z has z, z ′ for endpoints with z ∈ {x, y} and z ′ ∈ ∂K. Thus it suffices to complete Γ by the line segments [x 0 , x ′ ] and [x 0 , y ′ ] to obtain the reverse inequality. (Note that in this situation, [x ′ , y ′ ] is a diameter of K) If x 0 ∈ ∂ω, then the argument is similar taking 0 < r x0,x,y < 10 -2 min{|xx 0 |, |yx 0 |} sufficiently small s.t.:

• B(x 0 , r x0,x,y ) \ ∂ω has exactly two connected components,

• For all geodesic Γ joining x, y in (R 3 , d a 2 ), if x 0 ∈ Γ then (Γ ∩ K) \ ∂ω has exactly two connected components: one in ω and the other in R 3 \ ω.

Note that from Proposition ) + r n . It suffices to claim that in a metric space (X, d) which admits geodesic curves we have for x 0 , x, y three distinct points in X x 0 is on a geodesic joining x, y ⇐⇒ d(x, y) = d(x, x 0 ) + d(x 0 , y). Since x 0 is not on a geodesic curve joining x, y in (R 3 , d a 2 ), there is η > 0 s.t. d a 2 (x, y) + η < d a 2 (x, x 0 ) + d a 2 (x 0 , y) and thus long a 2 (Γ n

x ) + long a 2 (Γ n y ) = d Kn a 2 (x, y) < d a 2 (x, y) ≤ long a 2 (Γ n x ) + long a 2 (Γ n y ) + 2r nη.

Clearly we obtain a contradiction for n sufficiently large s.t. r n < η/2.