Vincent Millot

Yannick Sire
email: sire@cmi.univ-mrs.fr

ON A FRACTIONAL GINZBURG-LANDAU EQUATION AND 1/2-HARMONIC MAPS INTO SPHERES

Keywords: Fractional Ginzburg-Landau equations, Fractional harmonic maps, Square root Laplacian, Dirichletto-Neuman operator, Nonlinear boundary reactions, Partial regularity, Generalized varifolds

This paper is devoted to the asymptotic analysis of a fractional version of the Ginzburg-Landau equation in bounded domains, where the Laplacian is replaced by an integro-differential operator related to the square root Laplacian as defined in Fourier space. In the singular limit ε → 0, we show that solutions with uniformly bounded energy converge weakly to sphere valued 1/2-harmonic maps, i.e., the fractional analogues of the usual harmonic maps. In addition, the convergence holds in smooth functions spaces away from a countably H n-1 -rectifiable closed set of finite (n -1)-Hausdorff measure. The proof relies on the representation of the square root Laplacian as a Dirichlet-to-Neumann operator in one more dimension, and on the analysis of a boundary version of the Ginzburg-Landau equation. Besides the analysis of the fractional Ginzburg-Landau equation, we also give a general partial regularity result for stationary 1/2-harmonic maps in arbitrary dimension.

INTRODUCTION

Let n 1 and m 2 be given integers, and let ω ⊆ R n be a smooth bounded open set. In this paper we investigate the asymptotic behavior, as ε ↓ 0, of weak solutions vε : R n → R m of the fractional Ginzburg-Landau equation

(-∆) 1 2 vε = 1 ε (1 -|vε| 2 )vε in ω , (1.1)
possibly subject to an exterior Dirichlet condition

vε = g on R n \ ω , (1.2)
where g : R n → R m is a smooth function satisfying |g| = 1 in R n \ω. The action of the integro-differential operator (-∆) , and the notation p.v. means that the integral is taken in the Cauchy principal value sense. In Fourier space 1 , this operator has symbol 2π|ξ|, while 4π 2 |ξ| 2 is the symbol of -∆. In particular, (-∆)

1 2 is the square root of the standard Laplacian when acting on the Schwartz class S (R n ).

The weak sense for equation (1.1) will be understood variationaly using the distributional formulation of the fractional Laplacian in the open set ω. More precisely, we define the action of (-∆) In such a case we say that v is admissible, and (-∆)

1 2 v belongs to H -1/2 (ω; R m ), i.e., the topological dual space of H 1/2 00 (ω; R m ) (functions in H 1/2 (R n ; R m ) vanishing outside ω). When prescribing the exterior Dirichlet condition (1.2), the class of admissible functions is given by the affine space H 1/2 g (R n ; R m ) := 1 we consider here the ordinary frequency Fourier transform v → v given by v(ξ) := R n v(x)e -2iπx•ξ dξ.

1 g + H 1/2 00 (ω; R m ). Note that weak and strong definitions of (-∆) 1 2 are consistent, i.e., if v is smooth and bounded, then v is admissible, and the pointwise and distributional definitions of (-∆) 1 2 v agree, see e.g. [START_REF] Servadei | Weak and viscosity solutions of the fractional Laplace equation[END_REF]. Then observe that the derivative at v of E(•, ω) in the direction ϕ ∈ D(ω; R m ) is precisely given by (-∆) 1 2 v, ϕ ω . Hence the energy E(•, ω), which is built on the H 1/2 -seminorm, can be thought as Dirichlet 1/2-energy in ω associated to the operator (-∆)

1 2 . Integrating the potential in the right hand side of (1.1) we obtain the Ginzburg-Landau 1/2-energy associated to the equation in ω,

Eε(v, ω) := 1 2 ω e(v, ω) + 1 2ε (1 -|v| 2 ) 2 dx , (1.4)
where we have set e(v, ω) to be the nonlocal energy density in ω given by e v, ω := γn 2 ω |v(x)v(y)| 2 |x -y| n+1 dy + γn We define weak solutions of (1.1) as critical points of Eε with respect to perturbations supported in ω.

If -∆ is used instead of its square root and m = 1, equation (1.1) reduces to the well known Allen-Cahn equation. This scalar equation appears in the van der Waals-Cahn-Hilliard theory of phase transitions in fluids. From the geometrical point of view, the Allen-Cahn equation provides an approximation, as ε → 0, of codimension one minimal surfaces, i.e., limits of critical points are functions with values in {±1}, and the limiting interfaces between the two phases ±1 are generalized minimal hypersurfaces, see e.g. [START_REF] Hutchinson | Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory[END_REF][START_REF] Modica | The gradient theory of phase transitions and the minimal interface criterion[END_REF]. Since the work of ALBERTI, BOUCHITT É, & SEPPECHER [START_REF] Alberti | Un résultat de perturbations singulières avec la norme H 1/2[END_REF][START_REF] Alberti | Phase transition with the line-tension effect[END_REF], fractional generalizations of the Allen-Cahn equation involving (-∆) s with 0 < s < 1 have been investigated by many authors, see e.g. [START_REF] Cabr É | Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian[END_REF][START_REF] Cabr É | Sharp energy estimates for nonlinear fractional diffusion equations[END_REF][START_REF] Cabr É | Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates[END_REF][START_REF] Cabr É | Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions[END_REF][START_REF] Cabr É | SOL À-MORALES : Layer solutions in a halfspace for boundary reactions[END_REF][START_REF] Ález | Gamma convergence of an energy functional related to the fractional Laplacian[END_REF][START_REF] Palatucci | Local and global minimizers for a variational energy involving a fractional norm[END_REF][START_REF] Sire | Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result[END_REF], specially in the case s = 1/2 which is related to models for dislocations in crystals [START_REF] Garroni | A variational model for dislocations in the line tension limit[END_REF][START_REF] Ález | Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one[END_REF] (see also [START_REF] Dipierro | Strongly nonlocal dislocation dynamics in crystals[END_REF][START_REF] Dipierro | Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting[END_REF] for s = 1/2), and to thin-film micromagnetics [START_REF] Kurzke | A nonlocal singular perturbation problem with periodic well potential[END_REF][START_REF] Kurzke | Boundary vortices in thin magnetic films[END_REF]. The convergence as ε → 0 of minimizers of the fractional Ginzburg-Landau s-energy has been treated very recently in [START_REF] Savin | VALDINOCI : Γ-convergence for nonlocal phase transitions[END_REF][START_REF] Savin | Density estimates for a variational model driven by the Gagliardo norm[END_REF]. In these papers, Dirichlet and Ginzburg-Landau s-energies are built on the H s -seminorm and are defined as in (1.3)-(1.4)-(1.5) with the weight |x -y| -(n+2s) in place of |x -y| -(n+1) . It is proved in [START_REF] Savin | VALDINOCI : Γ-convergence for nonlocal phase transitions[END_REF][START_REF] Savin | Density estimates for a variational model driven by the Gagliardo norm[END_REF] that limits of minimizers still take values in {±1}, and the geometrical characterization of the associated interfaces depends on wether s 1/2 or s < 1/2. For s 1/2, limiting interfaces turn out to be (usual generalized) area-minimizing minimal surfaces, while the picture completely changes for s < 1/2, where limiting interfaces are the so-called nonlocal s-minimal surfaces introduced by CAF-FARELLI, ROQUEJOFFRE, & SAVIN [START_REF] Caffarelli | Nonlocal minimal surfaces[END_REF]. The interface induced by a map v taking values in {±1} is said to be a minimizing s-minimal surface if v itself is minimizing the Dirichlet s-energy in the class of maps with values in {±1}. In a sense, nonlocal s-minimal surfaces can be thought as fractional s-harmonic maps with values in S 0 ≃ {±1}. The dichotomy in the parameter s essentially comes from the fact that characteristic functions of (smooth) sets do not belong to H s for s 1/2, so that the s-energy density concentrates near a limiting interface very much like in the local case. For s < 1/2, the H s -regularity allows for (some) characteristic functions, and the energy of minimizers remains uniformly bounded as ε tends to 0. In this paper, we focus on the case s = 1/2 for some specific geometrical features that we shall explain shortly.

In the vectorial case m = 2, the local Ginzburg-Landau equation (i.e., with -∆ instead of its square root) has been widely studied because it shares many of the relevant features of more elaborated models of superconductivity or superfluidity, see e.g. [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF][START_REF]RIVI ÈRE : Line vortices in the U(1)-Higgs model[END_REF][START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF]. In the spirit of the classical Landau's theory of phase transtions, fractional Ginzburg-Landau equations have been recently suggested in the physics literature in order to incorporate a long-range dependence posed by a nonlocal ordering, as it might appear in certain high temperature superconducting compounds, see [START_REF] Milovanov | Fractional generalization of the Ginzburg-Landau equation: an unconventional approach to critical phenomena in complex media[END_REF][START_REF] Tarasov | Fractional Ginzburg-Landau equation for fractal media[END_REF][START_REF] Weitzner | Some applications of fractional equations[END_REF] and the references therein. In arbitrary dimensions n 1 and m 2, the local Ginzburg-Landau equation has a geometrical flavor like in the scalar case, and the limiting system in the asymptotic ε → 0 is given by the harmonic map equation into the sphere S m-1 (or into more general manifolds according to the potential well). Harmonic maps can be seen as higher dimensional generalizations of geodesics and are defined as critical points of the (1.6)

The equation in the right hand side of (1.6) arises when using test functions satisfying ϕ(x) ∈ T v * (x) S m-1

a.e., while the equation in the left hand side appears when ϕ is chosen with no constraint. The implication from right to left can be easily deduced (at least formally) writing ∆v * = (∆v * • v * )v * and using the identity 0 = ∆(|v * | 2 ) = 2∆v * • v * + 2|∇v * | 2 . A main feature of the harmonic map system is its critical structure concerning regularity. Indeed, the source term in the left hand side equation of (1.6) has a priori no better integrability than L 1 . This is precisely the borderline case where linear elliptic regularity does not apply. In dimension n = 2, harmonic maps are smooth by the famous result of H ÉLEIN [START_REF] Élein | Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne[END_REF].

However, equation (1.6) admits highly discontinuous solutions in dimensions n 3 [START_REF]RIVI ÈRE : Everywhere discontinuous harmonic maps into spheres[END_REF], and specific assumptions are required to obtain at least partial regularity (i.e., smoothness outside some "small set", see Subsection 4.1 for a more detailed discussion and references). In addition to this regularity issue, equation (1.6) turns out to be invariant under composition with conformal maps in dimension n = 2.

This conformal invariance implies that sequences of harmonic maps are usually non-compact in the energy space [START_REF] Lin | Gradient estimates and blow-up analysis for stationary harmonic maps[END_REF], and this lack of compactness is formally transferred to the local Ginzburg-Landau equation in the asymptotic ε → 0. The weak convergence of critical points of the local Ginzburg-Landau energy towards (weakly) harmonic maps has been proved by LIN &WANG [START_REF] Lin | Harmonic and quasi-harmonic spheres[END_REF][START_REF] Lin | Harmonic and quasi-harmonic spheres[END_REF][START_REF] Lin | Harmonic and quasi-harmonic spheres. III. Rectifiability of the parabolic defect measure and generalized varifold flows[END_REF] (see also [START_REF] Chen | Existence and partial regularity results for the heat flow for harmonic maps[END_REF]). In the spirit of the blow-up analysis for harmonic maps developped in [START_REF] Lin | Gradient estimates and blow-up analysis for stationary harmonic maps[END_REF], the authors have determined in [START_REF] Lin | Harmonic and quasi-harmonic spheres[END_REF][START_REF] Lin | Harmonic and quasi-harmonic spheres[END_REF][START_REF] Lin | Harmonic and quasi-harmonic spheres. III. Rectifiability of the parabolic defect measure and generalized varifold flows[END_REF] the geometrical nature of the possible defect measure arising in the weak convergence process, as well as the rectifiability of the energy concentration set.

The main object of the present article is to provide a careful analysis of critical points of the Ginzburg-Landau 1/2-energy Eε as ε → 0 in the spirit of [START_REF] Lin | Harmonic and quasi-harmonic spheres[END_REF][START_REF] Lin | Harmonic and quasi-harmonic spheres[END_REF][START_REF] Lin | Harmonic and quasi-harmonic spheres. III. Rectifiability of the parabolic defect measure and generalized varifold flows[END_REF], and as a byproduct to extend the previous works on the scalar case to the vectorial setting. By analogy with the local case, we expect solutions of the fractional Ginzburg-Landau equation (1.1) with uniformly bounded energy in ε to converge weakly as ε → 0 to critical points of the Dirichlet 1/2-energy under the constraint to be S m-1 -valued. In other words, a weak limit v * of critical points of the Ginzburg-Landau 1/2-energy should satisfy

d dt E v * + tϕ |v * + tϕ| , ω t=0 = 0 for all ϕ ∈ D(ω; R m ) . (1.7)
Computing the associated Euler-Lagrange equation, we obtain

(-∆) 1 2 v * ⊥ Tv * S m-1 in ω , (1.8)
in the distributional sense. In the case where |v * | = 1 in R n , the Lagrange multiplier related to the constraint takes a quite simple form and leads to the equation

(-∆) 1 2 v * (x) = γn 2 R n |v * (x) -v * (y)| 2 |x -y| n+1 dy v * (x) in ω , (1.9)
which is in clear analogy with (1.6). We shall refer to as weak 1/2-harmonic map into S m-1 in ω any map taking values into S m-1 in ω and satisfying condition (1.7).

The notion of 1/2-harmonic map into a manifold has been introduced in the case ω = R by DA LIO & RIVI ÈRE [START_REF] Lio | RIVI ÈRE : Three-term commutator estimates and the regularity of 1 2 -harmonic maps into spheres[END_REF][START_REF] Lio | RIVI ÈRE : Sub-criticality of non-local Schr ödinger systems with antisymmetric potential and applications to half harmonic maps[END_REF]. More precisely, weak 1/2-harmonic maps from R into S m-1 are defined in [START_REF] Lio | RIVI ÈRE : Three-term commutator estimates and the regularity of 1 2 -harmonic maps into spheres[END_REF] as critical points of the line energy

L(v) := 1 2 R (-∆) 1 4 v 2 dx ,
under the S m-1 -constraint as in (1.7). Here (-∆)

1 4 is the operator with symbol 2π|ξ|. A classical computation in Fourier space gives L(v) = E(v, R), so that the two notions of 1/2-harmonic maps indeed coincide. Recently, 1/2-harmonic maps into a manifold have found applications in geometrical problems such as minimal surfaces with free boundary (see [START_REF] Lio | RIVI ÈRE : Fractional harmonic maps and free boundaries problems[END_REF][START_REF] Fraser | Minimal surfaces and eigenvalue problems[END_REF][START_REF] Struwe | On a free boundary problem for minimal surfaces[END_REF], and Remark 4.28), and they are intimately related to harmonic maps with partially free boundary (see Remark 4.5,[START_REF] Gulliver | Harmonic maps which solve a free-boundary problem[END_REF][START_REF] Hamilton | Harmonic maps of manifolds with boundary[END_REF][START_REF] Moser | Intrinsic semiharmonic maps[END_REF]). While the main purpose of [START_REF] Lio | RIVI ÈRE : Three-term commutator estimates and the regularity of 1 2 -harmonic maps into spheres[END_REF][START_REF] Lio | RIVI ÈRE : Sub-criticality of non-local Schr ödinger systems with antisymmetric potential and applications to half harmonic maps[END_REF] was to show the regularity of 1/2-harmonic maps for n = 1, it was suggested there that they arise as limits of the fractional Ginzburg-Landau equation. Our primary objective here is to prove this assertion in arbitrary dimensions and for arbitrary domains, prescribing eventually an exterior Dirichlet condition like in (1.2). Working on such a program, it is natural to face the regularity problem for 1/2-harmonic maps in arbitrary dimension, and this is our secondary objective. Looking at the 1/2-harmonic map equation (1.8)-(1.9), one immediately realizes that it has precisely the same critical structure than the one of the local case, for both regularity and compactness. Here, equation (1.8) turns out to be invariant in dimension n = 1 under composition with traces on R of conformal maps from the half plane R 2 + into itself. By analogy with the local case again, it suggests that solutions of the fractional Ginzburg-Landau equation (1.1) are not strongly compact as ε → 0 in general. Describing the possible defect of strong convergence shall be one of the main concern in our asymptotic analysis of the fractional Ginzburg-Landau equation.

Our first main result, stated in Theorem 1.1 below, deals with the fractional Ginzburg-Landau equation supplemented with the Dirichlet exterior condition (1.2), and provides an answer to the questions above. We assume in this theorem the uniform energy bound with respect to ε which easily holds in most cases whenever the exterior condition g admits an S m-1 -valued extension in ω of finite energy. Note that, if g does not admits a smooth S m-1 -valued extension in ω, singularities have to appear in the limit ε → 0 as in usual Ginzburg-Landau problems. If n = m, this is the case when ∂ω ≃ S n-1 and g : ∂ω → S n-1 has a non vanishing topological degree.

Theorem 1.1. Let ω ⊆ R n be a smooth bounded open set, and let g : R n → R m be a smooth map satisfying |g| = 1 in R n \ ω. Let ε k ↓ 0 be an arbitrary sequence, and let {v k } k∈N ⊆ H 1/2 g (ω; R m ) ∩ L 4 (ω) be such that for each k ∈ N, v k weakly solves

     (-∆) 1 2 v k = 1 ε k (1 -|v k | 2 )v k in ω , v k = g in R n \ ω .
If sup k Eε k (v k , ω) < ∞, then there exist a (not relabeled) subsequence and v * ∈ H 1/2 g (ω; R m ) a weak 1/2harmonic map into S m-1 in ω such that v kv * ⇀ 0 weakly in H 1/2 00 (ω). In addition, there exist a finite nonnegative Radon measure µsing on ω, a countably H n-1 -rectifiable relatively closed set Σ ⊆ ω of finite (n -1)dimensional Hausdorff measure, and a Borel function θ : Σ → (0, ∞) such that (i) e(v k , ω) L n ω * ⇀ e(v * , ω) L n ω + µsing weakly* as Radon measures on ω;

(ii) (1 -|v k | 2 ) 2 ε k → 0 in L 1 loc (ω); (iii) 1 -|v k (x)| 2 ε k ⇀ γn 2 R n |v * (x) -v * (y)| 2
|x -y| n+1 dy + µsing in D ′ (ω);

(iv) µsing = θH n-1 Σ;

(v) v * ∈ C ∞ (ω \ Σ) and vn → v * in C ℓ loc (ω \ Σ) for every ℓ ∈ N;
(vi) if n 2, the limiting 1/2-harmonic map v * and the defect measure µsing satisfy the stationarity relation

d dt E v * • φt, ω t=0 = 1 2 Σ divΣX dµsing (1.10)
for all vector fields X ∈ C 1 (R n ; R n ) compactly supported in ω, where {φt} t∈R denotes the flow on R n generated by X;

(vii) if n = 1, the set Σ is finite and v * ∈ C ∞ (ω).
The proof of this theorem rests on the representation of (-∆)

:= R n × (0, ∞)
given by the convolution product with the Poisson kernel, see (2.9). More precisely, denoting by v → v e this harmonic extension, we shall prove that it is well defined on the space of functions of finite Dirichlet 1/2-energy, and that ∂ν v e = (-∆) 1 2 v as distributions on ω. Here, ∂ν denotes the exterior normal differentiation on ∂R n+1 + ≃ R n . When applying the extension procedure to a solution vε of the fractional Ginzburg-Landau equation (1.1), we end up with the following system of Ginzburg-Landau boundary reactions

       ∆v e ε = 0 in R n+1 + , ∂v e ε ∂ν = 1 ε (1 -|v e ε | 2 )v e ε on ω .
(1.11)

The asymptotic analysis of this system as ε → 0 will lead us to the main conclusions stated in Theorem 1.1. To this purpose, we shall first establish an epsilon-regularity type of estimate for (1.11) in the spirit of the regularity theory for harmonic maps [START_REF] Bethuel | On the singular set of stationary harmonic maps[END_REF][START_REF] Schoen | UHLENBECK : A regularity theory for harmonic maps[END_REF] or usual Ginzburg-Landau equations [START_REF] Chen | Existence and partial regularity results for the heat flow for harmonic maps[END_REF]. This estimate is the key to derive the convergence result and the rectifiability of the defect measure. Note that Theorem 1.1 actually shows that µsing is a (n -1)-rectifiable varifold in the sense of ALMGREN, see e.g. [START_REF] Simon | Lectures on geometric measure theory[END_REF]. We emphasize that identity (1.10) is precisely the coupling equation between the limiting 1/2-harmonic v * and the defect measure µsing. It states that the first inner variation of the Dirichlet 1/2energy of v * is equal to -1 2 times the first inner variation of the varifold µsing, see [70, formulas 15.7 and 16.2]. We shall achieve (1.10) in two independent steps. The first step consists in proving an analogous identity when passing to the limit ε → 0 in system (1.11). In the spirit of [START_REF] Lin | Harmonic and quasi-harmonic spheres. III. Rectifiability of the parabolic defect measure and generalized varifold flows[END_REF], the convenient way to let ε → 0 in the first inner variation of the Dirichlet energy of v e ε is to use the notion of generalized varifold of AMBROSIO & SONER [START_REF] Ambrosio | A measure-theoretic approach to higher codimension mean curvature flows[END_REF], once adapted to the boundary setting. In turn, the second step allows us to return to the original formulation on R n . It shows that the first inner variation of the Dirichlet 1/2-energy of an arbitrary map v is equal to the first inner variation up to ω of the Dirichlet energy of its harmonic extension v e , see Lemma 4.16. Note that this crucial observation also implies that an harmonic extension v e is stationary up to ω for the Dirichlet energy whenever the original function v is stationary for the Dirichlet 1/2-energy. Here, stationarity means stationarity with respect to inner variations (i.e., variations of the domain).

Concerning the regularity issue for 1/2-harmonic maps, it is also fruitful to rephrase the problem in terms of harmonic extensions. Applying the extension procedure to a weak S m-1 -valued 1/2-harmonic map v * in ω leads to the system

       ∆v e * = 0 in R n+1 + , ∂v e * ∂ν ⊥ Tve * S m-1 on ω .
(1.12)

This system turns out to be (almost) included in the class of harmonic maps with free boundary (see Remark 4.5) for which a regularity theory do exist [START_REF] Duzaar | A partial regularity theorem for harmonic maps at a free boundary[END_REF][START_REF] Duzaar | An optimal estimate for the singular set of a harmonic map in the free boundary[END_REF][START_REF] Hardt | Partially constrained boundary conditions with energy minimizing mappings[END_REF][START_REF] Scheven | Partial regularity for stationary harmonic maps at a free boundary[END_REF]. This theory provides partial regularity results, and it requires minimality, or at least stationarity up to the free boundary. As already mentioned in the discussion above, if v * is assumed to be stationary, then the extension v e * is stationary up to ω. Similarly, we shall see that harmonic extensions also inherit minimality up to ω. This is then enough to derive a general regularity result for 1/2-harmonic maps under a minimality or stationarity assumption.

We finally point out that the relation between fractional harmonic maps and harmonic maps with a free boundary has been previously noticed by MOSER [START_REF] Moser | Intrinsic semiharmonic maps[END_REF] for a (non-explicit) operator slightly different from the square root Laplacian (-∆) 1 2 , and leading to slightly weaker results.

Our regularity results for 1/2-harmonic maps can be summarized in the following theorem. For simplicity we only state it for a sphere target, but we would like to stress that it actually holds for more general target manifolds, see Remark 4.24. Notice that it is an interior regularity result. The regularity at the boundary when prescribing an exterior Dirichlet condition (1.2) remains an open question.

Theorem 1.2. Let ω ⊆ R n be a smooth bounded open set, and let v * ∈ H 1/2 (ω; R m ) ∩ L ∞ (R n ) be a weak 1/2-harmonic map into S m-1 in ω. Then v * ∈ C ∞ ω \ sing(v * ) , where sing(v * ) denotes the complement of the largest open set on which v * is continuous. Moreover, (i) if n = 1, then sing(v * ) ∩ ω = ∅; (ii) if n 2 and v * is stationary, then H n-1 sing(v * ) ∩ ω = 0; (iii) if v * is minimizing, then dim H sing(v * ) ∩ ω n -2 for n 3, and sing(v * ) ∩ ω is discrete for n = 2.
Before concluding this introduction, let us briefly comment on some possible extensions of the present results and further open questions. First of all, in the energy (1.4) one could replace the Ginzburg-Landau potential (1 -|u| 2 ) 2 by a more general nonnegative potential W (u) having a zero set {W = 0} given by a smooth compact submanifold N of R m without boundary, and then consider the corresponding fractional Ginzbug-Landau equation. In this context, the singular limit ε → 0 leads to the 1/2-harmonic map system into N (see Remark 4.24). If the codimension of N is equal to 1 (plus some non degeneracy assumptions on W ), the proof of Theorem 1.1 can certainly be reproduced with minor modifications.

However, the higher codimension case seems to require additional analysis since our espilon-regularity estimate (Theorem 5.1) strongly uses the codimension 1 structure. It could be interesting to have a proof handling both cases. In another direction, one could consider the fractional Ginzburg-Landau equation in some open subset ω of a more general (smooth) complete Riemannian manifold (M, g) with a fractional 1/2-Laplacian (-∆g)

1 2
given by the square root of the Laplace-Beltrami operator on M as defined through spectral theory and functional calculus. In the most difficult case of a noncompact manifold, the construction of (-∆g) 1 2 (or any fractional power s ∈ (0, 1)) can be found in [START_REF] Banica | Some constructions for the fractional Laplacian on noncompact manifolds[END_REF] (and the references therein). Under some geometrical assumption on M (e.g. M has bounded geometry), [START_REF] Banica | Some constructions for the fractional Laplacian on noncompact manifolds[END_REF] follows the approach of [START_REF] Stinga | Extension problem and Harnack's inequality for some fractional operators[END_REF] to construct the Poisson kernel for the harmonic extension in M × R+ (endowed with the product metric g + |dt| 2 ) of functions defined on M, and it is shown that the fractional Laplacian (-∆g) 1 2 coincides with the Dirichlet-to-Neumann operator associated to this harmonic extension. This is then enough to rephrase the fractional Ginzburg-Landau equation on M (and 1/2-harmonic maps) as a system of Ginzburg-Landau boundary reactions on ∂(M × R+) ≃ M (respectively, harmonic maps with partially free boundary), and we believe that most of our results should extend to this setting. Note that this approach was successfully used in [START_REF] Ález | Layer solutions for the fractional Laplacian on hyperbolic space: existence, uniqueness and qualitative properties[END_REF] in the scalar case m = 1 and for any fractional power s ∈ (0, 1) to study the so-called layer solutions in the hyperbolic space H n . As already mentioned, we have considered here only the case s = 1/2 for the specific relations with the geometrical problems of harmonic maps with partially free boundary and minimal surfaces with free boundary. However, it could be interesting to investigate in the case s = 1/2 the analogues of this geometric objects, and then perform the asymptotic analysis as ε → 0 of the corresponding Ginzburg-Landau equation. For this question, some aspects of the present analysis can actually be used for an arbitrary s ∈ (0, 1) by means of the 2s-harmonic extension procedure discovered in [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF]. In a forthcoming article [START_REF] Millot | Asymptotics for a fractional Allen-Cahn equation and stationary nonlocal minimal surfaces[END_REF], we shall consider the scalar Ginzburg-Landau (Allen-Cahn) equation in the case s ∈ (0, 1/2) and the related nonlocal s-minimal surfaces in the spirit of [START_REF] Caffarelli | Nonlocal minimal surfaces[END_REF][START_REF] Savin | VALDINOCI : Γ-convergence for nonlocal phase transitions[END_REF]. Interpreting nonlocal s-minimal surfaces as fractional s-harmonic maps with values in S 0 ≃ {±1}, and taking advantage of the 2s-harmonic extension, we shall prove in [START_REF] Millot | Asymptotics for a fractional Allen-Cahn equation and stationary nonlocal minimal surfaces[END_REF] the convergence, as ε → 0, of arbitrary solutions of the fractional Allen-Cahn equation with equibounded energy towards nonlocal s-minimal surfaces (possibly non-minimizing), thus extending a result in [START_REF] Savin | VALDINOCI : Γ-convergence for nonlocal phase transitions[END_REF] to the non-minimizing case.

The paper is organized as follows. In Section 2, we present all the needed material on the functional aspects of the square root Laplacian/Dirichlet-to-Neumann operator, as well as some key estimates related to harmonic extensions. Actually, we believe that this section can be of independent interest. Section 3 is devoted to the qualitative analysis of the fractional Ginzburg-Landau equation for ε fixed, and provides regularity results for this equation. Section 4 is entirely devoted to the analysis of 1/2-harmonic maps. In a first part, we give the regularity theory and prove Theorem 1.2. The second part deals with explicit examples of 1/2-harmonic maps underlying their geometrical nature, and stressing the analogies with standard harmonic maps. In Section 5, we prove the aforementioned epsilon-regularity estimate for the system of Ginzburg-Landau boundary reactions, and its asymptotic analysis as ε → 0 follows in Section 6. Then we return to the fractional Ginzburg-Landau equation in Section 7 where we perform its asymptotic analysis with and without exterior Dirichlet condition. The special case of energy minimizers under exterior Dirichlet condition is also treated. Finally, we collect in Appendix A the proofs of several statements from Section 2, and Appendix B provides some elliptic estimates coming into play in the proof of the epsilon-regularity result.

Notations. Throughout the paper, R n is identified with ∂R n+1 • the open set ∂ 0 Ω ⊆ R n defined by

+ = R n × {0}. We denote by Br(x) the open ball in R n+1 of radius r centered at x, while Dr(x) := Br(x) ∩ R n is the open disc in R n centered at x ∈ R n . For a set A ⊆ R n+1 , we write A + := A ∩ R n+1 + , ∂ + A := ∂A ∩ R n+1 + ,
∂ 0 Ω := x ∈ ∂Ω ∩ R n : B + r (x) ⊆ Ω for some r > 0 ,
is non empty and has Lipschitz boundary;

• ∂Ω = ∂ + Ω ∪ ∂ 0 Ω .
Finally, we shall always denote by C a generic positive constant which only depends on the dimensions n and m, and possibly changing from line to line. If a constant depends on additional given parameters, we shall write those parameters using the subscript notation.

ENERGY SPACES AND LOCAL

REPRESENTATION OF THE 1/2-LAPLACIAN 2.1. Basics on H 1/2 -spaces. Given an open set ω ⊆ R n , the fractional Sobolev space H 1/2 (ω; R m ) is defined by H 1/2 (ω; R m ) := v ∈ L 2 (ω; R m ) : [v] H 1/2 (ω) < ∞ ,
where

[v] 2 H 1/2 (ω) := γn 4 ω×ω |v(x) -v(y)| 2 |x -y| n+1 dxdy , γn := Γ (n + 1)/2 π (n+1)/2 ,
and we recall that H 1/2 (ω; R m ) is an Hilbert space normed by

v H 1/2 (ω) := v L 2 (ω) + [v] H 1/2 (ω) .
Then H

1/2 loc (R n ; R m ) denotes the class of functions which belongs to ∈ H 1/2 (ω; R m ) for every bounded open set ω ⊆ R n . The linear space H 1/2 00 (ω; R m ) is defined by H 1/2 00 (ω; R m ) := v ∈ H 1/2 (R n ; R m ) : v = 0 a.e. in R n \ ω ⊆ H 1/2 (R n ; R m ) .
Endowed with the norm induced by H 1/2 (R n ; R m ), the space H 1/2 00 (ω; R m ) is also an Hilbert space, and

for v ∈ H 1/2 00 (ω; R m ), [v] 2 H 1/2 (R n ) = γn 4 ω×ω |v(x) -v(y)| 2 |x -y| n+1 dxdy + γn 2 ω×ω c |v(x) -v(y)| 2 |x -y| n+1 dxdy . (2.1)
The topological dual space of H 1/2 00 (ω; R m ) will be denoted by H -1/2 (ω; R m ). We recall that if the boundary of ω is smooth enough (e.g. if ∂ω is Lipschitz regular), then

H 1/2 00 (ω; R m ) = D(ω; R m ) • H 1/2 (R n ) , (2.2)
see [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]Theorem 1.4.2.2].

Throughout the paper we shall be interested in functions for which the right hand side of (2.1) is finite.

We denote this class of funtions by

H 1/2 (ω; R m ) := v ∈ L 2 loc (R n ; R m ) : E(v, ω) < ∞ , where E(•, ω) is the 1/2-Dirichlet energy defined in (1.

3). The following properties hold for any bounded

open subsets ω and ω ′ of R n :

• H 1/2 (ω; R m ) is a linear space; • H 1/2 (ω; R m ) ⊆ H 1/2 (ω ′ ; R m ) whenever ω ′ ⊆ ω, and E(v, ω ′ ) E(v, ω) ; • H 1/2 (ω; R m ) ∩ H 1/2 loc (R n ; R m ) ⊆ H 1/2 (ω ′ ; R m ) ; • H 1/2 loc (R n ; R m ) ∩ L ∞ (R n ) ⊆ H 1/2 (ω; R m ) .
An elementary property concerning H 1/2 (ω; R m ) is given in Lemma 2.1 below whose proof can be found in Appendix A. Using this lemma, it is then elementary to see that H 1/2 (ω; R m ) is a Hilbert space for the scalar product associated to the norm

v → v L 2 (ω) + (E(v, ω)) 1/2 .
Lemma 2.1. Let x0 ∈ ω and ρ > 0 be such that Dρ(x0) ⊆ ω. There exists a constant Cρ > 0, independent of x0,

such that R n |v(x)| 2 (|x -x0| + 1) n+1 dx Cρ E v, Dρ(x0) + v 2 L 2 (Dρ(x 0 )) for all v ∈ H 1/2 (ω; R m ).

Remark 2.2 (Exterior Dirichlet condition). We observe that for any

v ∈ H 1/2 (ω; R m ), v + H 1/2 00 (ω; R m ) ⊆ H 1/2 (ω; R m ) . (2.3) Moreover, if v = g a.e. in R n \ ω for some g ∈ H 1/2 (ω; R m ), then v -g ∈ H 1/2 00 (ω; R m ).
As a consequence, the affine subspace

H 1/2 g (ω; R m ) := v ∈ H 1/2 (ω; R m ) : v = g a.e. in R n \ ω is characterized by H 1/2 g (ω; R m ) = g + H 1/2 00 (ω; R m ) . (2.4)
Finally, we shall keep in mind that

H 1/2 g (ω; R m ) ⊆ H 1/2 loc (R n ; R m ) (2.5) whenever g ∈ H 1/2 (ω; R m ) ∩ H 1/2 loc (R n ; R m ).
2.2. The fractional Laplacian. Let ω ⊆ R n be a bounded open set. We define the fractional 1/2-Laplacian (-∆)

1 2 : H 1/2 (ω; R m ) → H 1/2 (ω; R m )
′ as the continuous linear operator induced by the quadratic form E(•, ω). More precisely, for a given map v ∈ H 1/2 (ω; R m ), we define the distribution (-∆)

1
2 v through its action on elements of H 1/2 (ω; R m ) by setting

(-∆) 1 2 v, ϕ ω := γn 2 ω×ω (v(x) -v(y)) • (ϕ(x) -ϕ(y)) |x -y| n+1 dxdy + γn ω×ω c (v(x) -v(y)) • (ϕ(x) -ϕ(y)) |x -y| n+1 dxdy . (2.6)
Note that for every v ∈ H 1/2 (ω; R m ), the restriction of (-∆)

1 2 v to H 1/2 00 (ω; R m ) belongs to H -1/2 (ω; R m ) with the estimate (-∆) 1 2 v 2 H -1/2 (ω) 2E(v, ω) , (2.7)
which obviously follows from (2.1) and Cauchy-Schwarz Inequality. In addition, formula (2.6) being precisely twice the symmetric bilinear form associated to E(•, ω), we observe that the restriction of (-∆)

⊆ ω. Since H 1/2 (ω; R m ) ⊆ H 1/2 (ω ′ ; R m ) and H 1/2 00 (ω ′ ; R m ) ⊆ H 1/2 00 (ω; R m ), direct computations from (2.6) yield (-∆) 1 2 v, ϕ ω = (-∆) 1 2 v, ϕ ω ′ for all ϕ ∈ H 1/2 00 (ω ′ ; R m ). Remark 2.4. If v is a smooth bounded function, the distribution (-∆) 1 2
v can be rewritten from (2.6) as a pointwise defined function through the formula

(-∆) 1 2 v(x) = p.v. γn R n v(x) -v(y) |x -y| n+1 dy := lim δ↓0 γn R n \D δ (x) v(x) -v(y) |x -y| n+1 dy .
In turn, this last formula can be written in a non-singular form, that is

(-∆) 1 2 v(x) = γn 2 R n v(x) -v(x + h) -∇v(x) • h |h| n+1 dh .
or

(-∆) 1 2 v(x) = γn 2 R n -v(x + h) + 2v(x) -v(x -h) |h| n+1 dh .
Finally, as a motivation to the following subsection, we recall that (-∆)

(|x ′ -z| 2 + x 2 n+1 ) n+1 2 dz , (2.9)
where we write x = (x ′ , xn+1) ∈ R n+1

+ = R n × (0, ∞).
Notice that v e is well defined if v belongs to the Lebesgue space L p over R n with respect to the finite measure

m := (1 + |z|) -(n+1) dz (2.10)
for some 1 p ∞. In particular, v e can be defined whenever v ∈ H 1/2 (ω; R m ) for some open bounded

set ω ⊆ R n by Lemma 2.1. Moreover, if v ∈ L ∞ (R n ), then v e ∈ L ∞ (R n+1 + ) and v e L ∞ (R n+1 + ) v L ∞ (R n ) . (2.11)
For an admissible function v, the extension v e is harmonic in R n+1 + . In addition, it has a pointwise trace on ∂R n+1 + = R n which is equal to v at every Lebesgue point. In other words, v e solves  



∆v e = 0 in R n+1 + , v e = v on R n .
(2.12)

The operator v → v e enjoys some useful continuity properties. Among them, we shall use the following lemma which is proven in Appendix A.

Lemma 2.5. For every R > 0, the linear operator PR :

L 2 (R n , m) → L 2 (B + R ) defined by PR(v) := v e |B + R , (2.13)
is continuous.

If v ∈ Ḣ1/2 (R n ; R m ),
it is well known that the harmonic extension v e belongs to Ḣ1 (R n+1 + ; R m ), and the H 1/2 -seminorm of v can be computed from the Dirichlet energy of v e (here Ḣ1/2 and Ḣ1 denote the homogeneous Sobolev spaces). Moreover, v e turns out to be the extension of v of minimal energy. This result, summarized in Lemma 2.6 below, is a classical application of Fourier Transform (see e.g. [START_REF] Nezza | VALDINOCI : Hitchhiker's guide to the fractional Sobolev spaces[END_REF]). Lemma 2.6. Let v ∈ Ḣ1/2 (R n ; R m ), and let v e be its harmonic extension to R n+1 + given by (2.9). Then v e belongs to Ḣ1 (R n+1 + ; R m ) and

[v] 2 H 1/2 (R n ) = 1 2 R n+1 + |∇v e | 2 dx = inf 1 2 R n+1 + |∇u| 2 dx : u ∈ Ḣ1 (R n+1 + ; R m ) , u = v on R n .
For a bounded open set ω ⊆ R n , we have the following analogous result whose proof is postponed to Appendix A. In the following statement and hereafter, we denote by

H 1 loc (R n+1 + ∪ ω; R m ) the family of maps which belongs to H 1 (Ω; R m ) for every bounded open set Ω ⊆ R n+1 + such that Ω ⊆ R n+1 + ∪ ω.
Lemma 2.7. Let ω ⊆ R n be a bounded open set. For every v ∈ H 1/2 (ω; R m ), the harmonic extension v e given by (2.9) belongs to

H 1 loc (R n+1 + ∪ ω; R m ) ∩ L 2 loc (R n+1
+
). In addition, for every x0 ∈ ω, R > 0 and ρ > 0 such that D3ρ(x0) ⊆ ω, there exist constant CR,ρ > 0 and CR > 0, independent of v and x0, such that

v e 2 L 2 (B + R (x 0 )) CR,ρ E v, D2ρ(x0) + v 2 L 2 (D 2ρ (x 0 ))
, and

∇v e 2 L 2 (B + ρ (x 0 )) Cρ E v, D2ρ(x0) + v 2 L 2 (D 2ρ (x 0 ))
.

Remark 2.8. By the previous lemma, for

any v ∈ H 1/2 (ω; R m ) ∩ H 1/2 loc (R n ; R m ), the harmonic extension v e belongs to H 1 loc (R n+1
+ ; R m ), and for any R > 0,

v e 2 H 1 (B + R ) CR E v, D2R + v 2 L 2 (D 2R ) . If v ∈ H 1/
that ∂ 0 Ω ⊆ ω. Let v ∈ H 1/2 (ω; R m ),
and let v e be its harmonic extension to R n+1

+
given by (2.9). Then,

1 2 Ω |∇u| 2 dx - 1 2 Ω |∇v e | 2 dx E(u, ω) -E(v, ω) (2.15
)

for all u ∈ H 1 (Ω; R m ) such that u -v e is compactly supported in Ω ∪ ∂ 0 Ω. In the right hand side of (2.15), the trace of u on ∂ 0 Ω is extended by v outside ∂ 0 Ω. Proof. Let u ∈ H 1 (Ω; R m ) such that u -v e is compactly supported in Ω ∪ ∂ 0 Ω. We extend u by v e outside Ω. Then h := u -v e ∈ H 1 (R n+1 + ; R m ) and h is compactly supported in Ω ∪ ∂ 0 Ω. Hence h |R n ∈ H 1/2 00 (∂ 0 Ω; R m ). Since v ∈ H 1/2 (∂ 0 Ω; R m ) we deduce from (2.
3) that u admits a trace on R n which belongs to H 1/2 (∂ 0 Ω; R m ).

Using Lemma 2.6 and Lemma 2.9, we now estimate

1 2 Ω |∇u| 2 dx - 1 2 Ω |∇v e | 2 dx = 1 2 R n+1 + |∇h| 2 dx + R n+1 + ∇v e • ∇h dx = 1 2 R n+1 + |∇h| 2 dx + (-∆) 1 2 v, h |R n ∂ 0 Ω h |R n 2 H 1/2 (R n ) + (-∆) 1 2 v, h |R n ∂ 0 Ω = E(h |R n , ∂ 0 Ω) + (-∆) 1 2 v, h |R n ∂ 0 Ω . (2.16) Using the fact that u |R n , v ∈ H 1/2 (∂ 0 Ω; R m ), we derive E(h |R n , ∂ 0 Ω) = E(u |R n , ∂ 0 Ω) + E(v, ∂ 0 Ω) -(-∆) 1 2 v, u |R n ∂ 0 Ω , (2.17)
and

(-∆) 1 2 v, h |R n ∂ 0 Ω = (-∆) 1 2 v, u |R n ∂ 0 Ω -2E(v, ∂ 0 Ω) . (2.18)
Gathering (2.16)-(2.17)-(2.18) yields

1 2 Ω |∇u| 2 dx - 1 2 Ω |∇v e | 2 dx E(u |R n , ∂ 0 Ω) -E(v, ∂ 0 Ω) . Since u |R n = v outside ∂ 0 Ω, we infer that E(u |R n , ∂ 0 Ω) -E(v, ∂ 0 Ω) = E(u |R n , ω) -E(v, ω) ,
and the conclusion follows.

THE FRACTIONAL GINZBURG-LANDAU EQUATION

We consider for the entire section a bounded open set ω ⊆ R n with Lipschitz boundary. We are interested in weak solutions vε ∈ H 1/2 (ω; R m ) ∩ L 4 (ω) of the fractional Ginzburg-Landau equation

(-∆) 1 2 vε = 1 ε (1 -|vε| 2 )vε in ω . (3.1)
Here the notion of weak solution is understood in the duality sense according to the formulation (2.6) of the fractional Laplacian, i.e., (-∆)

1 2 vε, ϕ ω = 1 ε ω (1 -|vε| 2 )v • ϕ dx for all ϕ ∈ H 1/2 00 (ω; R m ) ∩ L 4 (ω)
(or equivalently for all ϕ ∈ D(ω; R m )). By (2.8), such solutions correspond to critical points in ω of the

Ginzburg-Landau 1/2-energy Eε(v, ω) := E(v, ω) + 1 4ε ω (1 -|v| 2 ) 2 dx .
In other words, we are interested in maps vε

∈ H 1/2 (ω; R m ) ∩ L 4 (ω) satisfying d dt Eε(vε + tϕ, ω) t=0 = 0 for all ϕ ∈ H 1/2 00 (ω; R m ) ∩ L 4 (ω) .
Among all kinds of critical points are the minimizers. We say that vε

∈ H 1/2 (ω; R m )∩L 4 (ω) is a minimizer of Eε in ω if Eε(vε, ω) Eε(vε + ϕ, ω) for all ϕ ∈ H 1/2 00 (ω; R m ) ∩ L 4
(ω) (or equivalently for all ϕ ∈ D(ω; R m )). The most standard way to construct minimizing solutions (and in particular solutions of (3.1)) is to minimize Eε(•, ω) under an exterior Dirichlet condition. More precisely, given a map g ∈ H 1/2 (ω; R m ) ∩ L 4 (ω), one consider the minimization problem

min Eε(v, ω) : v ∈ H 1/2 g (ω; R m ) ∩ L 4 (ω) , (3.2)
whose resolution follows directly from the Direct Method of Calculus of Variations.

To investigate the qualitative properties of solutions of (3.1), we rely on the harmonic extension to the half space R n+1 + introduced in the previous section. According to Lemma 2.9,

if vε ∈ H 1/2 (ω; R m )∩L 4 (ω)
is a weak solution of (3.1), then its harmonic extension v e ε given by (2.9) weakly solves

       ∆v e ε = 0 in R n+1 + , ∂v e ε ∂ν = 1 ε 1 -|v e ε | 2 v e ε on ω . (3.3)
In view of Lemma 2.7 and (2.14), the weak sense for this system corresponds to

R n+1 + ∇v e ε • ∇Φ dx = 1 ε ω 1 -|v e ε | 2 v e ε • Φ dH n (3.4) for all Φ ∈ H 1 (R n+1 + ; R m ) ∩ L 4 (ω) compactly supported in R n+1 + ∪ ω. System (3.
3) also has a variational structure. Indeed, considering an admissible bounded open set Ω ⊆ R n+1 + such that ∂ 0 Ω ⊆ ω, and setting

for u ∈ H 1 (Ω; R m ) ∩ L 4 (∂ 0 Ω), Eε(u, Ω) := 1 2 Ω |∇u| 2 dx + 1 4ε ∂ 0 Ω (1 -|u| 2 ) 2 dH n , (3.5)
the weak formulation (3.4) can be rephrased as If vε turns out to be minimizing, we can transfer the minimality of vε to v e ε with the help of Corollary 2.10. In the following proposition, we say that u ∈ H

d dt Eε (v e ε + tΦ, Ω) t=0 = 0 for all Φ ∈ H 1 (Ω; R m ) ∩ L 4 (∂ 0 Ω) compactly supported in Ω ∪ ∂ 0 Ω. Hence v e is
1 (Ω; R m ) ∩ L 4 (∂ 0 Ω) is minimizer of Eε in Ω if Eε(u, Ω) Eε(u + Φ, Ω) for all Φ ∈ H 1 (Ω; R m ) ∩ L 4 (∂ 0 Ω) compactly supported in Ω ∪ ∂ 0 Ω. Proposition 3.1 (Minimality transfer). Let vε ∈ H 1/2 (ω; R m ) ∩ L 4 (ω)
Ω ⊆ R n+1 + such that ∂ 0 Ω ⊆ ω.
Concerning regularity for system (3.1), it suffices to investigate the boundary regularity of the harmonic extension which satisfies (3.3). For this issue we can rely on the well known regularity theory for linear Neumann boundary value problems, see e.g. [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]. The following theorem deals with regularity in the interior of the free boundary, and a simple proof can be found in [START_REF] Cabr É | SOL À-MORALES : Layer solutions in a halfspace for boundary reactions[END_REF]Lemma 2.2] (which actually deals with scalar valued equations, but a quick inpection of the proof shows that the result still holds in the vectorial case).

Theorem 3.2. Let R > 0, and let uε ∈ H 1 (B + 2R ; R m ) ∩ L ∞ (B + 2R ) be a weak solution of        ∆uε = 0 in B + 2R , ∂uε ∂ν = 1 ε (1 -|uε| 2 )uε on D2R . Then u ∈ C ∞ B + R ∪ DR .
Applying Theorem 3.2 to system (3.3) yields interior regularity for bounded variational solutions of the fractional Ginzburg-Landau system (3.1).

Corollary 3.3. Let vε ∈ H 1/2 (ω; R m ) ∩ L ∞ (R n ) be a weak solution of (3.1). Then vε ∈ C ∞ (ω).
Dealing with arbitrary critical points of Eε(•, ω) in H 1/2 g (ω; R m ) for a smooth domain ω and a smooth (exterior) Dirichlet data g, one may wonder if regularity holds accross the boundary ∂ω. If vε is such a critical point and v e ε is its harmonic extension, we end up with the mixed boundary value problem

                 ∆v e ε = 0 in R n+1 + , ∂v e ε ∂ν = 1 ε 1 -|v e ε | 2 v e ε on ω , v e ε = g on R n \ ω . (3.6)
Even if we obviously expect regularity in the interior of ω, it might not hold across the edge ∂ω, since solutions of linear mixed boundary value problems are usually not better than H ölder continous (and 1/2 is in general the best possible H ölder exponent, see e.g. [START_REF] Savar | Regularity and perturbation results for mixed second order elliptic problems[END_REF] and the references therein). H ölder continuity for variational solutions of (non homogeneous) linear mixed boundary value problems follows from the general results of STAMPACCHIA [START_REF] Stampacchia | Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie h ölderiane[END_REF][START_REF] Stampacchia | Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus[END_REF] (see also [START_REF] Chicco | H ölder regularity for solutions of mixed boundary value problems containing boundary terms[END_REF]), and an estimate on the order of regularity is given by the classical result of SHAMIR [START_REF] Shamir | Regularization of mixed second-order elliptic problems[END_REF]. In our case, it is enough to prove an a priori global L ∞ -bound on vε to derive from [START_REF] Shamir | Regularization of mixed second-order elliptic problems[END_REF] the regularity stated in the following theorem.

Theorem 3.4. Assume that ∂ω is smooth. Let g ∈ C 1 (R n ; R m ) ∩ L ∞ (R n ), and let vε ∈ H 1/2 g (ω; R m ) ∩ L 4 (ω) be a weak solution of (3.1). Then vǫ ∈ C ∞ (ω) and vε is α-Hölder continuous near ∂ω for every α ∈ (0, 1/2).
As mentioned in the few lines above, the proof of Theorem 3.4 rely on the boundedness of vε. Indeed, once the L ∞ -bound on vε is obtained, the right hand side in the Neumann boundary equation of (3.6) remains bounded. This is enough to apply the regularity result of [START_REF] Shamir | Regularization of mixed second-order elliptic problems[END_REF], and then deduce the H ölder continuity of v e ε (and thus of vǫ). The higher order regularity away from ∂ω follows from Theorem 3.2.

Lemma 3.5. Let ω and g be as in Theorem 3.4. Step 1. By (2.5) and Remark 2.8,

Let vε ∈ H 1/2 g (ω; R m ) ∩ L 4 (ω) be a weak solution of (3.1). Then vε ∈ L ∞ (R n ).
v e ε ∈ H 1 loc (R n+1
+ ; R m ) and v e ε weakly solves (3.6). By standard elliptic regularity for the Dirichlet problem, we have

v e ε ∈ C 1 (R n+1 + \ ω). Since dist(∂ + Ω, ω) > 0, we have M := v e ε L ∞ (∂ + Ω) + ∇v e ε L ∞ (∂ + Ω) < ∞ .
Let us now consider the scalar function

η := |v e ε | 2 + λ ∈ H 1 (Ω) ∩ L 4 (ω) with λ := max(1, g 2 L ∞ (R n \ω)
), and fix an arbitrary nonnegative function Φ ∈ C ∞ (Ω) with compact support in Ω ∪ ω ∪ ∂ + Ω. By the chain-rule formula for Sobolev functions we have

Ω ∇η • ∇Φ dx = n+1 j=1 Ω (v e ε • ∂jv e ε )∂jΦ |v e ε | 2 + λ dx . Since v e ε |v e ε | 2 + λ ∈ H 1 (Ω; R m ) ,
we deduce that

Ω ∇η • ∇Φ dx = Ω ∇v e ε • ∇ Φv e ε |v e ε | 2 + λ dx - Ω |∇v e ε | 2 |v e ε | 2 + λ - n+1 j=1 |v e ε • ∂jv e ε | 2 (|v e ε | 2 + λ) 3/2 Φ dx .
On the other hand Φ 0, so that

Ω ∇η • ∇Φ dx Ω ∇v e ε • ∇ Φv e ε |v e ε | 2 + λ dx .
Using equation (3.6), we infer that

Ω ∇η • ∇Φ dx ∂ + Ω vε • ∂v e ε ∂ν Φ |v e ε | 2 + λ dH n + 1 ε ω (1 -|v e ε | 2 ) |v e ε | 2 Φ |v e ε | 2 + λ dH n , whence Ω ∇η • ∇Φ dx ∂ + Ω vε • ∂v e ε ∂ν Φ η dH n - 1 ε ω |v e ε | 2 (η + √ 1 + λ) η (η - √ 1 + λ)Φ dH n . (3.7)
Then we conclude by approximation that (3.7) actually holds for any nonnegative Φ ∈ H 1 (Ω) ∩ L 4 (ω)

satisfying Φ = 0 H n -a.e. on ∂ 0 Ω \ ω.
Step 2. Given T > 0 and β > 0, we define

ρ := max{η - √ 2λ, 0} , ρT := min(ρ, T ) , Ψ T,β := ρ β T ρ , Φ T,β := ρ 2β T ρ .
Those functions belong to H 1 (Ω) ∩ L 4 (ω) and vanish on

∂ 0 Ω \ ω. Setting ΩT := {0 < ρ < T } ∩ Ω, straightforward computations yield Ω |∇Ψ T,β | 2 dx = Ω ρ 2β T |∇η| 2 dx + (β 2 + 2β) Ω T ρ 2β |∇η| 2 dx ,
and

Ω ∇η • ∇Φ T,β dx = Ω ρ 2β T |∇η| 2 dx + 2β Ω T ρ 2β |∇η| 2 dx .
From this two equalities we now infer that

Ω |∇Ψ T,β | 2 dx (β + 1) Ω ∇η • ∇Φ T,β dx .
Using Φ T,β as a test function in (3.7), we deduce that

Ω |∇Ψ T,β | 2 dx (β + 1) ∂ + Ω vε • ∂v e ε ∂ν ρ 2β T ρ η dH n - β + 1 ε ω |v e ε | 2 (η + √ 1 + λ) η ρ 2β T ρ 2 dH n , which leads to Ω ∇ ρ β T ρ 2 dx (β + 1)H n (∂ + Ω)M 2β+2 . Applying the Poincaré Inequality in [80, Corollary 4.5.2] to ρ β T ρ yields ∂Ω ρ β T ρ 2 dx CΩ,ω(β + 1)M 2β+2 ,
for a constant CΩ,ω > 0 which only depends on Ω and ω. Next we let T → ∞ in this last inequality to obtain

ρ L 2(β+1) (∂Ω) C 1/(2β+2) Ω,ω (β + 1) 1/(2β+2) M . Letting now β → ∞ leads to ρ L ∞ (∂Ω) M , which in turn implies vε ∈ L ∞ (ω). Since vε = g outside ω,
we have thus proved that vε ∈ L ∞ (R n ).

We now deduce from the maximum principle the following upper bound on the modulus of a critical point vε satisfying an exterior Dirichlet condition.

Corollary 3.6. Let ω and g be as in Theorem 3.4.

Let vε ∈ H 1/2 g (ω; R m ) ∩ L 4 (ω) be a weak solution of (3.1). Then vε L ∞ (R n ) max(1, g L ∞ (R n \ω) ). Proof. By Theorem 3.4, v e ε is smooth in R n+1 + ∪ ω and continous in R n+1 + . We consider the function mε := λ 2 -|v e ε | 2 with λ := max(1, g L ∞ (R n \ω) ). Then mε satisfies                -∆mε = 2|∇v e ε | 2 0 in R n+1 + , ∂mε ∂ν = - 2 ε |v e ε | 2 (mε + 1 -λ 2 ) on ω , mε 0 on R n \ ω .
Assume that mε achieves its minimum over R n at a point x0 ∈ ω. Then x0 is a point of maximum of |vε|, and hence x0 is an absolute minima of mε over the whole half space R sphere S m-1 as well as the related notion of boundary harmonic map. We shall then discuss their regularity in view of the existing theory for classical harmonic maps. To simplify slightly the presentation, we made the choice to focus on the case where the target manifold is a sphere. However definitions and results extend to more general compact manifolds without boundary, see Remark 4.24 at the end of this subsection.

Definition 4.1. Let v ∈ H 1/2 (ω; R m ) be such that |v| = 1 a.e. in ω. We say that v is weakly 1/2-harmonic into S m-1 in ω if d dt E v + tϕ |v + tϕ| , ω t=0 = 0 (4.1) for all ϕ ∈ H 1/2 00 (ω; R m ) ∩ L ∞ (ω) compactly supported in ω.
Writing explicitly the criticality condition (4.1), we find the variational formulation of the Euler-Lagrange equation for 1/2-harmonic into S m-1 as stated in the following proposition.

Proposition 4.2. Let v ∈ H 1/2 (ω; R m ) be such that |v| = 1 a.e. in ω. Then v is a weak 1/2-harmonic into S m-1
in ω if and only if

(-∆) 1 2 v, ϕ ω = 0 (4.2) for all ϕ ∈ H 1/2 00 (ω; R m ) satisfying ϕ(x) ∈ T v(x) S m-1 a.e. in ω .
Proof.

Step 1. We start proving that (4.2) implies (4.1). By the density of compactly supported smooth functions stated in (2.2), it is enough to consider the case where ϕ ∈ D(ω; R m ). Then we observe that

(ϕ • v)v ∈ H 1/2 00 (ω; R m ). Hence ψ := ϕ -(ϕ • v)v belongs to H 1/2 00 (ω; R m ), and it satisfies ψ • v = 0 a.e. in ω. Noticing that v(x) + tϕ(x) |v(x) + tϕ(x)| = v(x) + tψ(x) + O(t 2 ) as t → 0 ,
we obtain by dominated convergence and assuming (4.2) that

d dt E v + tϕ |v + tϕ| , ω t=0 = (-∆) 1 2 v, ψ ω = 0 ,
and this first step is complete.

Step 2. We now prove the reverse implication. Let ϕ ∈ H 1/2 00 (ω; R m ) be such that ϕ • v = 0 a.e. in ω. By standard truncations and cut-off arguments, we may assume without loss of generality that ϕ is compactly supported in ω and that ϕ ∈ L ∞ (ω). Then,

v(x) + tϕ(x) |v(x) + tϕ(x)| = v(x) + tϕ(x) 1 + t 2 |ϕ(x)| 2 = v(x) + tϕ(x) + O(t 2 ) as t → 0 ,
and we deduce again by dominated convergence and assumption (4.1) that

0 = d dt E v + tϕ |v + tϕ| , ω t=0 = (-∆) 1 2 v, ϕ ω ,
which ends the proof.

Remark 4.3. Let v ∈ H 1/2 (ω; R m ) be a weak 1/2-harmonic map into S m-1 in ω.
In view of Proposition 4.2, the Euler-Lagrange equation can be written in the form

(-∆) 1 2 v ⊥ Tv S m-1 in H -1/2 (ω) , (4.3)
which is the clear analogue of the classical (weak) harmonic map system. Of course this equation is not completely explicit, but one can rewrite it computing the Lagrange multiplier associated to the constraint.

In the case where |v| = 1 a.e. in all of R n , it takes a quite simple form. Indeed, for ϕ ∈ D(ω; R m ), Proposition 4.2 yields

(-∆) 1 2 v, ϕ ω = (-∆) 1 2 v, (ϕ • v)v ω .
Using the representation (2.6) and the pointwise identity (based on the fact that |v| = 1),

v(x) -v(y) • (ϕ(x) • v(x))v(x) -(ϕ(y) • v(y))v(y) = 1 2 |v(x) -v(y)| 2 ϕ(x) • v(x) + ϕ(y) • v(y) ,
we obtain

(-∆) 1 2 v, (ϕ • v)v ω = γn 2 ω×R n |v(x) -v(y)| 2 |x -y| n+1 v(x) • ϕ(x) dxdy .
The identity above then leads to

(-∆) 1 2 v(x) = γn 2 R n |v(x) -v(y)| 2 |x -y| n+1 dy v(x) in D ′ (ω) ,
which is again in clear analogy with the standard harmonic map system into a sphere.

We now introduce what we call S m-1 -boundary harmonic maps. Those maps correspond to critical points of the Dirichlet energy under the constraint to be S m-1 -valued on a prescribed portion of the boundary.

Definition 4.4.

Let Ω ⊆ R n+1 + be an admissible bounded open set, and let u ∈ H 1 (Ω; R m ) be such that |u| = 1 H n -a.e. on ∂ 0 Ω. We say that u is weakly harmonic in Ω with respect to the (partially) free boundary

condition u(∂ 0 Ω) ⊆ S m-1 if Ω ∇u • ∇Φ dx = 0 (4.4) for all Φ ∈ H 1 (Ω; R m ) ∩ L ∞ (Ω) compactly supported Ω ∪ ∂ 0 Ω and satisfying Φ(x) ∈ T u(x) S m-1 H n -a.e. on ∂ 0 Ω . (4.5)
In short, we shall say that u is a weak (S m-1 , ∂ 0 Ω)-boundary harmonic map in Ω.

Defintion 4.4 can be motivated by the fact that

d dt 1 2 Ω |∇ut| 2 dx t=0 = Ω ∇u • ∇Φ dx
for variations ut of the form

ut := u + tΦ 1 + t 2 |Φ| 2 ,
with Φ satisfying (4.5). Such variations ut belong to H 1 (Ω; R m ), and they satisfy the constraint |ut| = 1

H n -a.e. on ∂ 0 Ω by (4.5). Actually, if u ∈ H 1 (Ω; R m ) is a weak (S m-1 , ∂ 0 Ω)-boundary harmonic map
in Ω, a standard truncation argument shows that (4.4) holds for any Φ ∈ H 1 (Ω; R m ) with compact support support in Ω ∪ ∂ 0 Ω and satisfying (4.5). Moreover, choosing Φ with compact support in Ω in formula (4.4) shows that ∆u = 0 in Ω. Integrating by parts in (4.4) then yields

∂ 0 Ω ∂u ∂ν • Φ dH n = 0
for all smooth functions Φ compactly supported in Ω ∪ ∂ 0 Ω and satisfying (4.5). In general the integral above has of course to be understood in the

H -1/2 -H 1/2
00 duality sense. Therefore, the weak (S m-1 , ∂ 0 Ω)-boundary harmonic map system can be reformulated as

       ∆u = 0 in Ω , ∂u ∂ν ⊥ Tu S m-1 in H -1/2 (∂ 0 Ω) .
= R m /rZ m with factor r > 2. Indeed, if u is a (S m-1 , ∂ 0 Ω)-boundary harmonic map in Ω satisfying u L ∞ (Ω) < r/2
, then it can be considered as a M-valued harmonic map. We finally point out that such L ∞ -bound is not too restrictive in many applications. For instance, if we consider an additional Dirichlet boundary condition u = g on ∂ + Ω with g : ∂ + Ω → S m-1 smooth, then we usually obtain u L ∞ (Ω) 1 through some maximum principle.

As a consequence of (2.14), Lemma 2.9, and Proposition 4.2, we can relate 1/2-harmonic maps into S m-1 to (S m-1 , ∂ 0 Ω)-boundary harmonic maps, as we already did for the fractional Ginzburg-Landau equation.

Proposition 4.6 (Criticality transfer).

Let v ∈ H 1/2 (ω; R m ) be a weak 1/2-harmonic map into S m-1 in ω, and let v e be its harmonic extension to R n+1

+

given by (2.9). Then v e is a weak

(S m-1 , ∂ 0 Ω)-boundary harmonic map in Ω for every admissible bounded open set Ω ⊆ R n+1 + such that ∂ 0 Ω ⊆ ω.
For what concerns regularity, let us emphasize that a full regularity result cannot hold in general neither for weakly 1/2-harmonic maps into S m-1 nor for weak (S m-1 , ∂ 0 Ω)-boundary harmonic maps due to topological constraints. In the case of boundary harmonic maps, the regularity issue is of course the regularity at the free boundary ∂ 0 Ω, since the system is the usual Laplace equation in Ω (so that boundary harmonic maps are smooth in the interior of the domain). At the end of the next subsection, we

shall give examples of weakly 1/2-harmonic maps from R 2 into S 1 which are not continuous at the origin (see Proposition 4.30), the prototypical one being v(x) = x |x| . Obviously, such examples do not exclude some partial regularity to hold, but dealing with general weak solutions we actually do not expect any reasonable kind of regularity by analogy with the classical harmonic map equation into a manifold and the famous counterexample of RIVI ÈRE [START_REF]RIVI ÈRE : Everywhere discontinuous harmonic maps into spheres[END_REF] (of a weak harmonic map from the three dimensional ball into S 2 which is everywhere discontinuous). In the context of 1/2-harmonic maps into spheres, we are not aware of an analoguous counterexample to regularity, but we believe that it should exist. As it is the case for harmonic maps into a manifold, it is then reasonnable to ask for an extra assumption on a weak 1/2-harmonic map to derive at least partial regularity. The (usual) assumption we make is either energy minimality or stationarity, i.e., criticality under inner variations. We now recall these notions for 1/2-harmonic maps and for boundary harmonic maps.

Definition 4.7. Let v ∈ H 1/2 (ω; R m ) be such that |v| = 1 a.e. in ω. We say that v is a minimizing 1/2- harmonic map into S m-1 in ω if E(v, ω) E(ṽ, ω)
for all ṽ ∈ H 1/2 (ω; R m ) such that |ṽ| = 1 a.e. in ω, and ṽv is compactly supported in ω. (Ω; R m ) be such that |u| = 1 H n -a.e. on ∂ 0 Ω. We say that u is a minimizing harmonic map in Ω with respect to the (partially) free

boundary condition u(∂ 0 Ω) ⊆ S m-1 if 1 2 Ω |∇u| 2 dx 1 2 Ω |∇ũ| 2 dx for all ũ ∈ H 1 (Ω; R m ) such that |ũ| = 1 H n -a.e. on ∂ 0 Ω, and ũ -u is compactly supported in Ω ∪ ∂ 0 Ω.
In short, we shall say that u is a minimizing (S m-1 , ∂ 0 Ω)-boundary harmonic map in Ω.

Obviously, if v ∈ H 1/2 (ω; R m ) is a minimizing 1/2-harmonic map into S m-1 in ω, then v is a weak 1/2-harmonic map into S m-1 in ω.
In the same way, minimizing (S m-1 , ∂ 0 Ω)-boundary harmonic maps are weak (S m-1 , ∂ 0 Ω)-boundary harmonic maps. As already pursued for the fractional Ginzburg-Landau equation, the minimality of a 1/2-harmonic map can be transfered to its harmonic extension with the help of Corollary 2.10.

Proposition 4.9 (Minimality transfer). Let v ∈ H 1/2 (ω; R m ) be a minimizing 1/2-harmonic map into S m-1 in ω. Let v e be the harmonic extension of v in R n+1

+

given by (2.9). Then v e is a minimizing (S m-1 , ∂ 0 Ω)boundary harmonic map in Ω for every admissible bounded open set

Ω ⊆ R n+1 + such that ∂ 0 Ω ⊆ ω.
As mentioned above, the regularity theory for harmonic maps also deals with stationary harmonic maps. For boundary harmonic maps, the stationarity criteria has to allow inner variations up to the partially free boundary ∂ 0 Ω, and it leads to the following definition.

u ∈ H 1 (Ω; R m ) be a weak (S m-1 , ∂ 0 Ω)-boundary harmonic map in Ω. We say that u is stationary if d dt 1 2 Ω ∇(u • Φt) 2 dx t=0 = 0 (4.7)
for any differentiable 1-parameter group of smooth diffeomorphisms Φt : R n+1 → R n+1 satisfying (∂iu • ∂ju)∂jXi dx = 0 (4.8) for all vector fields X = (X1, . . . , Xn+1) ∈ C 1 (Ω; R n+1 ) compactly supported in Ω ∪ ∂ 0 Ω and satisfying Xn+1 = 0 on ∂ 0 Ω (the left hand side of (4.8) being exactly minus twice the value of the derivative in (4.7)).

• Φ0 = id R n+1 ; • Φt(R n ) ⊆ R n ; • Φt -id R n+1 is compactly supported, and supp(Φt -id R n+1 ) ∩ R n+1 + ⊆ Ω ∪ ∂ 0 Ω. Remark 4.
Moreover, using the so-called stress-energy tensor T = (Ti) n+1 i=1 given by (Ti)j(u) := |∇u| 2 δij -2(∂iu • ∂ju) , 1 j n + 1 , identity (4.8) can be rewritten as the following system of conservation laws

   div Ti = 0 in Ω for each i ∈ {1, . . . , n + 1} , Ti • ν = 0 on ∂ 0 Ω for each i ∈ {1, . . . , n} , (4.9)
in the sense of distributions.

Remark 4.12. For any u ∈ H 1 (Ω; R m ) satisfying ∆u = 0 in Ω, it turns out that the integral in the left hand side of (4.8) only depends on the trace of (X1, . . . , Xn) on ∂ 0 Ω. Indeed, if u is harmonic in Ω, then we have div T = 0 in Ω. Therefore, if X vanishes on ∂ 0 Ω, integrating by parts this integral shows that it is equal to zero (no matter what the trace of u on ∂ 0 Ω is).

Remark 4.13. Note that minimizing boundary harmonic maps are always stationary, but a stationary boundary harmonic map might not be minimizing. On the other hand, if a boundary harmonic map is smooth enough up to the free boundary ∂ 0 Ω, then it is stationary. Indeed, under such a smoothness assumption a simple computation shows that a solution of (4.6) satisfies (4.9).

Concerning the 1/2-Dirichlet energy, we define the notion of stationarity in the usual way through inner variations.

Definition 4.14.

Let v ∈ H 1/2 (ω; R m ) be a weak 1/2-harmonic map into S m-1 in ω. We say that v is stationary in ω if d dt E v • φt, ω t=0 = 0
for any differentiable 1-parameter group of smooth diffeomorphisms φt : R n → R n satisfying

• φ0 = id R n ;
• φtid R n is compactly supported, and supp(φt

-id R n ) ⊆ ω.
Another consequence of Corollary 2.10 is that stationarity for 1/2-harmonic maps implies stationarity for harmonic extensions as stated in the following proposition. given by (2.9). Then v e is a stationary weak

(S m-1 , ∂ 0 Ω)-boundary harmonic map in Ω for every admissible bounded open set Ω ⊆ R n+1 + such that ∂ 0 Ω ⊆ ω.
As matter of fact, Proposition 4.15 is directly implied by the following lemma together with Remarks 4.11 & 4.12.

Lemma 4.16. Let Ω ⊆ R n+1 + be an admissible bounded open set such that ∂ 0 Ω ⊆ ω. Given X ∈ C 1 (R n ; R n ) compactly supported in ∂ 0 Ω, let {φt} t∈R be the flow generated by X. For each v ∈ H 1/2 (ω; R m ) we have d dt E v • φt, ω t=0 = - 1 2 Ω |∇v e | 2 divX -2 n+1 i,j=1
(∂iv e • ∂jv e )∂jXi dx ,

where X = (X1, . . . , Xn+1) ∈ C 1 (Ω; R n+1 ) is any vector field compactly supported in Ω ∪ ∂ 0 Ω satisfying X = (X, 0) on ∂ 0 Ω.
Proof. Let X = (X1, . . . , Xn+1) ∈ C 1 (Ω; R n+1 ) be an arbitrary vector field compactly supported in Ω ∪ ∂ 0 Ω and satisfying X = (X, 0) on ∂ 0 Ω. Then consider a compactly supported C 1 -extension of X to the whole space R n+1 , still denoted by X, such that X = (X, 0) on R n . We define {Φt} t∈R to be the flow generated by X, i.e., for each x ∈ R n+1 the map t → Φt(x) is defined as the solution of

   d dt Φt(x) = X Φt(x) , Φ0(x) = x .
One can easily check that the family {Φt} t∈R is admissible for Definition 4.10. Noticing that Φt = (φt, 0) on R n , we now infer from Corollary 2.10 that

1 2 Ω ∇(v e • Φt) 2 dx - 1 2 Ω |∇v e | 2 dx E v • φt, ω -E(v, ω) .
Dividing both sides of this inequality by t > 0, and then letting t → 0, we obtain

- 1 2 Ω |∇v e | 2 divX -2 n+1 i,j=1
(∂iv e • ∂jv e )∂jXi dx

d dt E v • φt, ω t=0 . (4.10)
Here, straightforward computations yield

d dt E v • φt, ω t=0 = (n + 1)γn 4 ω×ω |v(x) -v(y)| 2 |x -y| n+1 (x -y) • (X(x) -X(y)) |x -y| 2 dxdy + (n + 1)γn 2 ω×ω c |v(x) -v(y)| 2 |x -y| n+1 (x -y) • X(x) |x -y| 2 dxdy - γn 2 ω×R n |v(x) -v(y)| 2 |x -y| n+1 div X(x) dxdy . (4.11)
Since X was chosen arbitrary, inequality (4.10) holds with (-X) and (-X) instead of X and X respectively, and we conclude that inequality (4.10) is in fact an equality.

Remark 4.17. From formula (4.11), one can actually deduce that any sufficiently smooth 1/2-harmonic map v in ω is stationary.

Proof. To see this, we write the first two terms in the right hand side of (4.11) as A and B, respectively.

We have

A = lim δ↓0 γn 2 ω ω\D δ (y) |v(x) -v(y)| 2 X(x) • ∇x -1 |x -y| n+1 dx dy =: lim δ↓0 A δ .
Since X is compactly supported in ω, we can write for δ > 0 small enough

B = γn 2 ω c ω\D δ (y) |v(x) -v(y)| 2 X(x) • ∇x -1 |x -y| n+1 dx dy .
Given y ∈ R n \ ∂ω and δ > 0 small enough, we integrate by parts to find that

ω\D δ (y) |v(x) -v(y)| 2 X(x) • ∇x -1 |x -y| n+1 dx = ω\D δ (y) |v(x) -v(y)| 2 |x -y| n+1 div X(x) dx + 2 ω\D δ (y) v(x) -v(y) |x -y| n+1 • X(x) • ∇v(x) dx + χω(y) δ n+1 ∂D δ (y) |v(x) -v(y)| 2 (x -y) • X(x) |x -y| dH n-1 x ,
where χω denotes the characteristic function of the set ω. Therefore, for δ small we have

A δ + B = γn 2 R n ω\D δ (y) |v(x) -v(y)| 2 |x -y| n+1 div X(x) dx dy + γn R n ω\D δ (y) v(x) -v(y) |x -y| n+1 • X(x) • ∇v(x) dx dy + γn 2δ n+1 ω ∂D δ (y) |v(x) -v(y)| 2 (x -y) • X(x) |x -y| dH n-1 x dy =: I δ + II δ + III δ . (4.12)
Obviously,

lim δ↓0 I δ = γn 2 ω×R n |v(x) -v(y)| 2 |x -y| n+1 div X(x) dxdy .
Then we rewrite

II δ = γn ω R n \D δ (x) v(x) -v(y) |x -y| n+1 dy • X(x) • ∇v(x) dx , so that lim δ↓0 II δ = ω (-∆) 1 2 v(x) • X(x) • ∇v(x) dx .
Finally, expanding v around y, we easily get that

lim δ↓0 III δ = γn 2 ω S n-1 |σ • ∇v(y)| 2 X(y) • σ dH n-1 σ .
We claim that the integral above vanishes, i.e., lim δ III δ = 0. Indeed, write for y ∈ ω fixed,

S n-1 |σ • ∇v(y)| 2 X(y) • σ dH n-1 σ = m i=1 S n-1 | -→ a i • σ| 2 -→ b • σ dH n-1 σ ,
where we have set -→ a i := ∇vi(y) and -→ b := X(y). Given i ∈ {1, . . . , m}, we can assume that -→ a i and -→ b belongs to R 2 × {(0, . . . , 0)} by invariance under rotation. Then,

S n-1 | -→ a i • σ| 2 -→ b • σ dH n-1 σ = Cn S 1 | -→ a i • σ| 2 -→ b • σ dH 1 σ ,
for a dimensional constant Cn. Now observe that the function in the right hand side is a homogeneous polynomial in σ of degree 3. Hence its integral over S 1 vanishes, and the claim is proved.

Gathering (4.11) with (4.12), we have thus shown that

d dt E v • φt, ω t=0 = ω (-∆) 1 2 v(x) • X(x) • ∇v(x) dx . (4.13)
Finally, since v(x) ∈ S n-1 for x ∈ ω, we have ∂jv(x) ∈ T v(x) S n-1 for x ∈ ω and j = 1, . . . , n. Then the Euler-Lagrange equation (4.3) yields (-∆)

1 2 v • ∂jv = 0 in ω for j = 1, . . . , n.
Therefore the function under the integral in (4.13) vanishes in ω, which shows that v is stationary.

Going back to our discussion on the regularity of harmonic maps, we now mention that there is only one case where full (interior) regularity holds for general critical points. It is the case where the starting dimension equals two (the conformal dimension), and this is a well known result due to H ÉLEIN [START_REF] Élein | Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne[END_REF].

In higher dimensions, the (optimal) partial regularity result for minimizing harmonic maps has been obtained in the pioneering work of SCHOEN & UHLENBECK [START_REF] Schoen | UHLENBECK : A regularity theory for harmonic maps[END_REF]. It has then been extended to stationary harmonic maps by BETHUEL [START_REF] Bethuel | On the singular set of stationary harmonic maps[END_REF] (see also [START_REF] Evans | Partial regularity for stationary harmonic maps into spheres[END_REF]). All this results have an analogue in the context of harmonic maps with partially free boundary, where the new issue is of course to determine the partial regularity at the relative interior of the free boundary ∂ 0 Ω. Higher order regularity starting from continuous solutions has been studied and proved by GULLIVER & JOST [START_REF] Gulliver | Harmonic maps which solve a free-boundary problem[END_REF], see also [START_REF] Baldes | Harmonic mappings with a partially free boundary[END_REF][START_REF] Hamilton | Harmonic maps of manifolds with boundary[END_REF]. The minimizing case has been handled independently by HARDT & LIN [START_REF] Hardt | Partially constrained boundary conditions with energy minimizing mappings[END_REF] and DUZAAR & STEFFEN [START_REF] Duzaar | A partial regularity theorem for harmonic maps at a free boundary[END_REF][START_REF] Duzaar | An optimal estimate for the singular set of a harmonic map in the free boundary[END_REF], while full regularity in the conformal dimension and partial regularity under the stationarity condition has been more recently proved by SCHEVEN [66, Theorem 2.2, Theorem 4.1]. All these results on partial regularity essentially deals with an estimate on the Hausdorff dimension of the so-called singular set. The singular set sing(u) of a map u is usually defined as the complement of the largest (relative) open set on which u is continuous. It is therefore a relatively closed subset of the domain where u is defined. For a harmonic map with partially free boundary it is then a relatively closed subset of Ω ∪ ∂ 0 Ω, and in case of a (S m-1 , ∂ 0 Ω)-boundary harmonic map it is a relatively closed subset of ∂ 0 Ω (since regularity holds in Ω).

In view of Remark 4.5, we can obtain a partial regularity theory for bounded S m-1 -boundary harmonic maps from the known results mentioned above about harmonic maps with partially free boundary. In the statements below, dim H denotes the Hausdorff dimension. Theorem 4.18 ([29, 30, 41, 43, 66]).

Let Ω ⊆ R n+1 + be an admissible bounded open set. If u ∈ H 1 (Ω; R m ) ∩ L ∞ (Ω) is a weak (S m-1 , ∂ 0 Ω)-boundary harmonic map in Ω, then u ∈ C ∞ (Ω ∪ ∂ 0 Ω) \ sing(u) . Moreover, (i) if n = 1, then sing(u) = ∅; (ii) if n 2 and u is stationary, then H n-1 (sing(u)) = 0; (iii) if u is minimizing, then dim H (sing(u)) n -2 for n 3, and sing(u) is discrete for n = 2.
Proof. As indicated in Remark 4.5, the sphere S m-1 can be considered as a submanifold of a flat torus M := R m /rZ m with r > 2 u L ∞ (Ω) . Then u is a weakly harmonic map in Ω into the compact (boundaryless) manifold M satisfying the partially free boundary condition u(∂ 0 Ω) ⊆ S m-1 . Items (i) and (ii) follow from [66, Theorem 2.2, Theorem 4.1] (note that for n = 1, stationarity is not required since the conclusion of [START_REF] Scheven | Partial regularity for stationary harmonic maps at a free boundary[END_REF]Corollary 1.3] trivially holds in this case), while item (iii) is proved in [START_REF] Duzaar | A partial regularity theorem for harmonic maps at a free boundary[END_REF][START_REF] Duzaar | An optimal estimate for the singular set of a harmonic map in the free boundary[END_REF][START_REF] Hardt | Partially constrained boundary conditions with energy minimizing mappings[END_REF]. Finally, the higher order regularity away from the singular set is obtained from [START_REF] Gulliver | Harmonic maps which solve a free-boundary problem[END_REF]Section 4].

Thanks to Theorem 4.18, we can now deduce the partial regularity theory for bounded 1/2-harmonic In the case where v is minimizing, the argument follows the same lines using Proposition 4.9.

maps into S m-1 . Proof of Theorem 1.2. Let v ∈ H 1/2 (ω; R m ) ∩ L ∞ (R n ) be
Remark 4.19. In recent papers, DA LIO & RIVI ÈRE have obtained a direct proof for the full regularity of weak 1/2-harmonic maps in the case ω = R, first for a sphere target in [START_REF] Lio | RIVI ÈRE : Three-term commutator estimates and the regularity of 1 2 -harmonic maps into spheres[END_REF], and then for more general target manifolds in [START_REF] Lio | RIVI ÈRE : Sub-criticality of non-local Schr ödinger systems with antisymmetric potential and applications to half harmonic maps[END_REF]. A main point of their proof is to rewrite the Euler-Lagrange equation (4.3) in a form that exhibits a special algebraic structure which allows to use some compensated compactness arguments (somehow in the spirit of [START_REF] Élein | Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne[END_REF]). Even if we have deduced here the regularity theory for 1/2harmonic maps from the existing literature, we believe that the analysis in [START_REF] Lio | RIVI ÈRE : Three-term commutator estimates and the regularity of 1 2 -harmonic maps into spheres[END_REF][START_REF] Lio | RIVI ÈRE : Sub-criticality of non-local Schr ödinger systems with antisymmetric potential and applications to half harmonic maps[END_REF] could be of first importance for further investigations.

Remark 4.20. The relation between 1/2-harmonic maps and harmonic maps with partially free boundary was first noticed by MOSER [START_REF] Moser | Intrinsic semiharmonic maps[END_REF] for a different, non-explicit, integro-differential operator (which coincides with (-∆)

1
2 only in the case ω = R n ). Roughly speaking, the Dirichlet-to-Neumann operator considered in [START_REF] Moser | Intrinsic semiharmonic maps[END_REF] is associated to a "homogeneous Neumann type condition" on the exterior of ω, while we are dealing with Dirichlet exterior conditions. In this Neumann framework and under an additional technical condition, [START_REF] Moser | Intrinsic semiharmonic maps[END_REF] provides a similar partial regularity result from the regularity theory for harmonic maps with partially free boundary. Remark 4.21. In the context of harmonic map with partially free boundary, DUZAAR & GROTOWSKI [START_REF] Duzaar | A mixed boundary value problem for energy minimizing harmonic maps[END_REF] have studied regularity for the mixed boundary value problem which consists in prescribing a Dirichlet data on the remaining part ∂ + Ω of the boundary of Ω. Under suitable smoothness conditions on the Dirichlet data, ∂ + Ω and ∂ 0 Ω, and assuming that ∂ + Ω and ∂ 0 Ω meet orthogonaly, they have proved that minimizing solutions are continuous accross the edge ∂ + Ω ∩ ∂ 0 Ω. Unfortunately, this result cannot be used for 1/2-harmonic maps with exterior Dirichlet condition (even for minimizing maps) since the two parts of the boundary would have to meet tangentially. Up to our knowledge, there is no regularity result at the boundary ∂ω for the 1/2-harmonic map problem with a prescribed exterior Dirichlet condition.

Remark 4.22 (Energy Monotonicity).

For usual harmonic maps, it is well known that the stationarity property yields a monotonic control on the energy on balls with respect to the radius. For a stationary boundary harmonic map u ∈ H 1 (Ω; R m ), the relation (4.8) leads to the monotonicity of the function

r ∈ 0, dist(x0, ∂ + Ω) → 1 2r n-1 B + r (x 0 ) |∇u| 2 dx ,
for every x0 ∈ ∂ 0 Ω.

Remark 4.23 (Liouville property).

As a consequence of the energy monotonicity above, if n 2 and

u ∈ Ḣ1 (R n+1 + ; R m
) is an entire stationary boundary harmonic map (i.e., a stationary boundary harmonic map in B + r for every r > 0), then u is constant. As a byproduct, if n 2 and v ∈ Ḣ1/2 (R n ; R m ) is an entire stationary 1/2-harmonic map (i.e., a stationary 1/2-harmonic map in Dr for every r > 0), then v is constant.

Remark 4.24 (General target manifold). The definition of 1/2-harmonic maps extends to a more general target submanifold N ⊆ R m in the following way. Assuming that N is smooth, compact, and without boundary, the nearest point retraction πN on N is smooth in a small tubular neighborhood of N . We

then say that a map v ∈ H 1/2 (ω; R m ) satisfying v(x) ∈ N for a.e. x ∈ ω, is weakly 1/2-harmonic into N in ω if d dt E (πN (v + tϕ), ω) t=0 = 0 for all ϕ ∈ H 1/2 00 (ω; R m ) ∩ L ∞ (ω) compactly supported in ω.
The associated Euler-Lagrange equation reads (-∆)

1 2 v ⊥ TvN in H -1/2 00 (ω) ,
i.e., (-∆)

1 2 v, ϕ ω = 0 for all ϕ ∈ H 1/2 00 (ω; R m ) satisfying ϕ(x) ∈ T v(x)
N for a.e. x ∈ ω. Concerning the definition of (N , ∂ 0 Ω)-valued boundary harmonic maps, we simply reproduce Definition 4.4 replacing S m-1 by the manifold N . The notions of minimality and stationarity for both boundary harmonic and 1/2-harmonic maps remain unchanged. With these definitions, all the results of this subsection do hold. In particular, we have the same partial regularity theory for (N , ∂ 0 Ω)-boundary harmonic maps as stated in Theorem 4.18 by the general regularity results in [START_REF] Hardt | Partially constrained boundary conditions with energy minimizing mappings[END_REF][START_REF] Duzaar | A partial regularity theorem for harmonic maps at a free boundary[END_REF][START_REF] Duzaar | An optimal estimate for the singular set of a harmonic map in the free boundary[END_REF][START_REF] Scheven | Partial regularity for stationary harmonic maps at a free boundary[END_REF]. As a consequence, we have the same partial regularity theory for 1/2-harmonic maps into N as stated in Theorem 1.2. 4.2. 1/2-harmonic lines into S 1 . We provide in this subsection some explicit examples of 1/2-harmonic maps into S 1 enlightening the geometric flavour of the 1/2-harmonic map equation, as well as its analogy with usual harmonic maps. We start with an explicit representation formula for all entire 1/2-harmonic maps from the line R into S 1 with finite energy. In the sequel, we identify R 2 with the complex plane C

writing z = x1 + ix2.
The open unit disc of C is denoted by D, and ∂D is identified with S 1 . Theorem 4.25. Let v ∈ Ḣ1/2 (R; S 1 ) be a non-constant entire 1/2-harmonic map into S 1 , and let v e be its harmonic extension to R 2 + given by (2.9). There exist some d ∈ N, θ ∈ R, {λ k } d k=1 ⊆ (0, ∞), and {a k } d k=1 ⊆ R such that v e (z) or its complex conjugate equals

e iθ d k=1 λ k (z -a k ) -i λ k (z -a k ) + i . (4.14)
In addition,

E(v, R) = [v] 2 H 1/2 (R) = 1 2 R 2 + |∇v e | 2 dz = πd . (4.15)
Remark 4.26. Theorem 4.25 shows that the map

x ∈ R → x 2 -1 x 2 + 1 , -2x x 2 + 1 ∈ S 1
is a 1/2-harmonic map. Indeed, it corresponds to the case θ = 0, d = 1, λ1 = 1, and a1 = 0 in (4.14).

Note that this map is precisely the inverse of the stereographic projection with pole at (1, 0) going from the circle into the line.

The proof of Theorem 4.25 is based on an observation due to MIRONESCU & PISANTE [START_REF] Mironescu | PISANTE : A variational problem with lack of compactness for H 1/2 (S 1 ; S 1 ) maps of prescribed degree[END_REF] together with the following preliminary lemma.

Lemma 4.27. Let v ∈ Ḣ1/2 (R; R 2 ) be a nontrivial entire 1/2-harmonic map into S 1 , and let v e be its harmonic extension to R 2 + given by (2.9). Then v e is either a conformal or an anti-conformal transformation of R It is well-known that v e is conformal or anti-conformal if and only if H ≡ 0. We thus have to prove that H vanishes identically.

First notice that H is holomorphic since v e is harmonic. Since v(x) ∈ S 1 we find that ∂1v e (x) ∈ T v(x) S 1 for every x ∈ R. By the boundary equation in (4.6), it implies that ∂1v e • ∂2v e vanishes on R. Setting g to be the imaginary part of H, the function g is harmonic in R 2 + by holomorphicity of H. Since g vanishes on R, we can extend g to the whole plane by odd reflection (i.e., setting g(z) := -g(z) if Im(z) < 0), and the resulting function (still denoted by g) is harmonic in R 2 . On the other hand, v e ∈ Ḣ1 (R 2 + ; R 2 ) by Lemma 2.6. Hence g ∈ L 1 (R 2 ), which leads to g ≡ 0 by the mean value property of harmonic functions.

Therefore H takes real values, and it must be constant by holomorphicity. We then conclude that H ≡ 0 since H ∈ L 1 (R

C(z) := z -i z + i .
It is well known that C is a conformal map which is one-to-one from R 2 + into D, and one-to-one from R into S 1 \ {(1, 0)}. Setting

w := v e • C -1 ,
we find that w is conformal in D, and thus holomorphic in D. Moreover, by conformality of C, we have w ∈ H 1 (D; C), and more precisely

D |∇w| 2 dz = R 2 + |∇v e | 2 dz . (4.16)
In particular g

:= w |S 1 ∈ H 1/2 (S 1 ; C). But g is smooth on S 1 \ {(0, 1)} and g(z) = v • C -1 (z) for every z ∈ S 1 \ {(0, 1)}, so that g ∈ H 1/2 (S 1 ; S 1
). Now, w being holomorphic, it is also harmonic in D. It is therefore the harmonic extension to the disc of the map g. By a result of BREZIS & NIRENBERG [START_REF] Brezis | NIRENBERG : Degree theory and BMO. I. Compact manifolds without boundaries[END_REF][START_REF] Brezis | NIRENBERG : Degree theory and BMO. II. Compact manifolds with boundaries[END_REF],

it implies that |w(z)| → 1 uniformly as |z| → 1. For a nonconstant holomorphic map this latter property implies that it is a (finite) Blaschke product, see [START_REF] Mironescu | PISANTE : A variational problem with lack of compactness for H 1/2 (S 1 ; S 1 ) maps of prescribed degree[END_REF] and the references therein. In other words, we can find a positive d ∈ N, θ ∈ R, and α1, . . . , α d ∈ D such that

w(z) = e i θ d k=1 z -α k 1 -α k z . (4.17)
Then (4.14) follows from the previous formula with

λ k := |1 -α k | 2 1 -|α k | 2 , a k := - 2Im(α k ) |1 -α k | 2 , θ := θ + d k=1 θ k where θ k ∈ R and e iθ k = 1 -α k 1 -α k .
On the other hand, it is easy to check from (4.17) (see e.g. [START_REF] Mironescu | PISANTE : A variational problem with lack of compactness for H 1/2 (S 1 ; S 1 ) maps of prescribed degree[END_REF]) that

D |∇w| 2 dz =
E(g, S 1 ) := γ1 4 S 1 ×S 1 |g(x) -g(y)| 2 |x -y| 2 dxdy .
It is well known that for every g ∈ H 1/2 (S 1 ; C),

E(g, S 1 ) = 1 2 D |∇wg| 2 dx ,
where wg ∈ H 1 (D; C) denotes the harmonic extension of g to the disc. In view of identity (4.16) and Lemma 2.6, we have

E(g, S 1 ) = E(g • C, R) for all g ∈ H 1/2 (S 1 ; C) ,
where C is the Caley transform. Defining (weak) 1/2-harmonic maps from S 1 into S 1 as critical points of E(•, S 1 ) with respect to perturbations in the target (as in Definition 4.1, see (4.19) below), we deduce that g is such a 1/2-harmonic map if and only if g • C is a (finite energy) 1/2-harmonic map from R into S 1 . In terms of the harmonic extension wg, the Euler-Lagrange equation for 1/2-harmonic maps from S 1 into

S 1 writes ∂wg ∂ν ∧ g = 0 on S 1 , (4.18)
where ∂ ∂ν is the exterior normal derivative on ∂D ≃ S 1 . As a consequence of Theorem 4.25, we find that g is a 1/2-harmonic map from S 1 into S 1 if and only if g is the restriction to S 1 of a (finite) Blaschke product (i.e., g is of the form (4.17)). This fact fact has also been discovered independently by BERLYAND, MIRONESCU, RYBALKO, & SANDIER [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semistiff boundary conditions: Existence and bubbling[END_REF] (with essentially the same proof), and by DA LIO & RIVI ÈRE [START_REF] Lio | RIVI ÈRE : Fractional harmonic maps and free boundaries problems[END_REF].

Note that the explicit formula (4.17) shows that the energy is quantized by the topological degree, i.e., E(g, S 1 ) = π|deg(g)| , for any 1/2-harmonic map g from S 1 into S 1 . By the result of MIRONESCU & PISANTE [START_REF] Mironescu | PISANTE : A variational problem with lack of compactness for H 1/2 (S 1 ; S 1 ) maps of prescribed degree[END_REF], it shows in particular that every 1/2-harmonic map from S 1 into S 1 is minimizing in its own homotopy class. Here the homotopy classes are classified by the topological degree for H 1/2 (S 1 ; S 1 )-maps as defined in [START_REF] Brezis | NIRENBERG : Degree theory and BMO. I. Compact manifolds without boundaries[END_REF][START_REF] Brezis | NIRENBERG : Degree theory and BMO. II. Compact manifolds with boundaries[END_REF]. These properties of 1/2-harmonic circles into S 1 are in clear analogy with the theory of usual harmonic maps where it is well known that harmonic 2-spheres into S 2 are minimizing in their own homotopy class and have an energy quantized by the degree.

With the help of Theorem 4.25 we can now give an explicit representation of 0-homogeneous maps wich are weakly 1/2-harmonic from the unit disc of R 2 into S 1 . Those maps provide examples of singular weak 1/2-harmonic maps. In particular, it shows that x |x| is a weak 1/2-harmonic map into S 1 . It would be interesting to determine which maps are minimizing or stationary, and then to compare the result with [START_REF] Brezis | Harmonic maps with defects[END_REF] (which deals with 0-homogeneous harmonic maps from the unit ball in R 3 into S 2 ). Proposition 4.30.

For n = 2, let v ∈ H 1/2 (D1; R 2 ) be a 0-homogeneous map in all of R 2 such that |v| = 1 a.e. in R 2 . Then v is a weak 1/2-harmonic map into S 1 in D1 if and only if v(x) = g( x |x|
) for a 1/2-harmonic circle g into S 1 in the sense of Remark 4.29.

Proof. Step 1. Let us consider an arbitrary map v ∈ H 1/2 (D1; R 2 ) which is 0-homogeneous. By a rescaling argument we first deduce that v ∈ H 1/2 loc (R 2 ; R 2 ), and in view of Remark 2.8, v e ∈ H 1 loc (R

3 + ; R 2 ).
Since v e clearly inherits the 0-homogeneity of v, we infer that the map v e := v e

|S 2 + belongs to H 1 (S 2 + ; R 2 ) where S 2 + := S 2 ∩ R 3 + . Therefore g := v e |∂S 2 + ∈ H 1/2 (S 1 ; R 2 ) identifying ∂S 2 + with S 1 . Obviously, we have v = g( x |x|
) and v e (x) = v e ( x |x| ). From the harmonicity of v e , we deduce that v e is the unique solution of

   ∆ S 2 v e = 0 on S 2 + , v e = g on ∂S 2 + ,
where ∆ S 2 denotes the Laplace-Beltrami operator on S 2 .

We now introduce the one-to-one conformal mapping S : D → S 2 + defined by

S(x) := 2x1 |x| 2 + 1 , 2x2 |x| 2 + 1 , 1 -|x| 2 |x| 2 + 1 .
Note that S is the identity on S 1 . By conformality of S the map wg := v e • S ∈ H 1 (D : R 2 ) is harmonic in D, and wg = g on S 1 in the trace sense. It is therefore the harmonic extension of g to the disc D, and we infer from Remark 4.29 and the conformality of S that

1 2 S 2 + |∇T v e | 2 dH 2 = 1 2 D |∇wg| 2 dx = E(g, S 1 ) ,
where ∇T denotes the tangential gradient.

Step 2. Assume that v ∈ H 1/2 (D1; R 2 ) is a 0-homogeneous 1/2-harmonic map into S 1 in D1. We claim that the associated map g as defined in Step 1 is a 1/2-harmonic circle into S 1 , i.e.,

d dt E g + tϕ |g + tϕ| , S 1 t=0 = 0 (4.19) for all ϕ ∈ H 1/2 (S 1 ; R 2 ) ∩ L ∞ (S 1
). Arguing as in the proof of Proposition 4.2, we infer that it is enough to prove (4. [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF]) for all ϕ ∈ H 1/2 (S 1 ; R 2 ) ∩ L ∞ (S 1 ) such that ϕ • g = 0 a.e. on S 1 . For such a function ϕ, we have

d dt E g + tϕ |g + tϕ| , S 1 t=0 = d dt E g + tϕ , S 1 t=0 = d dt 1 2 D |∇(wg + twϕ)| 2 dx t=0 ,
where wϕ is the harmonic extension of ϕ to the disc D (notice that wϕ ∈ H Let us now consider a function η ∈ C 1 ((0, +∞)) with compact support in the interval (0, 1) satisfying andΦ • v e = 0 H 2 -a.e. on D1. Since v is assumed to be 1/2-harmonic, we deduce from Proposition 4.6 that

1 (D; R 2 ) ∩ L ∞ (D)). Setting Φ := wϕ • S -1 ∈ H 1 (S 2 + ; R 2 ) ∩ L ∞ (S 2
1 0 η(r)r 2 dr = 1. Define for x ∈ R 3 + , Φ := η(|x|)Φ( x |x| ). Then Φ ∈ H 1 (B + 1 ; R 2 ) ∩ L ∞ (B + 1 ), Φ has compact support in B + 1 ∪ D1,
B + 1 ∇v e • ∇ Φ dx = 0 . (4.21)
On the other hand, by homogeneity of v e , we have Step 3. Consider a 1/2-harmonic circle g : S 1 → S 1 , and set v := g( x |x| ). From the smoothness of g, we infer that v ∈ H From Lemma 2.9 we then infer that (-∆)

B + 1 ∇v e • ∇ Φ dx = 1 0 η(r)r 2 dr S 2 + ∇T v e • ∇T Φ dH 2 = S 2 + ∇T v e • ∇T Φ dH 2 . ( 4
1/2 loc (R 2 ; R 2 ) ∩ L ∞ (R 2 ) ⊆ H 1/2 (D1; R 2 ),
1 2 v ∧ v = 0 in R 2 \ {0}
, and thus v satisfies (4.2) in D1 \ Dρ for every 0 < ρ < 1. Since v belongs to H 1/2 (D1; R 2 ), a standard capacity argument shows that (4.2) actually holds in D1. Hence v is a weak 1/2-harmonic map in D1 by Proposition 4.2.

SMALL ENERGY ESTIMATE FOR GINZBURG-LANDAU BOUNDARY REACTIONS

In this section we perform a preliminary analysis on the asymptotic, as ε → 0, of critical points of the Ginzburg-Landau boundary energy Eε defined in (3.5). The first step we make here is to prove an epsilonregularity type of estimate which allows to control the regularity of solutions under the assumption that the energy, suitably renormalized, is small. This is the purpose of the following theorem which is a corner stone of the present paper. The entire section is devoted to its proof. Theorem 5.1. Let ε > 0 and R > 0 such that ε R. There exist constants η0 > 0 and C0 > 0 independent of ε and R such that for each map uε

∈ C 2 (B + R ∪ DR; R m ) satisfying |uε| 1 and solving      ∆uε = 0 in B + R , ∂uε ∂ν = 1 ε (1 -|uε| 2 )uε on DR , (5.1
)

the condition Eε(uε, B + R ) η0R n-1 implies sup B + R/4 |∇uε| 2 + sup D R/4 (1 -|uε| 2 ) 2 ε 2 C0 R 2 η0 . (5.
2)

The proof of Theorem 5.1 is divided into three main parts according to the following subsections.

For the local Ginzburg-Landau equation, the argument leading to the analoguous estimate is essentially based on the so-called Bochner inequality satified by the energy density, see e.g. [START_REF] Chen | Existence and partial regularity results for the heat flow for harmonic maps[END_REF]. In our case, such inequality does not seem to be avaible, and to prove (5.2) we better use a compactness approach in the spirit of [START_REF] Wang | Limits of solutions to the generalized Ginzburg-Landau functional[END_REF].

5.1. Energy monotonicity and clearing-out lemma. As usual in Ginzburg-Landau or harmonic map problems, the first main ingredient to derive regularity estimates is a useful monotonicity formula. This is the purpose of the following lemma.

Lemma 5.2 (Monotonicity formula). Let R > 0 and uε ∈ C 2 (B + R ∪ DR; R m ) satisfying (5.1). Then, for every x0 ∈ DR and every 0 < ρ < r < dist(x0, ∂DR),

1 r n-1 Eε uε, B + r (x0) - 1 ρ n-1 Eε uε, B + ρ (x0) = r ρ 1 t n-1 ∂ + B t (x 0 ) ∂uε ∂ν 2 dH n dt + 1 4ε r ρ 1 t n D t (x 0 ) (1 -|uε| 2 ) 2 dH n dt .
Proof.

Step 1. Consider a smooth vector field X = (X1, . . . , Xn+1) : R n+1 → R n+1 compactly supported in BR and satisfying Xn+1 = 0 on R n . For t ∈ R small, the map Φt(x) := x -tX(x) defines a smooth diffeomorphism from BR into BR satisfying Φt(DR) ⊆ DR, and

Φt(B + R ) ⊆ B + R . Setting ut := uε • Φt, standard computations (see e.g. [71, Chapter 2.2]) yield d dt 1 2 B + R ∇ut 2 dx t=0 = 1 2 B + R |∇uε| 2 divX -2 n+1 i,j=1 (∂iuε • ∂juε)∂jXi dx , (5.3)
and d dt

1 4ε D R (1 -|ut| 2 ) 2 dH n t=0 = 1 4ε D R (1 -|uε| 2 ) 2 div R n X dH n .
Using (5.1) we integrate by parts in (5.3) to find

d dt 1 2 B + R ∇ut 2 dx t=0 = - 1 4ε D R (1 -|uε| 2 ) 2 div R n X dH n , whence d dt Eε(ut, B + R ) t=0 = 0 . (5.4)
Step 2. Let x0 ∈ DR, and 0 < r < dist(x0, ∂DR). Without loss of generality we may assume that x0 = 0 (to simplify the notation). Let η ∈ C ∞ (R; [0, 1]) be an even function such that η(t) = 0 for |t| r. Using the vector field X(x) = η(|x|)x in Step 1, we find n -1 2

B + R |∇uε| 2 η(|x|) dx + 1 2 B + R |∇uε| 2 η ′ (|x|)|x| dx - B + R ∂uε ∂|x| 2 η ′ (|x|)|x| dx + n 4ε D R (1 -|uε| 2 ) 2 η(|x|) dH n + 1 4ε D R (1 -|uε| 2 ) 2 η ′ (|x|)|x| dH n = 0 . (5.5)
Then, given an arbitrary t ∈ (0, r), we consider a sequence {η k } of functions as above such that η k converges weakly* in BV as k → ∞ to the characteristic function of the interval [-t, t]. Using η k as a test function in (5.5) and letting k → ∞ leads to n -1 2

B + t |∇uε| 2 dx - t 2 ∂ + B t |∇uε| 2 dH n + t ∂ + B t ∂uε ∂ν 2 dH n + n 4ε D t (1 -|uε| 2 ) 2 dH n - t 4ε ∂D t (1 -|uε| 2 ) 2 dH n-1 = 0 .
Dividing by t n and rearranging terms, we end up with

d dt 1 t n-1 Eε(uε, B + t ) = 1 t n-1 ∂ + B t ∂uε ∂ν 2 dH n + 1 4εt n D t (1 -|uε| 2 ) 2 dH n .
Integrating this equality between ρ > 0 and r yields to the announced result.

Remark 5.3 (Liouville property).

As a consequence of Lemma 5.2 and the stationarity equation (5.4), any entire (smooth) solution uε : R n+1 + → R m of the Ginzburg-Landau boundary equation (i.e., uε solves (5.1) for every radius R > 0) satisfying Eε(uε, R n+1 + ) < ∞, has to be constant. In view of (3.3) and Lemma 2.6, the same Liouville property holds for entire solutions of the fractional Ginzburg-Landau equation having finite energy. More precisely, if vε : R n → R m is a (smooth) solution of (3.1) in R n satisfying Eε(vε, R n ) < ∞, then vε is constant. The argument goes as follows.

Proof. If n 2, Lemma 5.2 implies Eε(uε, B + ρ ) C(r/ρ) 1-n for every 0 < ρ r, and the constancy of uε follows letting r → ∞. The case n = 1 is slightly more involved, and relies on (5.4). For n = 1 we

consider a nonnegative cut-off function ζ ∈ C ∞ (R 2 ) such that ζ = 1 on B1 and ζ = 0 on R 2 \ B2. Setting ζ k (x) := ζ(x/k), we use the vector field X(x) = ζ k (x)x in (5.4) to obtain 1 2 B + 2k \B + k |∇uε| 2 (x • ∇ζ k ) -2 2 i,j=1 (∂iuε • ∂juε)xi∂jζ k dx + 1 4ε D 2k (1 -|uε| 2 ) 2 ζ k dH 1 + 1 4ε D 2k \D k (1 -|uε| 2 ) 2 x1∂1ζ k dH 1 = 0 . Since |∇ζ k | C/k, this identity yields 1 ε D k (1 -|uε| 2 ) 2 dH 1 CEε(uε, B + 2k \ B + k ) -→ k→∞ 0 ,
and we deduce that |uε| ≡ 1. We then infer that uε is a bounded solution of

     ∆uε = 0 in R n+1 + , ∂uε ∂ν = 0 on R n ,
and the constancy of uε follows from Liouville Theorem on bounded entire harmonic functions through the usual reflection argument across ∂R n+1

+ ≃ R n .

The second key ingredient in Ginzburg-Landau problems is the so-called clearing-out property which essentially asserts that a solution must take values in a small neighborhood of the potential-well whenever the energy is small enough. This is precisely the object of the following lemma. The proof uses some ingredients from [66, Lemma 3.1] suitably modified to fit the Ginzburg-Landau setting.

Lemma 5.4 (Clearing-out). Let ε ∈ (0, 1]. There exists a constant η1 > 0 independent of ε such that for each

map uε ∈ C 2 (B + 1 ∪ D1; R m )
(x k )| < 1/2.
Step 2. We now consider the case ε < 1/2. Let us fix an arbitrary point x0 = (x ′ 0 , (x0)n+1) in B + 1/2 , and set x1 := (x ′ 0 , 0) ∈ D 1/2 . By the Monotonicity Formula in Lemma 5.2 we have

1 r n-1 Eε(uε, B + r (x1)) (1 -|x1|) 1-n Eε(uε, B + 1-|x 1 | (x1)) 2 n-1 Eε(uε, B + 1 ) 2 n-1 η1 (5.7)
for every 0 < r < 1 -|x1|. We set

R := 1 3 |x1 -x0| .
Case 1) Let us first assume that R ε/7. Since ε < 1/2 < 1 -|x1| we have

1 ε N -1 Eε(uε, B + ε (x1)) 2 n-1 η1 .
Next we define for y ∈ B + 1 ∪ D1, vε(x) := uε(x1 + εy) ,

so that        ∆vε = 0 in B + 1 , ∂vε ∂ν = (1 -|vε| 2 )vε on D1 , and
E1(vε, B + 1 ) = 1 ε n-1 Eε(w, B + ε (x1)) 2 n-1 η1 .
Choosing η1 such that uε(y) dy .

We consider for x ∈ B2R(x0

) ⊆ B + 5R (x1), vε (x)

We estimate

|vε(x0)| 2 = B 2R (x 0 ) Gx 0 ∆(|vε| 2 ζ 2 ) dx = - B 2R (x 0 ) ∇Gx 0 • ∇(|vε| 2 ζ 2 ) dx = -2 B 3R/2 (x 0 ) ζ 2 n j=1 ∂jGx 0 ∂jvε • vε dx -2 A R (x 0 ) |vε| 2 ζ ∇Gx 0 • ∇ζ dx vε L ∞ (B 3R/2 (x 0 )) B 3R/2 (x 0 ) |∇Gx 0 ||∇uε| dx + C R n+1 A R (x 0 ) |vε| 2 dx , (5.9)
where AR(x0) := B2R(x0) \ BR(x0).

By harmonicity of uε, ∇uε is harmonic in

B + 1 . Since B R/4 (x) ⊆ B + 5R (x1) for all x ∈ B 3R/2 (x0), we deduce that ∇uε(x) = 1 |B R/4 (x)| B R/4 (x)
∇uε(y) dy for all x ∈ B 3R/2 (x0) .

From Jensen's inequality and (5.7), we then infer that

|∇uε(x)| 2 1 |B R/4 (x)| B R/4 (x) |∇uε(y)| 2 dy C R n+1 Eε(uε, B + 5R (x1)) C R 2 η1
for all x ∈ B 3R/2 (x0). Consequently,

|∇uε(x)| C R √ η1 for all x ∈ B 3R/2 (x0) .
(5.10) Using (5.8) together with (5.10), we obtain that

B 3R/2 (x 0 ) |∇Gx 0 ||∇uε| dx C √ η1 .
(5.11) Estimate (5.10) also yields

vε L ∞ (B 3R/2 (x 0 )) |vε(x0)| + 3R 2 ∇uε L ∞ (B 3R/2 (x 0 )) |vε(x0)| + C √ η1 .
(5.12)

Next we infer from Poincaré Inequality that

B 2R (x 0 )\B R (x 0 ) |vε| 2 dx B + 5R (x 1 ) |uε -ūε| 2 dx CR 2 B + 5R (x 1 ) |∇uε| 2 dx , so that 1 R n+1 B 2R (x 0 )\B R (x 0 ) |vε| 2 dx C R n-1 Eε(uε, B + 5R (x1)) Cη1 .
(5.13) Gathering (5.9), (5.11), (5.12), and (5.13), we conclude that

|vε(x0)| 2 C √ η1 |vε(x0)| + √ η1 1 2 |vε(x0)| 2 + Cη1 ,
and thus

|uε(x0) -ūε| C √ η1 .
In particular,

1 -|uε(x0)| 1 -|ūε| + C √ η1 . ( 5
d(ūε) 1 |B + 5R (x1)| B + 5R (x 1 ) d(uε(x)) dx + 1 |B + 5R (x1)| B + 5R (x 1 )
|uε(x) -ūε| dx .

(

Then we use Jensen and Poincaré inequalities together with (5.7), to derive

1 |B + 5R (x1)| B + 5R (x 1 ) |uε(x) -ūε| dx 2 1 |B + 5R (x1)| B + 5R (x 1 ) |uε(x) -ūε| 2 dx C R n-1 B + 5R (x 1 )
|∇uε| 2 dx Cη1 . (5.17)

Using again Jensen and Poincaré inequalities for the function d • uε, estimate (5.7), and the facts that |∇d| 1 and ε/R < 7, we obtain

1 |B + 5R (x1)| B + 5R (x 1 ) d • uε dx 2 1 |B + 5R (x1)| B + 5R (x 1 ) |d • uε| 2 dx C 1 R n D 5R (x 1 ) |d • uε| 2 dH n + 1 R n-1 B + 5R (x 1 ) |∇(d • uε)| 2 dx C R n-1 1 ε D 5R (x 1 ) (1 -|uε| 2 ) 2 dH n + B + 5R (x 1 )
|∇uε| 2 dx Cη1 .

(5.18)

Combining (5.16), (5.17 Arguing as in the proof of Lemma 5.4, we infer that 1 r n-1 Eε(uε, B + r (x)) 2 n-1 η0 for all x ∈ D 1/2 and 0 < r < 1 -|x| .

(5.20)

We are going to prove that if η0 is small enough, smallness depending only on n and m, then

|∇uε| 2 Cη0 in B + 1/4 .
In view of the Neumann boundary condition, it clearly implies the full estimate (5.2). We shall achieve it considering two different cases. In the spirit of [START_REF] Chen | Existence and partial regularity results for the heat flow for harmonic maps[END_REF], we start with the localization of an interior region where |∇uε| 2 is large.

Since uε is smooth, we can find σε ∈ (0, 1/2) such that We now distinguish two cases.

(1/2 -σε) 2 sup B + σε |∇uε| 2 = max 0 σ 1/2 (1/2 -σ) 2 sup B + σ |∇uε| 2 , ( 5
Case 1). We first assume that Rε > ρε/7. Since ∇uε is harmonic in B + 1 , we have

∇uε(xε) = 1 |BR ε (xε)| B Rε (xε)
∇uε dx .

Then we deduce from the harmonicity of uε, Jensen's inequality, (5.20), and (5.22) that

eε 2 |BR ε (xε)| B Rε (xε) |∇uε| 2 dx C R n+1 ε B + 5Rε (xε) |∇uε| 2 dx C R 2 ε η0 C ρ 2 ε η0 .
Here we have also used that BR ε (xε We claim that rε 1 .

) ⊆ B + 5Rε (xε) ⊆ B + 1 . Consequently, ρ 2 ε eε = 1 4 (1/2 -σε)
(5.26)

The proof of (5.26) is postponed to the next subsection, and we complete now the argument. Then, taking (5.26) for granted, (5.21) yields max Proof. We argue by contradiction assuming that there exist a sequence ε k ↓ 0, and corresponding solutions {vε k } k∈N of (5.27) satisfying (5.28), (5.29), and (5.30). To simplify the notation, we shall write v k := vε k and z k := zε k . Note that v k is smooth in B + 1 ∪ D1 by Theorem 3.2. Thanks to (5.28) we can consider the smooth functions

0 σ 1/2 (1/2 -σ) 2 sup B + σ |∇uε| 2 = (1/2 -σε) 2 sup B + σε |∇uε| 2 = 4ρ
; R m ) solving        ∆vε = 0 in B + 1 , ∂vε ∂ν = 1 ε (1 -|vε| 2 )
a k := |v k | and w k := v k |v k | .
Noticing that

|∇v k | 2 = |∇a k | 2 + a 2 k |∇w k | 2 , (5.31)
we deduce from (5.28) that

1/2 a k 1 , |∇a k | 2 , |w k | = 1 , |∇w k | 4 .
(5.32)

In addition, system (5.27) yields

     -∆a k + |∇w k | 2 a k = 0 in B + 1 , ∂a k ∂ν = 1 ε k (1 -a 2 k )a k on D1 , (5.33) and      -div(a 2 k ∇w k ) = a 2 k |∇w k | 2 w k in B + 1 , ∂w k ∂ν = 0 on D1 .
(5.34)

In view of the boundary condition in (5.34), we can extend a k and w k to B1 by even reflection across D1, i.e.,

a k (x) = a k (x ′ , -xn+1) and w k (x) = w k (x ′ , -xn+1) for x ∈ B1 \ B + 1 , (5.35)
and then derive from (5.34),

-div(a 2 k ∇w k ) = a 2 k |∇w k | 2 w k in B1 .
(5.36)

From (5.32) and the boundary condition in (5.33), we infer that a k → a * in C 0,α (B1) for every 0 < α < 1 , (5.37) for a (not relabeled) subsequence and a function a * satisfying

1 2 a * 1 , |∇a * | 2 , a * = 1 on D1 .
(5.38)

From (5.32), (5.36), and standard elliptic regularity, we can find a further subsequence (not relabeled) such that (5.40)

w k → w * in C 1,α loc (B1) ∩ C 0,α (
In view of (5.38), we infer from [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]Theorem 8.32] that

w * C 1,α (Br ) Cα,r (5.41)
for every 0 < α < 1 and every 0 < r < 1, where Cα,r denotes a constant which only depends on n, m, α, and r.

In turn, we infer from the C 1,α loc -convergence of w k and (5.33) that for every 0 < α < 1 and every 0 < r < 1.

a k → a * in C 1,α loc (B + 1 ) for every 0 < α < 1 , ( 5
Finally the convergence of a k and w k implies that v k → v * := a * w * in C 1,α loc (B + 1 ) ∩ C 0,α (B + 1 ). In addition v * ∈ C 1,α loc (B + 1 ∪ D1) satisfies for every 0 < α < 1 and every 0 < r < 1,

v * C 1,α (B + r )
Cα,r , (5.45) thanks to (5.41) and (5.44).

We now claim that v k actually converges to v * in the C 1,α -topology locally up to the boundary D1.

Then we shall see that the estimates leading to the C 1,α -convergence can be iterated to obtain a C ℓ loc (B + 1 ∪ D1)-convergence for every ℓ ∈ N. We shall complete the proof of Proposition 5.5 right after proving the two following lemmas. Lemma 5.6. We have

v k → v * in C 1,α loc (B + 1 ∪ D1) for every 0 < α < 1.
Proof. By (5.39) we have

w k → w * in C 1,α loc (B + 1 ∪D1
), and it is enough to prove that a k → a * in C 1,α loc (B + 1 ∪ D1) for every 0 < α < 1. In view of (5.42), it remains to show the desired convergence near D1. We fix x0 ∈ D1, 0 < α < 1, and 0 < r < dist(x0, ∂D1) arbitrary. Without loss of generality we may assume that x0 = 0.

We introduce the function

h k := 1 -a k ,
and we notice that (5.32) and the boundary condition in (5.33) yield 0 h k 3ε k on D1 .

(5.46)

Then we define for y ∈ D r/4 fixed,

ϑ k,y (x) := h k (x + y) -h k (x) for x ∈ B + r/2 .
(5.47)

By (5.32) the Lipschitz constant of h k is bounded by 2, and hence

ϑ k,y L ∞ (B + r/2 )
2|y| .

(5.48)

In addition ϑ k,y satisfies        -∆ϑ k,y = f k,y in B + r/2 , ε k a k (1 + a k ) ∂ϑ k,y ∂ν + ϑ k,y = ε k g k,y on D r/2 , with f k,y (x) := |∇w k (x + y)| 2 a k (x + y) -|∇w k (x)| 2 a k (x) , and
g k,y (x) := 1 a k (x)(1 + a k (x)) - 1 a k (x + y)(1 + a k (x + y)) ∂h k (x + y) ∂ν .
From (5.32), (5.39), and (5.41), we infer that

f k,y L ∞ (B + r/2 )
Cα,r|y| α , (5.49) and

g k,y L ∞ (D r/2 ) Cα,r|y| α .
(5.50)

Next we consider the unique (variational) solution

ζ k ∈ H 1 (B + r/2 ) of              -∆ζ k = 1 in B + r/2 , ζ k = 1 on ∂ + B r/2 , 4ε k 3 ∂ζ k ∂ν + ζ k = 0 on D r/2 ,
i.e., ζ k is the unique critical point of the strictly convex and coercive functional Let us now define

J k (ζ) := 1 2 B +
κ k (r, y) := max ϑ k,y L ∞ (B + r/2 ) , f k,y L ∞ (B + r/2 )
Cα,r|y| α (5.52) (recall (5.48), (5.49), and that |y| r/4), and consider the functions

H + k (x) := κ k (r, y)ζ k (x) -ϑ k,y (x) + ε k g k,y L ∞ (D r/2 ) , and
H - k (x) := κ k (r, y)ζ k (x) + ϑ k,y (x) + ε k g k,y L ∞ (D r/2 ) . By construction H ± k satisfies                  -∆H ± k 0 in B + r/2 , H ± k 0 on ∂ + B r/2 , εn a k (1 + a k ) ∂H ± k ∂ν + H ± k 0 on D r/2 .
By the maximum principle and the Hopf boundary lemma, it follows that H ± k 0 in B + r/2 (see e.g. the proof of Lemma 9.1). Therefore,

ϑ k,y (x) κ k (r, y)ζ k (x) + ε k g k,y L ∞ (D r/2 ) .
(5.53)

Gathering (5.50), (5.51), and (5.52), together this last estimate, we deduce that

h k (x + y) -h k (x) ε k Cα,r|y| α for every x, y ∈ D r/4 .
In view of (5.46), we have thus proved that

1 -a k ε k C 0,α (D r/4 )
Cα,r .

(5.54)

From of (5.32) we conclude that

1 ε k (1 -a 2 k )a k C 0,α (D r/4 )
Cα,r .

(5.55)

Applying Lemma 9.2 in Appendix B to equation (5.33) together with estimates (5.39), (5.41), and (5.55),

we finally deduce that

a k C 1,α (B + r/8 )
Cα,r ,

and thus a k → a * in C 1,α ′ (B + r/8
) for every 0 < α ′ < α.

Lemma 5.7. We have

v k → v * in C ℓ loc (B + 1 ∪ D1) for every ℓ ∈ N.
Proof. In view of Lemma 5.6 we can argue by induction assuming that for an integer ℓ 1, w k → w * and a k → a * in C ℓ,α loc (B + 1 ∪ D1) for every α ∈ (0, 1). We aim to prove that w k → w * and a k → a * in C ℓ+1,α loc (B + 1 ∪ D1) for every α ∈ (0, 1). We proceed with the notations used in the proof of Lemma 5.6.

Step 1: Improved convergence of {w k }. Using the even reflection across D1 as stated in (5.35), we first rewrite equation (5.36) as

-∆w k = 2 a k ∇a k • ∇w k + |∇w k | 2 w k in B1 . (5.56) (Recall that a k 1/2.) Let us now fix a multi-index β = (β1, . . . βn) ∈ N n of length |β| := i βi = ℓ -1,
and set

∂ β := ∂ β 1 1 ∂ β 2 2 . . . ∂ βn n .
Using the boundary condition ∂ν w k = 0 on D1, we notice that the term 2 a k ∇a k • ∇w k is actually continuous across D1. As a consequence, {∂ β ( 2

a k ∇a k • ∇w k + |∇w k | 2 w k )} is bounded in C 0,α loc (
B1) for every 0 < α < 1, by the induction hypothesis. By elliptic regularity, it implies that {∂ β w k } is bounded in C 2,α loc (B1) for every 0 < α < 1.

If ℓ = 1 the required improved convergence readily follows, and we may now assume that ℓ 2. In view of the arbitrariness of the multi-index β above, we have showed that

(A0) {∂ β w k } is bounded in C 0,α loc (B + 1 ∪ D1) for β ∈ N n such that |β| = ℓ + 1 and 0 < α < 1; (B0) {∂n+1∂ β w k } is bounded in C 0,α loc (B + 1 ∪ D1) for β ∈ N n such that |β| = ℓ and 0 < α < 1; (C0) {∂ 2 n+1 ∂ β w k } is bounded in C 0,α loc (B + 1 ∪ D1) for β ∈ N n such that |β| = ℓ -1 and 0 < α < 1.
Next we want to estimate the remaining derivatives of w k of order ℓ + 1. We proceed by induction assuming that for an integer 0 γ ℓ -2,

(Aγ) {∂ γ n+1 ∂ β w k } is bounded in C 0,α loc (B + 1 ∪ D1) for β ∈ N n such that γ + |β| = ℓ + 1 and 0 < α < 1; (Bγ) {∂ γ+1 n+1 ∂ β w k } is bounded in C 0,α loc (B + 1 ∪ D1) for β ∈ N n such that γ + |β| = ℓ and 0 < α < 1; (Cγ) {∂ γ+2 n+1 ∂ β w k } is bounded in C 0,α loc (B + 1 ∪ D1) for β ∈ N n such that γ + |β| = ℓ -1 and 0 < α < 1, and we aim to show that {∂ γ+3 n+1 ∂ β w k } is bounded in C 0,α loc (B + 1 ∪ D1) for β ∈ N n such that γ + |β| = ℓ -2 and 0 < α < 1. Using equation (5.56), we have -∂ γ+3 n+1 ∂ β w k = ∆ ′ ∂ γ+1 n+1 ∂ β w k ) + ∂ γ+1 n+1 ∂ β 2 a k ∇a k • ∇w k + |∇w k | 2 w k in B + 1 ,
where ∆ ′ := n j=1 ∂ 2 j . By (Bγ) and the induction hypothesis, the right hand side of the above equation remains bounded in C 0,α loc (B + 1 ∪ D1), and thus

{∂ γ+3 n+1 ∂ β w k } is indeed bounded in C 0,α loc (B + 1 ∪ D1).
In conclusion, we have thus proved that {w k } remains bounded in C ℓ+1,α loc (B + 1 ∪ D1) for every exponent 0 < α < 1, and the improved convergence follows .

Step 2: Improved convergence of {a k }. By elliptic regularity (applied to (5.

= i βi = ℓ. Recalling that h k := 1 -a k , we notice that ∂ β h k solves        -∆ ∂ β h k = ∂ β |∇w k | 2 a k in B + 2r , ε k a k (1 + a k ) ∂ ∂ β h k ∂ν + ∂ β h k = ε k R β,k on D2r , (5.57)
where

R β,k := 1 a k (1 + a k ) ∂ ∂ β h k ∂ν -∂ β 1 a k (1 + a k ) ∂h k ∂ν
involves only derivatives of a k of order less than or equal to ℓ. Recalling that a k 1/2, we deduce from the induction hypothesis and the boundedness of

{w k } in C ℓ+1,α (B + 2r ) that m β := sup k ∂ β h k L ∞ (B + 2r ) + R β,k C 0,α (D 2r ) + ∂ β |∇w k | 2 a k C 0,α (B + 2r ) < ∞ . (5.58)
Arguing as in the proof of Lemma 5.6 to derive (5.53), we infer that

|∂ β h k | Crm β ε k on Dr, which then yields ∂ ∂ β h k ∂ν L ∞ (Dr ) Crm β . (5.59)
Next we consider for y ∈ D r/4 the function ϑ k,y defined in (5.47). By the induction hypothesis, we have

∂ β ϑ k,y L ∞ (B + r/2 )
c1|y| α for some constant c1 independent of k. On the other hand,

∂ β ϑ k,y solves        -∆ ∂ β ϑ k,y = f β k,y in B + r/2 , ε k a k (1 + a k ) ∂ ∂ β ϑ k,y ∂ν + ∂ β h k = ε k g β k,y on D r/2 , with f β k,y (x) := ∂ β |∇w k | 2 a k (x + y) -∂ β |∇w k | 2 a k (x) , and
g β k,y (x) := R β,k (x + y) -R β,k (x) + 1 a k (x)(1 + a k (x)) - 1 a k (x + y)(1 + a k (x + y)) ∂ ∂ β h k ) ∂ν (x + y) .
By the induction hypothesis and (5.59), we have

f β k,y L ∞ (B + r/2 ) + g β k,y L ∞ (D r/2 ) c2|y| α
for a constant c2 independent of k. Arguing precisely as in the proof of Lemma 5.6, we deduce that

∂ β h k (x + y) -∂ β h k (x) ε k c3|y| α for every x, y ∈ D r/4 ,
and a constant c3 independent of k. Whence,

sup k ∂ β h k ε k C 0,α (D r/4 ) < ∞ ,
and inserting this estimate together with (5.58) in equation (5.57), we conclude from Lemma 9.2 that

{∂ β h k } remains bounded in C 1,α (B + r/8
). In view of the arbitrariness of the multi-index β above, we have showed that

(D0) {∂ β h k } is bounded in C 0,α (B + r/8 ) for every β ∈ N n such that |β| = ℓ + 1; (E0) {∂n+1∂ β h k } is bounded in C 0,α (B + r/8
) for every β ∈ N n such that |β| = ℓ. To complete the proof we need to estimate the remaining derivatives of h k of order ℓ + 1. Again, we proceed by induction assuming that for an integer 0 γ ℓ -1,

(Dγ) {∂ γ n+1 ∂ β h k } is bounded in C 0,α (B + r/8 ) for every β ∈ N n such that γ + |β| = ℓ + 1; (Eγ) {∂ γ+1 n+1 ∂ β h k } is bounded in C 0,α (B + r/8 ) for every β ∈ N n such that γ + |β| = ℓ,
and we aim to show that

{∂ γ+2 n+1 ∂ β h k } is bounded in C 0,α (B + r/8
) for every multi-index β ∈ N n such that γ + |β| = ℓ -1. Using equation (5.57), we notice that

∂ γ+2 n+1 ∂ β h k = -∂ γ n+1 ∆ ′ ∂ β h k -∂ γ n+1 ∂ β |∇w k | 2 a k in B + r/8 . (5.60)
Clearly the right hand side of (5.60) is bounded in C 0,α (B + r/8 ) by (Dγ) and the induction hypothesis, and thus

{∂ γ+2 n+1 ∂ β h k } is indeed bounded in C 0,α (B + r/8
). In conclusion, the sequence {a k } is bounded in C ℓ+1,α (B + r/8 ), and hence a k → a * in C ℓ+1,α ′ (B + r/8 ) for every 0 < α ′ < α.

Proof of Proposition 5.5 completed. Up to a subsequence, we have z k → z * for some point z * ∈ B we infer that |∇v * | 2 1/4 in B + 1 ∩ B̺ 0 (z * ) for some radius 0 < ̺0 1/4 which only depends on n and m. Using assumption (5.30) we now estimate

̺ n+1 0 |B + 1 | 4 B + 1 |∇v * | 2 dx lim inf k→∞ B + 1 |∇v k | 2 dx 2 n η0 . (5.61)
We then find a contradiction if η0 is small enough, and the proposition is proved.

Proof of (5.26) completed. We argue by contradiction assuming that rε > 1. We consider the rescaled function (5.62)

ũε(x) := uε x √ eε + xε for x ∈ B + 1 , so that ũε ∈ C 2 (B + 1 ; R m ) solves        ∆ũε = 0 in B + 1 , ∂ ũε ∂ν = 1 ε (1 -|ũε| 2 )ũε on D1 , with ε := ε √ eε . Moreover, (5.19) and (5
On the other hand, (5.20) leads to

Eε(ũε, B1) = ( √ eε) n-1 Eε uε, B + ( √ eε) -1 (xε) 2 n-1 η0 . (5.63)
If η0 is small enough, we conclude from Proposition 5.5 that ε ς0 where ς0 > 0 only depends on n and m. Applying [17, Lemma 2.2] we infer that ũε C 2,α (B + 3/4 )

Cα for every 0 < α < 1, where Cα only depends on α, n, and m. Arguing as in (5.61), we then find a contradiction between (5.62) and (5.63) whenever η0 is sufficiently small.

ASYMPTOTICS FOR GINZBURG-LANDAU BOUNDARY REACTIONS

6.1. Convergence to boundary harmonic maps and defect measures. With Theorem 5.1 in hands, we are now able to give a preliminary description of both weak limits as ε ↓ 0 of critical points of Eε, and the possible defect measure arising in the weak convergence process. This is the object of the following theorem.

{u k } k∈N ⊆ H 1 (Ω; R m ) ∩ L ∞ (Ω) be such that for every k ∈ N, |u k | 1, and u k weakly solves        ∆u k = 0 in Ω , ∂u k ∂ν = 1 ε k (1 -|u k | 2 )u k on ∂ 0 Ω . If sup k Eε k (u k , Ω) < ∞,
(i) |∇u k | 2 L n+1 Ω ⇀ |∇u * | 2 L n+1 Ω + µsing weakly* as Radon measures on Ω ∪ ∂ 0 Ω; (ii) (1 -|u k | 2 ) 2 ε k → 0 in L 1 loc (∂ 0 Ω);
(iii) Σ = supp(µsing) ∪ sing(u * );

(iv) u k → u * in C ℓ loc (Ω ∪ ∂ 0 Ω) \ Σ for every ℓ ∈ N; (v) if n = 1 the set Σ is finite and u * ∈ C ∞ (Ω ∪ ∂ 0 Ω). Proof. Step 1. First notice that Theorem 3.2 yields u k ∈ C ∞ (Ω ∪ ∂ 0 Ω).
By the uniform energy bound and the assumption |u k | 1, we can find a (not relabeled) subsequence such that u k ⇀ u * weakly in H 1 (Ω)

for some map u * ∈ H 1 (Ω; R m ). Since |u k | 1 and u k is harmonic in Ω, we deduce that u k → u * in C ℓ loc (Ω) for every ℓ ∈ N, u * is harmonic in Ω, and |u * | 1 in Ω. On the other hand, |u k | → 1 in L 2 (∂ 0 Ω
), and we infer from the compact imbedding

H 1 (Ω) ֒→ L 2 (∂ 0 Ω) that |u * | = 1 H n -a.e. on ∂ 0 Ω.
It then remains to analyse the asymptotic behavior of u k near ∂ 0 Ω. Setting

µ k := |∇u k | 2 L n+1 Ω + 1 2ε k (1 -|u k | 2 ) 2 H n ∂ 0 Ω , we have sup k µ k (Ω ∪ ∂ 0 Ω) < ∞.
Hence we can find a further subsequence such that

µ k ⇀ µ := |∇u * | 2 L n+1 Ω + µsing , (6.1)
weakly* as Radon measures on Ω ∪ ∂ 0 Ω for some nonnegative µsing on Ω ∪ ∂ 0 Ω. Notice that the local

smooth convergence of u k to u * in Ω implies that supp(µsing) ⊆ ∂ 0 Ω (6.2)
(here supp(µsing) denotes the relative support of µsing in Ω ∪ ∂ 0 Ω). By Lemma 5.2, we have

ρ 1-n µ k (Bρ(x)) r 1-n µ k (Br(x)) (6.3)
for every x ∈ ∂ 0 Ω and every 0 < ρ < r < dist(x, ∂ + Ω). Therefore,

ρ 1-n µ(Bρ(x)) r 1-n µ(Br(x)) (6.4)
for every x ∈ ∂ 0 Ω and every 0 < ρ < r < dist(x, ∂ + Ω). As a consequence, the (n -1)-dimensional

density Θ n-1 (µ, x) := lim r↓0 µ(Br(x))
ωn-1r n-1 (6.5) exists and is finite at every point x ∈ ∂ 0 Ω. Here ωn-1 denotes the volume of the unit ball in R n-1 if n 2, and ω0 = 1. Note that (6.1) and (6.3) yield

Θ n-1 (µ, x) C dist(x, ∂ + Ω) n-1 sup k∈N Eε k (u k , Ω) < ∞ for all x ∈ ∂ 0 Ω . (6.6)
On the other hand, by the smooth convergence of

u k toward u * in Ω, Θ n-1 (µ, x) = 0 for all x ∈ Ω .
In addition, we observe that x ∈ ∂ 0 Ω → Θ n-1 (µ, x) is upper semicontinuous.

Next we define the concentration set

Σ := x ∈ ∂ 0 Ω : inf r lim inf k→∞ r 1-n µ k (Br(x)) : 0 < r < dist(x, ∂ + Ω) η0 ,
where η0 > 0 is the constant given by Theorem 5.1. From (6.3) and (6.4) we infer that

Σ = x ∈ ∂ 0 Ω : lim r↓0 lim inf k→∞ r 1-n µ k (Br(x)) η0 = x ∈ ∂ 0 Ω : lim r↓0 r 1-n µ(Br(x)) η0 ,
and consequently,

Σ = x ∈ ∂ 0 Ω : Θ n-1 (µ, x) η0 ωn-1 . (6.7)
In particular, Σ is a relatively closed subset of ∂ 0 Ω (since Θ(µ, •) is upper semicontinuous). Moreover, by a well known property of upper densities (see e.g. [3, Theorem 2.56]), we have η0

ωn-1 H n-1 (Σ) µ(Σ) < ∞ . (6.8)
If n = 1, it obviously implies that Σ is finite.

Step 2. Let us now show that

u k → u * in C ℓ loc (Ω ∪ ∂ 0 Ω) \ Σ for every ℓ ∈ N.
In view of the local smooth convergence of u k in Ω, it suffices to prove the claim near ∂ 0 Ω. To this purpose, let us fix x0 ∈ ∂ 0 Ω \ Σ and 0 < r < dist(x0, ∂ + Ω ∪ Σ) such that r 1-n µ(Br(x0)) < η0. By the monotonicity in (6.4), we may assume without loss of generality that µ(∂Br(x0)) = 0. Then lim k µ k (Br(x0)) = µ(Br(x0)), which in turn implies that r 1-n µ k (Br(x0)) η0 for k sufficiently large. Taking k even larger we have ε k r, and we can then apply Theorem 5.1 and Lemma 5.4 to deduce that |∇u k | Cr and 1/2

|u k | 1 in B + r/4 (x0)
. This is now enough to reproduce the convergence proof in Proposition 5.5, Lemma 5.6, and Lemma 5.7. It that

u k → u * in C ℓ loc B + r/4 (x0) ∪ D r/4 (x0)
for every ℓ ∈ N. We finally notice that Theorem 5.1 also provides the estimate

(1 -|u k | 2 ) 2 Crε 2 k in D r/4 (x0). As a consequence, lim k→∞ 1 ε k D r/4 (x 0 ) (1 -|u k | 2 ) 2 dH n = 0 , (6.9)
a fact that we shall use later on.

Step 3. Let us now prove that u * is a weak S m-1 -boundary harmonic map in Ω ∪ ∂ 0 Ω. We distinguish the two cases n = 1 and n 2.

Case 1, n = 1. Let Φ ∈ H 1 (Ω; R m )∩L ∞ (Ω) with compact support in Ω∪∂ 0 Ω such that Φ(x) ∈ T u * (x) S m-1
for H n -a.e. x ∈ ∂ 0 Ω. By Step 1 the set supp Φ ∩ Σ contains finitely many points b1, . . . , bL. Then, fix an

arbitray cut-off function ζ ∈ C ∞ (R n+1 ; [0, 1]) such that ζ = 0 in a small neighborhood of each b l . We set Φ := ζΦ, so that Φ has compact support in (Ω ∪ ∂ 0 Ω) \ Σ. From the convergence of u k established in
Step 2, we have

|(1 -|u k | 2 )|| Φ| Cε k on ∂ 0 Ω, and thus lim k→∞ 1 ε k ∂ 0 Ω (1 -|u k | 2 )u k • Φ dH n = lim k→∞ 1 ε k ∂ 0 Ω (1 -|u k | 2 )(u k -u * ) • Φ dH n = 0 ,
by dominated convergence. On the other hand,

lim k→∞ Ω ∇u k • ∇ Φ dx = Ω ∇u * • ∇ Φ dx ,
and we deduce that

Ω ∇u * • ∇ Φ dx = 0 .
Given an arbitrary δ > 0, we now choose the cut-off function

ζ of the form ζ(x) = χ δ (x) ζ(x) where ζ ∈ C ∞ (R n+1 ; [0, 1]) satisfies ζ = 0 in a small neighborhood of b l only for l 2, and χ δ ∈ C ∞ c (R n+1 ; [0, 1]) satisfies χ δ = 0 in B δ (b1), χ δ = 1 outside B 2δ (b1), and |∇χ δ | C/δ. Setting Φ = ζΦ, we have Ω χ δ ∇u * • ∇ Φ dx + Ω∩B + 2δ (b 1 ) Φ • (∇χ δ • ∇u * ) dx = 0 . (6.10)
Using Cauchy-Schwarz inequality we estimate

Ω∩B + 2δ (b 1 ) Φ • (∇χ δ • ∇u * ) dx C Φ L ∞ (Ω) Ω∩B + 2δ (b 1 ) |∇u * | 2 dx 1/2 -→ δ↓0 0 .
Therefore, letting δ ↓ 0 in (6.10) leads to

Ω ∇u * • ∇ Φ dx = 0 .
We then repeat the argument for each point b l to reach the conclusion

Ω ∇u * • ∇Φ dx = 0 . (6.11)
Hence u * is a weak S m-1 -boundary harmonic map in Ω ∪ ∂ 0 Ω, and u * ∈ C ∞ (Ω ∪ ∂ 0 Ω) by Theorem 4.18.

Case 2, n 2. Consider an arbitrary Φ as in Case 1, and write K := supp Φ. From Step 1 we know that

H n-1 (Σ ∩ K) < ∞.
By [32, Theorem 3, p.154] it implies that cap 2 (Σ ∩ K) = 0 where cap 2 denotes the Newtonian capacity. Moreover, the proof of [32, Theorem 3, p.154] provides a sequence of functions

{χ l } l∈N such that χ l ∈ Ḣ1 (R n+1 ), 0 χ l 1, χ l = 0 in a neighborhood of Σ ∩ K, χ l → 1 a.e. as l → ∞, and
lim l→∞ R n+1 |∇χ l | 2 dx = 0 .
Arguing as in Case 1, we obtain that

0 = Ω ∇u * • ∇(χ l Φ) dx = Ω χ l ∇u * • ∇Φ dx + Ω Φ • (∇χ l • ∇u * ) dx , (6.12)
and we estimate

Ω Φ • (∇χ l • ∇u * ) dx C Φ L ∞ (Ω) ∇u * L 2 (Ω) R n+1 |∇χ l | 2 dx 1/2 -→ l→∞ 0 .
Letting l → ∞ in (6.12) then shows that (6.11) holds, whence u * is a weak S m-1 -boundary harmonic map in Ω ∪ ∂ 0 Ω.

Step 4. We conclude the proof by showing that Σ = supp(µsing) ∪ sing(u * ). If x0 does not belong to supp(µsing) ∪ sing(u * ), we can find r0 > 0 such that µsing(Br(x0)) = 0 for all r < r0. Hence,

r 1-n µ(Br(x0)) = lim n→∞ r 1-n µ k (Br(x0)) = r 1-n 2 Br (x 0 )∩Ω |∇u * | 2 dx
for all r < r0. Since u * is smooth in a neighborhood of x0 by Theorem 4.18, we deduce that Θ n-1 (µ, x0) = 0, and thus x0 ∈ Σ by (6.7).

Let us now assume that x0 ∈ Σ. If x0 ∈ Ω then x0 ∈ supp(µsing)∪sing(u * ) by ( 6.2) and the smoothness of u * in Ω. If x0 ∈ ∂ 0 Ω we deduce from (6.9) and the convergence established in Step 2 that x0 ∈ sing(u * ), and

µ(Br(x0)) = lim k→∞ µ k (Br(x0)) = 1 2 Br (x 0 )∩Ω |∇u * | 2 dx
for a radius r > 0 sufficiently small. Therefore, µsing(Br(x0)) = 0, and thus x0 does not belong to supp(µsing).

Step 5. In view of (6.1), it only remains to prove (ii). Let K be a compact subset of ∂ 0 Ω, and set δ0 := 1 2 dist(K, ∂ + Ω). For δ ∈ (0, δ0/2), we define Σ δ := {x ∈ ∂ 0 Ω : dist(x, Σ) δ}. Then, K ∩ Σ δ is a compact set. From Step 4 and the local smooth convergence of u k toward u * outside Σ, we deduce that ε -1

k K\Σ δ (1 -|u k | 2 ) 2 dH n → 0.
On the other hand, for any x0 ∈ K ∩ Σ δ we have B + δ (x0) ⊆ Ω, and we infer from the monotonicity formula in Lemma 5.2 that

1 4ε k 2δ δ 1 t n D t (x 0 ) (1 -|u k | 2 ) 2 dH n dt 1 δ n-1 0 Eε k (u k , B + δ 0 (x0)) .
Hence,

lim sup k→∞ 1 4ε k D δ (x 0 ) (1 -|u k | 2 ) 2 dH n 2 n δ n δ k-1 0 sup n∈N Eε k (u k , Ω) .
Then, by a standard covering argument, we deduce that

lim k→∞ 1 ε k Σ δ (1 -|u k | 2 ) 2 dH n = 0 , and thus ε -1 k K (1 -|u k | 2 ) 2 dH n → 0 as k → ∞.
To complete this subsection, we now prove the (n-1)-rectifiability of the defect measure µsing through the celebrated PREISS criteria [START_REF] Preiss | Geometry of measures in R n : distributions, rectifiability, and densities[END_REF].

Proposition 6.2. Assume that n 2. In Theorem 6.1 the set Σ is countably H n-1 -rectifiable, and the defect measure µsing satisfies µsing = θH n-1 Σ (6.13)

for some positive Borel function θ : Σ → (0, ∞).

Proof. By a well known property of Sobolev functions (see e.g. [80, (3.3.28)]), we have

lim r↓0 1 r n-1 Ω∩B + r (x)
|∇u * | 2 dx = 0 for H n-1 -a.e. x ∈ Σ . (6.14) Therefore (6.5) yields Θ n-1 (µsing, x) := lim r↓0 µsing(Br(x))

ωn-1r n-1 = Θ n-1 (µ, x) for H n-1 -a.e. x ∈ Σ . (6.15)

On the other hand, we derive from (6.6) that Θ * ,n-1 (µsing, x) := lim sup r↓0 µsing(Br(x))

ωn-1r n-1 C dist(x, ∂ + Ω) n-1 sup k∈N Eε k (u k , Ω) < ∞
for all x ∈ Σ. Since supp(µsing) ⊆ Σ, we infer from [3, Theorem 2.56] that µsing is absolutely continuous with respect to H n-1 Σ. Then we deduce from (6.15) and (6.5)-(6.6)-(6.7) that

Θ n-1 (µsing, •) = Θ n-1 (µ, •) ∈ (0, ∞) µsing-a.e. ,
and according to PREISS rectifiability criteria [START_REF] Preiss | Geometry of measures in R n : distributions, rectifiability, and densities[END_REF], it implies that µsing is a (n -1)-rectifiable measure (see e.g. [START_REF] Ambrosio | PALLARA : Functions of Bounded Variation and Free Discontinuity Problems[END_REF]Definition 2.59]). Next we infer from [3, Theorem 2.83], (6.15), and (6.6)-(6.7) that µsing is of the form (6.13) with θ(x) := Θ n-1 (µ, x) ∈ (0, ∞). As a consequence, Σ is a countably H n-1 -rectifiable set.

Remark 6.3 (Approximate tangent space). As a consequence of Proposition 6.2 and the fact that Σ ⊆ ∂ 0 Ω ⊆ R n , the measure µsing admits an (n -1)-dimensional approximate tangent space TxΣ ⊆ R n at

x with multiplicity θ(x) for H n-1 -a.e. x ∈ Σ (see e.g. [START_REF] Ambrosio | PALLARA : Functions of Bounded Variation and Free Discontinuity Problems[END_REF]Theorem 2.83]). More precisely, if we denote by Gn,n-1 the Grassmann manifold of unoriented (n -1)-dimensional planes in R n , then for H n-1 -a.e.

x ∈ Σ there exists a plane TxΣ ∈ Gn,n-1 such that

lim r↓0 1 r n-1 ∂ 0 Ω φ y -x r dµsing(y) = θ(x) TxΣ φ(y) dH n-1 (y)
for all φ ∈ C 0 c (R n ).

Stationarity defect and generalized varifolds.

In this subsection we assume that n 2, and our discussion starts from Theorem 6.1. In this theorem, we point out that the limiting (S m-1 , ∂ 0 Ω)-boundary harmonic map u * is a priori only weakly harmonic, and might not be stationary, i.e., it might not satisfy (4.8). On the other hand, u * arises as a weak limit of (smooth) critical points of the boundary Ginzburg-Landau energy Eε, so that the lack of stationarity of u * should be quantified. For the classical Ginzburg-Landau system, the analoguous question is treated by LIN & WANG in [START_REF] Lin | Harmonic and quasi-harmonic spheres. III. Rectifiability of the parabolic defect measure and generalized varifold flows[END_REF] where they show that the possible stationarity defect is related to the defect measure through an explicit formula.

The main objective in this subsection is to prove that a similar formula holds in the Ginzburg-Landau boundary context, and this is the object of the following theorem. To recall from [START_REF] Ambrosio | A measure-theoretic approach to higher codimension mean curvature flows[END_REF] the concept of generalized varifolds, we need to introduce the following (compact and convex) set of matrices

An-1 := A ∈ R (n+1)×(n+1) : A is symmetric, trace(A) = n -1 , -(n + 1)In+1 A In+1 ,
where In+1 denotes the identity square matrix of size n + 1. Definition 6.5. A (n -1)-dimensional generalized varifold V on Ω is a nonnegative Radon measure on Ω × An-1. The class of all generalized (n -1)-varifolds on Ω is denoted by V * n-1 ( Ω). For V ∈ V * n-1 ( Ω) we denote by V the weight of V defined as the first marginal of V , i.e., V := π ♯ V where π : Ω × An-1 → Ω is the canonical projection. (Notice that V is a Radon measure on Ω.) The first variation δV of a generalized varifold V ∈ V * n-1 ( Ω) is the element of (C 1 c ( Ω; R n+1 )) * defined by δV, X := -

Ω×A n-1 A : ∇X dV for all X ∈ C 1 c ( Ω; R n+1 ) .
If δV = 0, then V is said to be stationary.

Remark 6.6 (Weak convergence of varifolds). The convergence in V * n-1 ( Ω) is understood as weak* convergence of Radon measures on Ω × An-1. In particular, if

V k ⇀ V in V * n-1 ( Ω), then δV k ⇀ δV weakly* in (C 1 c ( Ω; R n+1 )) * .
Remark 6.7 (Disintegration and barycenter). Given V ∈ V * n-1 ( Ω), we denote by {Vx} x∈ Ω a disintegration of V , i.e., {Vx} x∈ Ω is a family of probability measures on An-1 such that x → Vx is V -measurable, and

Ω×A n-1 f (x, A) dV = Ω A n-1 f (x, A) dVx d V (6.16)
for any bounded Borel function f : Ω × An-1 → R. The measurability condition on {Vx} x∈ Ω means that x → Vx(B) is V -measurable for every Borel set B ⊆ An-1. This fact ensures that the inner integral in the right hand side of (6.16) is V -measurable, so that its integral is well defined. We refer to [START_REF] Ambrosio | PALLARA : Functions of Bounded Variation and Free Discontinuity Problems[END_REF] for the existence and the uniqueness of {Vx} x∈ Ω modulo V -null sets. Throughout the subsection, we may use the disintegrated notation V = Vx V . We also denote by ĀV (x) the barycenter of V at x defined by

ĀV (x) :=

A n-1

A dVx .

Then x → ĀV (x) is V -measurable, and ĀV (x) ∈ An-1 since Vx is a probability measure. Moreover, we can rewrite the action of the first variation δV as δV, X := -

Ω ĀV (x) : ∇X d V for all X ∈ C 1 c ( Ω; R n+1 ).
We may now present the way to relate our problem to generalized varifolds. We start with the construction of a generalized varifold starting from a Sobolev map. For u ∈ H 1 ( Ω; R m ), we denote by Vu ∈ V * n-1 ( Ω) the generalized varifold given by

Vu := 1 2 δA u |∇u| 2 L n+1 Ω ,
where δ Au(x) is the Dirac mass concentrated at the matrix Au(x) defined by

Au(x) :=        In+1 -2 (∇u(x)) T (∇u(x)) |∇u(x)| 2 if |∇u(x)| = 0 , In-1 otherwise ,
and In-1 is the matrix of the orthogonal projection on R n-1 ≃ R n-1 ×{(0, 0)}. One may easily check that Au(x) ∈ An-1, so that Vu indeed belongs to V * n-1 ( Ω). Next we add the Ginzburg-Landau potential in the construction above to build generalized varifolds based on the Ginzburg-Landau boundary energy.

More precisely, for ε > 0 and an arbitrary map u ∈ H 1 ( Ω; R m ) ∩ L 4 (∂ 0 Ω), we set

V ε u := Vu + 1 2ε δI n-1 (1 -|u| 2 ) 2 H n ∂ 0 Ω ∈ V * n-1 ( Ω) , so that V ε u (B) = 1 2 B |∇u| 2 dx + 1 2ε B∩∂ 0 Ω (1 -|u| 2 ) 2 dH n
for any open subset B ⊆ Ω. The first variation of V ε u is then given by

δV ε u , X = - 1 2 Ω |∇u| 2 divX -2 n+1 i,j=1 (∂iu • ∂ju)∂jXi dx - 1 2ε ∂ 0 Ω (1 -|u| 2 ) 2 div R n-1 X dH n
for all X ∈ C 1 c ( Ω; R n+1 ). In the case where u = ũε and uε is a critical point of the Ginzburg-Landau boundary energy Eε, the first variation reduces to the following expression. Lemma 6.8.

Given ε > 0, if uε ∈ H 1 (Ω; R m ) ∩ L ∞ (Ω) is a critical point of Eε in Ω, then δV ε ũε , X = 1 2ε ∂ 0 Ω (1 -|uε| 2 ) 2 ∂nXn dH n for all X ∈ C 1 c ( Ω; R n+1 ).
Proof.

If uε ∈ H 1 (Ω; R m ) ∩ L ∞ (Ω) is a critical point of Eε in Ω, then uε ∈ C ∞ (Ω ∪ ∂ 0 Ω) by Theorem 3.2. It obviously implies u - ε ∈ C ∞ (Ω -∪ ∂ 0 Ω), and        ∆u - ε = 0 in Ω -, ∂u - ε ∂ν = 1 ε (1 -|uε| 2 )uε on ∂ 0 Ω . (6.17)
Let us now consider a vector field X = (X1, . . . , Xn+1) ∈ C 1 c ( Ω; R n+1 ). Arguing as in the proof of Lemma 5.2, we integrate by parts to find

Ω |∇uε| 2 divX -2 n+1 i,j=1 (∂iuε • ∂juε)∂jXi dx = - 1 2ε ∂ 0 Ω (1 -|uε| 2 ) 2 div R n X dH n - ∂ 0 Ω |∇uε| 2 Xn+1 dH n + 2 ε 2 ∂ 0 Ω (1 -|uε| 2 ) 2 |uε| 2 Xn+1 dH n .
Similarly, (6.17) yields

Ω - |∇u - ε | 2 divX -2 n+1 i,j=1 (∂iu - ε • ∂ju - ε )∂jXi dx = - 1 2ε ∂ 0 Ω (1 -|uε| 2 ) 2 div R n X dH n + ∂ 0 Ω |∇u - ε | 2 Xn+1 dH n - 2 ε 2 ∂ 0 Ω (1 -|uε| 2 ) 2 |uε| 2 Xn+1 dH n .
Therefore,

δV ε ũε , X = 1 2ε ∂ 0 Ω (1 -|uε| 2 ) 2 ∂nXn dH n + 1 2 ∂ 0 Ω |∂n+1uε| 2 -|∂n+1u - ε | 2 Xn+1 dH n .
Since ∂n+1u - ε = -∂n+1uε by (6.17), the conclusion follows from this last equality.

We now relate a weak limit u * and its defect measure µsing to the weak limit of the generalized varifolds V ε ũε , which has to be stationary by Lemma 6.8 and the vanishing property of the Ginzburg-Landau potential as ε → 0. Corollary 6.9. Let {u k } k∈N be the subsequence given by Theorem 6.1, and let µsing be the defect measure represented in (6.13). Up to a further subsequence (not relabeled), V ε k ũk ⇀ V * for some stationary V * ∈ V * n-1 ( Ω). In addition, V * = Vũ * + Vsing , (6.18) where Vsing ∈ V * n-1 ( Ω) is supported by Σ × An-1, and Vsing = µsing.

Proof.

Step 1. By symmetry with respect to {xn+1 = 0}, we have V

ε k ũk ( Ω) = 2Eε k (u k , Ω)
, and thus

sup k∈N V ε k ũk ( Ω) < ∞ .
Hence we can find a (not relabeled) further subsequence such that V ε k ũk ⇀ V * for some V * ∈ V * n-1 ( Ω). Then δV ε k ũk ⇀ δV * as distributions. We claim that δV * = 0. Indeed, for an arbitrary X ∈ C 1 c ( Ω; R n+1 ), we deduce from item (ii) in Theorem 6.1 and Lemma 6.8 that δV * , X = lim

k→∞ 1 2ε k ∂ 0 Ω (1 -|u k | 2 ) 2 ∂nXn dH n = 0 .
Step 2. For Φ ∈ C 0 c ( Ω), we denote by Φ the reflection of Φ with respect to the hyperplane {xn+1 = 0}, i.e., Φ(x ′ , xn+1) := Φ(x ′ , -xn+1) for x ∈ Ω. Noticing that

Ω Φ d V ε k ũk = 1 2 Ω |∇u k | 2 (Φ + Φ) dx + 1 2ε k ∂ 0 Ω (1 -|u k | 2 ) 2 Φ dH n ,
we infer from Theorem 6.1 that

lim k→∞ Ω Φ d V ε k ũk = 2 Ω |∇u * | 2 (Φ + Φ) dx + Σ Φ dµsing = 1 2 Ω |∇ũ * | 2 Φ dx + Σ Φ dµsing .
On the other hand, V ε k ũk ⇀ V * weakly* as Radon measures on Ω, and hence

V * = Vũ * + µsing . (6.19) Next consider Ψ ∈ C 0 c ( Ω × An-1) with compact support in ( Ω \ Σ) × An-1. Then, Ω×A n-1 Ψ dV ε k ũk = 1 2 Ω Ψ x, Aũ k (x) |∇ũ k | 2 dx + 1 4ε k ∂ 0 Ω Ψ(x, In-1)(1 -|u k | 2 ) 2 dH n .
From the convergences established in items (ii) & (iv) of Theorem 6.1, we deduce that

Ω×A n-1 Ψ dV * = lim k→∞ Ω×A n-1 Ψ dV ε k ũk = lim k→∞ 1 2 Ω∩{|∇ũ * | =0} Ψ x, Aũ k (x) |∇ũ k | 2 dx = 1 2 Ω Ψ x, Aũ * (x) |∇ũ * | 2 dx = Ω×A n-1 Ψ dVũ * .
As a consequence, V * (K) = Vũ * (K) for every compact set K ⊆ ( Ω \ Σ) × An-1. By inner regularity, it implies

V * B \ (Σ × An-1) = Vũ * B \ (Σ × An-1) = Vũ * (B)
for all Borel sets B ⊆ Ω × An-1 (in the last equality, we have used the fact that Vũ * is absolutely continuous with respect to L n+1 ). Therefore,

V * ( Ω \ Σ) × An-1 = Vũ * .

Finally, setting

Vsing := V * (Σ × An-1) , we clearly have (6.18), and Vsing = µsing holds by (6.19).

The last main step to establish Theorem 6.4 is the following geometrical property on the singular part of the limiting varifold.

Lemma 6.10. In Corollary 6.9, the barycenter of generalized varifold Vsing satisfies

ĀV sing (x) = AT x Σ for H n-1 -a.e. x ∈ Σ ,
where AT x Σ is the matrix of the orthogonal projection on the approximate tangent plane TxΣ of Σ at x according to Remark 6.3.

Proof. Let x0 ∈ Σ be such that Σ admits an approximate tangent plane Tx 0 Σ at x0 with multiplicity θ(x0), x0 is a Lebesgue point of AV sing with respect to µsing, and (6.14) holds at x0. Those properties are satisfied H n-1 -a.e. on Σ.

Next consider an arbitrary X ∈ C 1 c (R n+1 ; R n+1 ), and set Xr(x) := rX( x-x 0 r ). From Corollary 6.9 and the choice of x0, we infer that 0 = lim Since θ(x0) > 0 and X is arbitrary, we deduce that

ĀV sing (x0) Tx 0 Σ ∇φ dH n-1 = 0 for all φ ∈ C 1 c (R n+1 ) . (6.20)
We claim that

F := Tx 0 Σ ∇φ dH n-1 : φ ∈ C 1 c (R n+1 ) = Ker AT x 0 Σ . (6.21)
Before proving (6.21) we complete the argument. From the last two identities we derive that at least two eigenvalues of ĀV sing (x0) vanish. On the other hand, ĀV sing (x0) ∈ An-1 so that trace( ĀV sing (x0)) = n-1

and ĀV sing (x0) In+1. It implies that the remaining eigenvalues are actually equal to one. Hence ĀV sing (x0) is a matrix of an orthogonal projection over an (n -1)-dimensional plane. Now (6.20) and (6.21) show that ĀV sing (x0) and AT x 0 Σ have the same kernel, whence ĀV sing (x0) = AT x 0 Σ.

To prove (6.21), we argue as follows. First notice that we may assume without loss of generality that Tx 0 Σ = R n-1 × {(0, 0)}. Since the admissible φ's are compactly supported, we obtain Remark 6.11. In view of Proposition 6.2 and Lemma 6.10, the generalized varifold Vsing := δ ĀV sing µsing is actually a real (n -1)-rectifiable varifold in the classical sense, see [START_REF] Simon | Theorems on regularity and singularity of energy minimizing maps[END_REF].

Proof of Theorem 6.4. We first infer from Corollary 6.9, Lemma 6.10, and (6.13) that

δV * , X = - 1 2 Ω |∇ũ * | 2 divX -2 n+1 i,j=1 (∂i ũ * • ∂j ũ * )∂jXi dx - Σ AT x Σ : ∇X dµsing = - 1 2 Ω |∇ũ * | 2 divX -2 n+1 i,j=1 (∂i ũ * • ∂j ũ * )∂jXi dx - Σ θ divΣX dH n-1 = 0 , (6.22)
for all X ∈ C 1 c ( Ω; R n+1 ) (we recall that divΣX(x) := AT x Σ : ∇X(x)). Let us now consider an arbitrary vector field X = (X1, . . . , Xn+1) ∈ C 1 (Ω; R n+1 ) compactly supported in Ω ∪ ∂ 0 Ω and satisfying Xn+1 = 0 on ∂ 0 Ω. We then extend X to Ω by setting X(x) := ( X1(x), . . . , Xn(x), -Xn+1(x)) , where we recall that Xj is the extension of Xj to Ω obtained by even reflection accross ∂ 0 Ω. Then X is Lipschitz continuous and compactly supported in Ω. Notice also that ∂i Xj is continuous in Ω for all indices i, j ∈ {1, . . . , n}. Then, by a standard mollification argument, we can find a sequence

{X k } k∈N ⊆ C 1 c ( Ω; R n+1 ) such that X k → X uniformly on Ω, ∂iX k j → ∂i Xj uniformly on Ω for all i, j ∈ {1, . . . , n}, ∇X k → ∇ X a.e. on Ω with ∇X k L ∞ ( Ω)
∇ X L ∞ ( Ω) . Applying (6.22) to each X k and letting k → ∞, we derive by dominated convergence that

1 2 Ω |∇ũ * | 2 div X -2 n+1 i,j=1 (∂i ũ * • ∂j ũ * )∂i Xj dx + Σ θ divΣ X dH n-1 = 0 .
Finally, by symmetry of ũ * and X, we have

1 2 Ω |∇ũ * | 2 div X -2 n+1 i,j=1 (∂i ũ * • ∂j ũ * )∂j Xi dx = Ω |∇u * | 2 divX -2 n+1 i,j=1
(∂iu * • ∂ju * )∂jXi dx , and

Σ θ divΣ X dH n-1 = Σ θ divΣX dH n-1 ,
and the conclusion follows.

ASYMPTOTICS FOR THE FRACTIONAL GINZBURG-LANDAU EQUATION

The purpose of this section is to apply our previous results on the Ginzburg-Landau boundary equation to the asymptotic analysis, as ε ↓ 0, of solutions of the fractional Ginzburg-Landau equation. We start with the analogue of Theorem 6.1 and Theorem 6.4 in the fractional setting when no exterior condition is imposed. We then prove Theorem 1.1. We conclude this section with the particular case of minimizers of the Ginzburg-Landau 1/2-energy under a Dirichlet exterior condition.

7.1. Asymptotics without exterior Dirichlet condition. We start in this subsection with a general case where no exterior condition is imposed. We obtain here the most important convergence results.

} k∈N ⊆ H 1/2 (ω; R m ) ∩ L ∞ (R n ) be such that for every k ∈ N, |v k | 1, and v k weakly solves (-∆) 1 2 v k = 1 ε k (1 -|v k | 2 )v k in ω . ( 7
(ii) (1 -|v k | 2 ) 2 ε k → 0 in L 1 loc (ω); (iii) µsing = θH n-1 Σ; (iv) v * ∈ C ∞ (ω \ Σ) and v k → v * in C ℓ loc (ω \ Σ) for every ℓ ∈ N; (v) if n 2,
(R n , m) such that v k ⇀ v * weakly in L 2 (R n , m). In particular, v k ⇀ v * weakly in L 2 loc (R n ).
On the other hand, the uniform energy bound also shows that |v k | 2 → 1 in L 2 (ω), and {v k } is bounded in H 1/2 (ω). Hence v k ⇀ v * weakly in H 1/2 (ω), and from the compact embedding H 1/2 (ω) ֒→ L 2 (ω), it implies that v k → v * strongly in L 2 (ω). In particular, |v * | = 1 a.e. in ω.

We now claim that v * ∈ H 1/2 (ω; R m ), and more precisely that

E(v * , ω) lim inf k→∞ E(v k , ω) < ∞ .
In view of the weak convergence of {v k } in H 1/2 (ω), it remains to show that

ω×ω c |v * (x) -v * (y)| 2 |x -y| n+1 dxdy lim inf k→∞ ω×ω c |v k (x) -v k (y)| 2 |x -y| n+1 dxdy . (7.2)
We fix R > 0 and 0 < δ < R/2 such that ω ⊆ D R/2 . Set ω δ := {x ∈ R n : dist(x, ω) < δ}, and write

ω×(ω δ ) c ∩D R |v k (x) -v k (y)| 2 |x -y| n+1 dxdy = ω |v k (x)| 2 w1(x) dx + (ω δ ) c ∩D R |v k (x)| 2 w2(x) dx -2 (ω δ ) c ∩D R v k (x) • f k (x) dx =: I k + II k + III k , (7.3)
where ∪ ω), whence the announced weak convergence.

w1(x) := (ω δ ) c ∩D R dy |x -y| n+1 , w2(x) := ω dy |x -y| n+1 , f k (x) := ω v k (y) dy |x -y| n+1 . By the strong L 2 (ω)-convergence of v k toward v * , f k converges strongly in L 2 ((ω δ ) c ∩DR) to the function f * (x) := ω v * (y) dy |x-y| n+1 . From the weak L 2 loc (R n )-convergence of v k , we deduce that ω |v * (x)| 2 w1(x) dx lim inf k→∞ I k , (ω δ ) c ∩D R |v * (x)| 2 w2(x) dx lim inf k→∞ II k , and
lim k→∞ III k = -2 (ω δ ) c ∩D R v * (x) • f * (x) dx . Therefore, ω×(ω δ ) c ∩D R |v * (x) -v * (y)| 2 |x -y| n+1 dxdy lim inf k→∞ ω×ω c |v k (x) -v k (y)| 2 |x -
Step 2. Let us now consider an increasing sequence {Ω l } l∈N of bounded admissible open sets such that ∂ 0 Ω l ⊆ ω for every l ∈ N, ∪ l Ω l = R n+1 + , and ∪ l ∂ 0 Ω l = ω. By (2.11), Step 1, and the results in Section 3,

v e k ∈ H 1 (Ω l ; R m ) ∩ L ∞ (Ω l ) satisfies |v e k | 1 and solves        ∆v e k = 0 in Ω l , ∂v e k ∂ν = 1 ε k (1 -|v e k | 2 )v e k on ∂ 0 Ω l ,
for every l ∈ N. In addition, we have proved in Step 1 that sup k Eε k (v e k , Ω l ) < ∞ for every l ∈ N. Therefore, we can find a further subsequence such that the conclusions of Theorems 6.1 & 6.4 hold in every Ω l , and v e * is the limiting (S m-1 , ∂ 0 Ω l )-boundary harmonic map in each Ω l by Step 1. This yields the announced conclusions on the defect measure µsing and on the concentration set Σ stated in (i), (ii), (iii), and (iv). The stationarity relation stated in (v) between v * and µsing is in turn a direct consequence of Theorem 6.4 and Lemma 4.16. Then it only remains to prove that v * is a weak 1/2-harmonic map into S m-1 in ω. By Proposition 4.2, it is enough to check that (-∆)

1 2 v * , ϕ ω = 0 for every ϕ ∈ H 1/2 00 (ω; R m ) ∩ L ∞ (ω) compactly supported in ω satisfying v * • ϕ = 0 a.e. in ω. Given such a test function ϕ, we consider an arbitrary extension Φ ∈ H 1 (R n+1 + ; R m ) ∩ L ∞ (R n+1 + ) of ϕ which is compactly supported in R n+1 + ∪ ω.
Then supp Φ ⊆ Ω l ∪ ∂ 0 Ω l for l large enough. Since v e * is a weak (S m-1 , ∂ 0 Ω l )-boundary harmonic map in Ω l , we infer from Lemma 2.9 that

(-∆) 1 2 v * , ϕ ω = Ω l ∇v e * • ∇Φ dx = 0 ,
and the proof is complete.

Remark 7.2. We notice that Theorem 6.1 actually implies that v

e k → v e * in C ℓ loc R n+1 + ∪ (ω \ Σ) for every ℓ ∈ N.
In view of Lemma 2.9, it shows that (-∆) Given an arbitrary sequence ε k ↓ 0, we consider {v k } k∈N ⊆ H 1/2 g (ω; R m ) ∩ L 4 (ω) such that for each k ∈ N, v k weakly solves (7.1), and sup k Eε k (v k , ω) < ∞. We recall that under such assumptions, we have proved in Section 3 that v k ∈ C 0,α loc (R n ) ∩ C ∞ (R n \ ∂ω) for every 0 < α < 1/2, and that |v k | 1 in R n . Therefore, we can apply Theorem 7.1 to find a (not relabeled) subsequence such that v k ⇀ v * weakly in H 1/2 (ω) ∩ L 2 loc (R n ) for some map v * ∈ H 1/2 (ω; R m ) which is a bounded weak 1/2-harmonic map into S m-1 in ω, and such that all the conclusions of Theorem 7.1 hold. Since v k is constantly equal to g outside ω, we also infer that v * ∈ H

1 2 v k → (-∆) 1 2 v * in C ℓ loc (ω \ Σ) for every ℓ ∈ N.

Proof of (a). We first notice that

v k -v * ∈ H 1/2 00 (ω; R m ) for every k ∈ N by (2.4). Hence, [v k -v * ] 2 H 1/2 (R n ) = E(v k -v * , ω) 2 E(v k , ω) + E(v * , ω) , so that {v k -v * } in bounded in H 1/2 00 (ω). Since v k -v * ⇀ 0 weakly in H 1/2 (ω), it remains to show that ω×ω c (v k (x) -v * (x)) • ϕ(x) |x -y| n+1 dxdy → 0 for every ϕ ∈ H 1/2 00 (ω; R m ). First notice that v k → v * strongly in L 2 (ω) by the compact embedding H 1/2 (ω) ֒→ L 2 (ω).
Then, by density of smooth maps compactly supported in ω, it suffices to consider the case where ϕ ∈ D(ω). For such a test function ϕ, the assertion is easily proved using the dominated convergence Theorem and the fact that v k → v * a.e. in ω (up to a further subsequence if necessary).

Proof of (b).

From the uniform energy bound, we may find a further subsequence such that e(v k , ω) L n ω * ⇀ e(v * , ω) L n ω + µ def weakly* as Radon measures on ω for some finite nonnegative measure µ def . We thus have to prove that µ def ≡ µsing. It will then show that µsing is finite, and that H n-1 (Σ) < ∞ (since H n-1 (Σ) is controlled by the total variation of µsing, see (6.8)).

Let us now fix ϕ ∈ D(ω) arbitrary. We notice that ω e(v k , ω)ϕ dx = (-∆)

1 2 v k , ϕv k ω - γn 2 ω×ω (v k (x) -v k (y)) • v k (y)(ϕ(x) -ϕ(y)) |x -y| n+1 dxdy -γn ω×ω c (v k (x) -g(y)) • g(y)ϕ(x) |x -y| n+1 dxdy =: I k -II k -III k . We consider a function Φ ∈ C ∞ (R n+1
+ ) compactly supported in R n+1 +
∪ ω such that Φ |R n = ϕ, and set K := supp(Φ). We observe that Φv e k is a smooth function compactly supported by K, so that Lemma 2.9 yields

(-∆) 1 2 v k , ϕv k ω = R k+1 + ∇v e k • ∇(Φv e k ) dx = R n+1 + |∇v e k | 2 Φ dx + K ∇v e k • (v e k ∇Φ) dx .
From (a) we deduce that v e k ⇀ v e * weakly in H 1 loc (R n+1 + ), so that ∇v e n ⇀ ∇v e * weakly in L 2 (K) and v e k → v e * strongly in L 2 (K). Together with item (i) in Theorem 7.1, it yields

(-∆) 1 2 v k , ϕv k ω -→ k→∞ R n+1 + |∇v e * | 2 Φ dx + ω ϕ dµsing + K ∇v e * • (v e * ∇Φ) dx = R n+1 + ∇v e * • ∇(Φv e * ) dx + ω ϕ dµsing .
By Lemma 2.9 again, we have thus proved that

(-∆) 1 2 v k , ϕv k ω -→ k→∞ (-∆) 1 2 v * , ϕv * ω + ω ϕ dµsing , (7.4)
where we have used the (elementary) fact that ϕv * ∈ H 1/2 00 (ω; R m ). On the other hand, we have that v k → v * a.e. on R n , eventually after the extraction of a further subsequence. Using ϕ ∈ D(ω) and the weak convergence of v k in H 1/2 (ω), we deduce that

II k → γn 2 ω×ω (v * (x) -v * (y)) • v * (y)(ϕ(x) -ϕ(y)) |x -y| n+1 dxdy (7.5)
and by dominated convergence, To derive item (iii), we just notice that

III k → γn ω×ω c (v * (x) -g(y)) • g(y)ϕ(x) |x -y| n+1 dxdy ( 7
ω (1 -|v k | 2 ) ε k ϕ dx = ω (1 -|v k | 2 ) 2 ε k ϕ dx + 1 ε k ω (1 -|v k | 2 )v k • (ϕv k ) dx = ω (1 -|v k | 2 ) 2 ε k ϕ dx + (-∆) 1 2 v k , ϕv k ω ,
so that the announced convergence follows from item (ii), (7.4), and Remark 4.3.

Remark 7.3. In view of Remarks 4.3 & 7.2, we have

1 -|v k (x)| 2 ε k → γn 2 R n |v * (x) -v * (y)| 2 |x -y| n+1 dy in C ℓ loc (ω \ Σ)
for every ℓ ∈ N. Let ε k ↓ 0 be an arbitrary sequence, and let

{v k } k∈N ⊆ H 1/2 g (ω; R m ) ∩ L 4 (ω) be such that for each k ∈ N, v k ∈ argmin Eε k (v, ω) : v ∈ H 1/2 g (ω; R m ) ∩ L 4 (ω) . (7.8)
Then there exist a (not relabeled) subsequence and v * ∈ H

1/2 g (ω; R m ) a minimizing 1/2-harmonic map into S m-1 in ω such that v k -v * → 0 strongly in H 1/2 00 (ω). In addition, (i) Eε k (v k , ω) → E(v * , ω); (ii) 1 -|v k (x)| 2 ε k ⇀ γn 2 R n |v * (x) -v * (y)| 2 |x -y| n+1 dy in D ′ (ω); (iii) vn → v * in C ℓ loc (ω \ sing(v * )) for every ℓ ∈ N; (iv) if n = 1, then ω ∩ sing(v * ) = ∅, while dim H sing(v * ) ∩ ω n -2 for n 3, and sing(v * ) ∩ ω is discrete for n = 2.
Proof. By assumption (7.7) there exists g ∈ H 1/2 g (ω; R m ) such that | g| = 1 a.e. in ω. Then we infer from (7.8) that for all k ∈ N,

Eε k (v k , ω) Eε k ( g, ω) = E( g, ω) .
Therefore sup k Eε k (v k , ω) < ∞, and we can extract a (not relabeled) subsequence such that the conclusions of Theorem 1.1 do hold for a map v * ∈ H 1/2 g (ω; R m ) which is a bounded weak 1/2-harmonic map into S m-1 in ω, and a finite measure µsing on ω. By the compact embedding H 1/2 (ω) ֒→ L 2 (ω), we may assume that v k → v * strongly in L 2 (ω). Since v k = v * = g outside ω, we deduce that v k → v * a.e. in R n , eventually up to a further subsequence. By (7.7) and the lower semicontinuity of the 1/2-Dirichlet energy with respect to a.e. pointwise convergence, we have

E(v, ω) lim inf k→∞ Eε k (v k , ω) lim inf k→∞ E(v k , ω) E(v * , ω)
for all v ∈ H 1/2 g (ω; S m-1 ). Hence v * is a minimizing 1/2-harmonic map into S m-1 in ω, and item (iv) follows from Theorem 1.2. On the other hand, the minimality of v k yields Eε k (v k , ω) Eε k (v * , ω) = E(v * , ω) .

Combining the two previous inequalities leads to item (i). In turn, it implies that µsing = 0, and then yields item (ii). The conclusion in item (iii) is a straightforward consequence of Theorem 6.1 -item (iii)together with the fact that µsing = 0. It now only remains to prove the strong convergence of v kv * to 0 in H 1/2 00 (ω). To this purpose, we have to prove that lim k→∞ (-∆) (7.9) Indeed, if (7.9) holds, then item (i) yields

[v k -v * ] 2 H 1/2 (R n ) = E(v k -v * , ω) = E(v k , ω) + E(v * , ω) -(-∆) 1 2 v k , v * ω -→ k→∞ 0 .
To show (7.9) we first write (-∆) , (8.3) which ends this first part. Remark 8.1 (Proof of Lemma 2.5). Notice that the first inequality in (8.3) shows the continuity of the linear operator PR defined in (2.13).

To complete the proof of Lemma 2.7, we shall need the following smooth approximation result. and define for 0 < δ < δ0, ω δ := {x ∈ ω : dist(x, ω ′ ) < δ} , ω -δ := {x ∈ ω ′ : dist(x, ∂ω ′ ) > δ} .

For 0 < ε k < δ < δ0, we estimate through Jensen's inequality,

ω ′ ×ω ′ |v k (x) -v k (
We have thus proved that

E(v k , ω ′ ) -→ k→∞ E(v, ω ′ ) .
In particular, v k ∈ H 1/2 (ω ′ ; R m ) for k large enough.

Setting V k (x, y) := (v k (x)-v k (y)), (8.5) implies that the sequence {V k } is bounded in L 2 (ω ′ ×(ω ′ ) c , Λ)

for the weighted measure Λ := |x -y| -n-1 L n x ⊗ L n y . Hence we can extract a subsequence such that V k ⇀ w weakly L 2 (ω ′ × (ω ′ ) c , Λ). On the other hand, from the convergence of v k to v in L 2 loc (R n ) we deduce that for all F ∈ L 2 (ω ′ × (ω ′ ) c , Λ) compactly supported in ω ′ × ω ′ c , Combing (8.6) with the convergence of vn in H 1/2 (ω ′ ; R m ), we infer that (-∆)

1 2 v k , v ω ′ -→ k→∞ (-∆) 1 2 v, v ω ′ = 2E(v, ω ′ ) ,
and we finally conclude

E(v k -v, ω ′ ) = E(v, ω ′ ) -(-∆) 1 2 v k , v ω ′ + E(v k , ω ′ ) -→ k→∞ 0 ,
which proves (ii).

It now remains to show (iii). We consider an arbitrary radius R > 0, x0 ∈ ω ′ and ρ > 0 such that D2ρ(x0) ⊆ ω ′ . Noticing that (v kv) e = v e kv e , we deduce from (8.3) that

B + R (x 0 ) |v e k -v e | 2 dx CR,ρ E v k -v, D2ρ(x0) + v k -v 2 L 2 (D 2ρ (x 0 )) CR,ρ E(v k -v, ω ′ ) + v k -v 2 L 2 (D 2ρ (x 0 )) → 0 ,
and the proof is complete. Using the fact that v ∈ H 1/2 (D2ρ(x0); R m ) and ϕ ∈ H

1 2 1 2 2 π n+1 2

 1122 on a smooth bounded function v : R n → R m is defined by(-∆) v(x) := p.v. γn R n v(x)v(y) |x -y| n+1 dywith γn := Γ n+1

1 2 1 2

 11 v on an element ϕ ∈ D(ω; R m ) by setting(-∆) v, ϕ ω := γn 2 ω×ω v(x)v(y) • ϕ(x)ϕ(y) |x -y| n+1 dxdy + γn ω×(R n \ω) v(x)v(y) • ϕ(x)ϕ(y) |x -y| n+1 dxdy .This formula turns out to define a distribution on ω wheneverv ∈ L 2 loc (R n ; R m ) satisfies E(v, ω) := γn 4 ω×ω |v(x)v(y)| 2 |x -y| n+1 dxdy + γn 2 ω×(R n \ω) |v(x)v(y)| 2 |x -y| n+1 dxdy < ∞ . (1.3)

  R n \ω |v(x)v(y)| 2|x -y| n+1 dy .(1.5)

  Dirichlet energy with respect to perturbations in the target. If the target is a sphere, it corresponds to variations of the Dirichlet energy of the form t → v * +tϕ |v * +tϕ| for bounded test functions ϕ and |t| ≪ 1. It leads to the Euler-Lagrange equation (in the sense of distributions) -∆v * = |∇v * | 2 v * ⇐⇒ -∆v * ⊥ Tv * S m-1 .

1 2 as

 2 Dirichlet-to-Neumann operator associated to the harmonic extension to the open half space R n+1 +

  and A + the closure of A + in R n+1 . If A ⊆ R n and no confusion arises, the complement of A in R n is simply denoted by A c . The Froebenius inner product between two matrices M = (Mij) and N = (Nij) is denoted by M : N := i,j MijNij. Concerning bounded open sets Ω ⊆ R n+1 + , we shall say that Ω is an admissible open set whenever • ∂Ω is Lipschitz regular;

Proof.

  Let us fix for the whole proof an admissible bounded open set Ω ⊆ R n+1 + with smooth boundary such that ω ⊆ ∂ 0 Ω. We shall use an argument in the spirit of the proof by BREZIS & KATO for the standard Laplacian.

Remark 3 . 7 . 1 , 1 . 4 . 1 / 2 - 1 .

 37114121 n+1 + by (2.11). If m(x0) < 0 we obtain ∂ν mε(x0) > 0. On the other hand, by the strong maximum maximum principle and the Hopf boundary lemma, we have ∂ν mε(x0) < 0 which leads to contradiction. In the case where the exterior condition g satisfies g L ∞ (R n \ω) Corollary 3.6 provides the estimate |vε| 1 which is very standard for the usual (local) Ginzburg-Landau equation. We emphasize that, in this case, we have |v e ε | HARMONIC MAPS: DEFINITIONS, REGULARITY, AND EXAMPLES 4.Definitions and regularity theory. In this subsection we assume that ω ⊆ R n is a bounded open set with Lipschitz boundary. We start introducing the concept of 1/2-harmonic map in ω with values into the

(4. 6 ) 4 . 5 (

 645 Remark Harmonic maps with partially free boundary). In view of the discussion above, let us mention that (S m-1 , ∂ 0 Ω)-boundary harmonic maps belong to a larger class of harmonic maps known in the literature as harmonic maps with partially free boundary, see[START_REF] Baldes | Harmonic mappings with a partially free boundary[END_REF][START_REF] Duzaar | A mixed boundary value problem for energy minimizing harmonic maps[END_REF][START_REF] Duzaar | A partial regularity theorem for harmonic maps at a free boundary[END_REF][START_REF] Duzaar | An optimal estimate for the singular set of a harmonic map in the free boundary[END_REF][START_REF] Gulliver | Harmonic maps which solve a free-boundary problem[END_REF][START_REF] Hamilton | Harmonic maps of manifolds with boundary[END_REF][START_REF] Hardt | Partially constrained boundary conditions with energy minimizing mappings[END_REF][START_REF] Scheven | Partial regularity for stationary harmonic maps at a free boundary[END_REF] and references therein. In most studies, one considers a smooth compact Riemannian manifold M without boundary (that we can assume to be isometrically embedded in some Euclidean space by the Nash embedding Theorem), and N a smooth closed submanifold of M. The boundary portion ∂ 0 Ω is called the partially free boundary, and N is the supporting manifold. Then M-valued (weak) harmonic maps in Ω with the partially free boundary condition u(∂ 0 Ω) ⊆ N are defined as critical points of the Dirichlet energy under the constraints u(x) ∈ M for a.e. x ∈ Ω and u(x) ∈ N for H n -a.e. x ∈ ∂ 0 Ω. For (S m-1 , ∂ 0 Ω)-boundary harmonic maps, we may consider the submanifold N = S m-1 of the target M = R m . However, to apply known results on harmonic maps, the compactness of M is usually required. Similarly to[START_REF] Moser | Intrinsic semiharmonic maps[END_REF] Sec. 3], a way to avoid this problem is to consider bounded (S m-1 , ∂ 0 Ω)-boundary harmonic maps, noticing that S m-1 can be viewed as a submanifold of a flat torus M

Definition 4 . 8 .

 48 Let Ω ⊆ R n+1 + be an admissible bounded open set, and let u ∈ H 1

Definition 4 . 10 .

 410 Let Ω ⊆ R n+1 + be an admissible bounded open set, and let

Proposition 4 . 15 (

 415 Stationarity transfer). Let v ∈ H 1/2 (ω; R m ) be a stationary weak 1/2-harmonic map into S m-1 in ω. Let v e be the harmonic extension of v in R n+1 +

  a weak 1/2-harmonic map into S m-1 in ω, and let Ω ⊆ R n+1 + be an arbitray admissible bounded open set such that ∂ 0 Ω ⊆ ω. By Proposition 4.6, v e is a weak (S m-1 , ∂ 0 Ω)-boundary harmonic map in Ω. If n = 1, then Theorem 4.18 shows that v e ∈ C ∞ (Ω ∪ ∂ 0 Ω), whence v ∈ C ∞ (ω) by arbitrariness of Ω. Let us now assume that n 2, and that v is stationary. By Proposition 4.15, v e is stationary. Since sing(v) ∩ ∂ 0 Ω = sing(v e ) ∩ ∂ 0 Ω, Theorem 4.18 yields H n-1 (sing(v) ∩ ∂ 0 Ω) = 0, and the conclusion follows by monotone convergence letting ∂ 0 Ω ↑ ω.

2 + 2 +

 22 into the closed unit disc D. Proof. First recall that v e is smooth in R by Theorem 4.18, and v e takes values in D by (2.11). Let us now consider the Hopf differential H of v e defined by H(z) := |∂1v e | 2 -|∂2v e | 2 -2i(∂1v e • ∂2v e ) .

  and v is smooth away from the origin. Moreover, we deduce from Step 1 thatv e (x) = wg • S -1 x |x| . on R 2 \ {0} ,where p := S -1 x |x| ∈ S 1 and νp denotes the exterior normal to ∂D ≃ S 1 at the point p. In view of (4.18) we have ∂v e ∂ν (x) ∧ v(x) = 1 |x| ∂wg ∂νp (p) ∧ g(p) = 0 on R 2 \ {0} .

  := uε(x)ūε . Denote by Gx 0 the fundamental solution in R n+1 of the Laplace equation with pole at x0. We recall that |∇Gx 0 (x)| C |x -x0| n for all x ∈ R n+1 \ {x0} . (5.8) Then consider a smooth cut-off function ζ ∈ C ∞ c (B2R(x0); [0, 1]) such that ζ ≡ 1 in BR(x0), ζ ≡ 0 in B2R(x0) \ B 3R/2 (x0), and |∇ζ| C/R.

. 14 )

 14 Now we claim that1 -|ūε| C √ η1 . (5.15) Setting for y ∈ R m , d(y) := |1 -|y||, the function d is 1-Lipschitz. Since d(ūε) d(uε(x)) + |uε(x) -ūε|, we have

5 . 2 . 1 , Step 1 .

 5211 ), and (5.18), yields (5.15) Now (5.14) and (5.15) imply that 1 -|uε(x0)| C √ η1, and thus |uε(x0)| 1/2 whenever η1 is chosen small enough, smallness depending only on the dimensions n and m. Proof of Theorem 5.By rescaling variables, it suffices to consider the case R = 1 and ε ∈ (0, 1]. We shall choose η0 η1 where η1 is given by Lemma 5.4, so that

. 21 )

 21 and xε ∈ B + σε such that 2|∇uε(xε)| 2 sup x∈B + σε |∇uε| 2 =: eε . (5.22) Denote by xε ∈ Dσ ε the orthogonal projection of xε on R n , and set

|∇uε| 2 4eε . ( 5 . 25 )

 525 Let us now introduce the quantity rε := ρε √ eε .

2 *

 2 B1) for every 0 < α < 1 , (5.39) for a map w * satisfying |w * | = 1, |∇w * | 4, and solving  ∇w * ) = a 2 * |∇w * | 2 w * in B

. 42 )

 42 again by elliptic regularity. As a consequence, a * solves     -∆a * + |∇w * | 2 a * = 0 in B + 1 , a * = 1 on D1 .

(5. 43 )

 43 Hence a * ∈ C 1,α loc (B + 1 ∪ D1) for every 0 < α < 1, and more precisely [36, Corollary 8.36] yields a * C 1,α (B + r )

ζ 2

 2 dH n defined over the affine space {ζ ∈ H 1 (B + r/2 ) :ζ = 1 on ∂ + B r/2 }. By Lemma 9.1 in Appendix B, ζ k ∈ C 0,β (B + r/2 ) ∩ C ∞ (B + r/2 \ ∂D r/2) for some β ∈ (0, 1), and0 ζ k Cr ε k on D r/4 ,(5.51)for a constant Cr which only depends on n and r.

  33)), the improved convergence of {w k } implies the convergence of a k in C ℓ+1,α loc (B + 1 ) for every 0 < α < 1. It then remains to improve the convergence of a k (or equivalently h k ) up to the boundary D1. As in the proof of Lemma 5.6, we consider an arbitrary x0 ∈ D1, a radius 0 < 2r < dist(x0, ∂D1), and α ∈ (0, 1). Again we can assume that x0 = 0 without loss of generality. Let us now fix a multi-index β = (β1, . . . βn) ∈ N n of length |β| :

+ 1 / 2 . 1 / 2 .

 1212 Thanks to Lemma 5.6 we have ∇v k (z k ) → ∇v * (z * ). Then (5.29) leads to |∇v * (z * )| 2 From (5.45)

3 7

 3 .25) imply that 1/2 |ũε| 1 and |∇ũε| 2 in B + 1 . Considering the point zε := √ eε(xεxε) ∈ B + 1/2 (which actually belongs to the half axis {0} × R+ and satisfies |zε| = 3Rε √ eε ), we observe that (5.22) yields |∇ũε(zε)| 2 1 2 .

Theorem 6 . 1 .

 61 Let Ω ⊆ R n+1 + be an admissible bounded open set. Let ε k ↓ 0 be an arbitrary sequence, and let

  then there exist a (not relabeled) subsequence and a bounded weak (S m-1 , ∂ 0 Ω)boundary harmonic map u * in Ω such that u k ⇀ u * weakly in H 1 (Ω) as k → ∞. In addition, there exist a finite nonnegative Radon measure µsing on ∂ 0 Ω and a relatively closed set Σ ⊆ ∂ 0 Ω of finite (n -1)-dimensional Hausdorff measure such that

Theorem 6 . 4 .

 64 Assume that n 2. In Theorem 6.1, the limiting (S m-1 , ∂ 0 Ω)-boundary harmonic map u * and the defect measure µsing represented in(6.13) satisfyΩ |∇u * | 2 divX -2 n+1 i,j=1 (∂iu * • ∂ju * )∂jXi dx + Σ θ divΣX dH n-1 = 0 for all vector fields X = (X1, . . . , Xn+1) ∈ C 1 (Ω; R n+1 ) compactly supported in Ω ∪ ∂ 0 Ω and satisfying Xn+1 = 0 on ∂ 0 Ω.In the spirit of[START_REF] Lin | Harmonic and quasi-harmonic spheres. III. Rectifiability of the parabolic defect measure and generalized varifold flows[END_REF], the proof of Theorem 6.4 relies on the notion of generalized varifold introduced by AMBROSIO-SONER[START_REF] Ambrosio | A measure-theoretic approach to higher codimension mean curvature flows[END_REF]. To simplify the proof of Theorem 6.4, we shall assume that the admissible open set Ω ⊆ R n+1 + satisfies Ω = Ω ∩ R n+1 + for some Lipschitz bounded open set Ω ⊆ R n+1 which is symmetric with respect to the hyperplane R n = {xn+1 = 0}. Since all the arguments are local in nature, the general case can be handled in a similar way with minor modifications. According to this symmetry assumption, we introduce some notations. For B ⊆ R n+1 , we write B + := B ∩ R n+1 + and B -:= B ∩ R n+1 -, where R n+1 -:= R n × (-∞, 0). For a map u ∈ H 1 (Ω; R m ), we denote by ũ ∈ H 1 ( Ω; R m ) the extension of u to Ω obtained by even reflection across ∂ 0 Ω, i.e., ũ(x ′ , xn+1) :=    u(x ′ , xn+1) for x = (x ′ , xn+1) ∈ Ω , u(x ′ , -xn+1) for x = (x ′ , xn+1) ∈ Ω -:= Ω -, and we write u -:= ũ|Ω -.

  ĀV sing (x0) : ∇X dH n-1 .

Tx 0 Σ

 0 ∂jφ dH n-1 = 0 for all j ∈ {1, . . . , n -1} , and the inclusion F ⊆ Ker AT x 0 Σ follows. To prove the reverse inclusion it suffices to use admissible functions of the form φ(x) = χ(xn)ψ(x ′′ ) or φ(x) = χ(xn+1)ψ(x ′′ ) where we write x = (x ′′ , xn, xn+1), χ ∈ C 1 c (R) satisfies χ ′ (0) = 1, and ψ ∈ C 1 c (R n-1 ) is such that R n-1 ψ = 1.

Theorem 7 . 1 .

 71 Let ω ⊆ R n be a bounded open set with Lipschitz boundary. Let ε k ↓ 0 be an arbitrary sequence, and let {v k

. 1 )

 1 If sup k Eε k (v k , ω) < ∞, then there exist a (not relabeled) subsequence and v * ∈ H 1/2 (ω; R m ) a bounded weak1/2-harmonic map into S m-1 in ω such that v k ⇀ v * weakly in H 1/2 (ω) ∩ L 2 loc (R n ) as k → ∞.In addition, there exist a nonnegative Radon measure µsing on ω, a countably H n-1 -rectifiable relatively closed set Σ ⊆ ω of locally finite (n -1)-dimensional Hausdorff measure in ω, and a Borel function θ : Σ → (0, ∞) such that (i) |∇v e k | 2 L n+1 R n+1 + * ⇀ |∇v e * | 2 L n+1 R n+1 + + µsing locally weakly* as Radon measures on R n+1 + ∪ ω;

7. 2 .

 2 Asymptotics with Dirichlet exterior condition. This section is devoted to proof of Theorem 1.1. To this aim, we consider for the rest of this subsection a smooth bounded open set ω ⊆ R n , and a smooth exterior Dirichlet condition g : R n → R m satisfying |g| = 1 in R n \ ω.

1 / 2 g

 12 (ω; R m ). It now remains to prove that(a) v kv * ⇀ 0 weakly in H 1/2 00 (ω); (b)µsing is a finite measure on ω, H n-1 (Σ) < ∞, and statements (i) and (iii) in Theorem 1.1 hold.

. 6 )

 6 as k → ∞. Gathering (7.4),(7.5), and (7.6) leads to ω e(v k , ω)ϕ dx -→ k→∞ ω e(v * , ω)ϕ dx + ω ϕ dµsing , and thus µ def = µsing by the arbitrariness of ϕ.

7. 3 .Theorem 7 . 4 .

 374 Asymptotics for Dirichlet minimizers. We finally consider solutions of the minimization problem (3.2), and we show that, in this case, no concentration occurs by minimality. Let ω ⊆ R n be a smooth bounded open set, and let g : R n → R m be a smooth map satisfying |g| = 1 in R n \ ω, and such that H 1/2 g (ω; S m-1 ) := v ∈ H 1/2 g (ω; R m ) : |v| = 1 a.e. in ω = ∅ . (7.7)

1 2 1 2

 11 v k , v * ω = (-∆) v * , v * ω = 2E(v * , ω) .

1 2 2 ω×ω 2 . 2 ( 2 L 2 ( 2 L 2 ( 2 ( 2 ( 1 ) 2 L 2 (

 1222222222122 v k , v * ω = γn 2 ω×ω (v k (x)v k (y)) • (v * (x)v * (y)) |x -y| n+1 dxdy + γn ω×ω c (v k (x)v * (y)) • (v * (x)v * (y))|x -y| n+1 dxdy =:I k + II k .Since v k ⇀ v * weakly in H 1/2 (ω), we haveI k -→ k→∞ γn |v * (x)v * (y)| 2 |x -y| n+1 dxdy . Next we set V k (x, y) := (v k (x)v * (y)) = (v k (x)v k (y)). Then {V k } is bounded in L 2 (ω × ω c , Λ) for the weighted measure Λ := |x -y| -n-1 L n x ⊗ L n y .Extracting a further subsequence if necessary, V k is thus weakly converging in L 2 (ω × ω c , Λ) to some function V . Using the strong convergence in L 2 (ω) of v k , we can argue as in the proof of Lemma 8.2 in Appendix A to show that V (x, y) = (v * (x)v * (y)).As a consequence,lim k→∞ γn ω×ω c (v k (x)v * (y)) • (v * (x)v * (y)) |x -y| n+1 dxdy = γn ω×ω c |v * (x)v * (y)| 2 |x -y| n+1 dxdy ,and (7.9) is proved.8. APPENDIX AThe purpose of this first appendix is to give detailed proofs of the different auxiliary results stated inSection Proof of Lemma 2.1. Let ω ⊆ R n be a bounded open set, and v ∈ H 1/2 (ω; R m ). Let us fix x0 ∈ ω and ρ > 0 such that D2ρ(x0) ⊆ ω. Then, D c 2ρ (x 0 )×Dρ(x 0 ) |v(x)v(y)| 2 |x -y| n+1 dxdy CE v, D2ρ(x0) .Next we estimateD c 2ρ (x 0 )×Dρ(x 0 ) |v(x)v(y)| 2 |x -y| n+1 dxdy D c 2ρ (x 0 )×Dρ(x 0 ) |v(x)| -|v(y)| |x -x0| + |y -x0|) n+1 dxdy Cρ D c 2ρ (x 0 )×Dρ(x 0 ) |v(x)| 2 -2|v(y)| 2 (|x -x0| + 1) n+1 dxdy , which yields D c 2ρ (x 0 ) |v(x)| 2 (|x -x0| + 1) n+1 dx Cρ E v, D2ρ(x0) + v Dρ(x 0 )) . Hence, R n |v(x)| 2 (|x -x0| + 1) n+1 dx Cρ E v, D2ρ(x0) + v D 2ρ (x 0 )), and the proof is complete.Proof of Lemma 2.7, Part 1. We prove in this first part that v e ∈ L 2 loc (R n+1 + ). We fix R > 0 arbitrary, x0 ∈ ω and ρ > 0 such that D2ρ(x0) ⊆ ω. We claim that v e ∈ L 2 (B + R (x0)). Using Jensen's inequality we estimate forx = (x ′ , xn+1) ∈ B + R (x0), |v e (x)| 2 γn R n xn+1|v(z)| 2 (|x ′ -z| 2 + x 2 n+1 ) |z -x0| 2 -2R|z -x0|) |x ′ -z| 2 + x 2 n+1 ) 2 (|z -x0| + 1) n+1 dz + γn D 5R (x 0 ) xn+1|v(z)| 2 (|x ′ -z| 2 + x 2 n+1 )Then we estimate for 0 < xn+1 < R,D R (x 0 ) D 5R (x 0 ) xn+1|v(z)| 2 (|x ′ -z| 2 + x 2 n+1 ) n+1 2 dz dx ′ = D R (x 0 ) D 5R (x ′ ) xn+1|v(x ′y + x0)| 2 (|y -x0| 2 + x 2 n+1 ) n+1 2 dy dx ′ D 6R (x 0 ) D R (x 0 ) xn+1|v(y + x0x ′ )| 2(|y -x0| 2 + x 2 n+1 ) and (8.2) we deduce from Lemma 2.1 that B + R (x 0 ) |v e (x)| 2 dx CR R n |v(z)| 2 (|z -x0| + 1) n+1 dz CR,ρ E v, D2ρ(x0) + v D 2ρ (x 0 ))

Lemma 8 . 2 .

 82 Let ω ⊆ R n be a bounded open set, and let v ∈ H 1/2 (ω; R m ). There exists a sequence {v k } k∈N of smooth functions such that v k → v strongly in L 2 loc (R n ), and such that for any relatively compact open subset ω ′ ⊆ ω with Lipschitz boundary,(i) v k ∈ H 1/2 (ω ′ ; R m ) for k large enough; (ii) E(v kv, ω ′ ) → 0 as k → ∞; (iii) v e k → v e in L 2 loc (Rn+1+ ) as k → ∞.Proof. Let us fix a sequence ε k ↓ 0, and a nonnegative function̺ ∈ C ∞ c (R n ) compactly supported in D1 satisfying D 1 ̺ dx = 1. Setting ̺ k (x) := ε -n k ̺(x/ε k ), we define v k (x) := Dε k ̺ k (z)v(x + z) dz . Then v k ∈ C ∞ (R n ; R m ), v k → v strongly in L 2 loc (R n ).Extracting a subsequence if necessary, we may assume that v k → v a.e. in R n . Let us now consider a relatively compact open subset ω ′ ⊆ ω with Lipschitz boundary. Set δ0 := min dist(ω ′ , ∂ω), sup x∈ω ′ dist(x, ∂ω ′ ) ,

ω

  ′ ×(ω ′ ) c |v k (x)v k (y)| 2 |x -y| n+1 dxdy ω δ ×ω c -δ |v(x)v(y)| 2 |x -y| n+1 dxdy .Arguing as above, we apply Fatou's lemma as k → ∞, and then the monotone convergence theorem as δ ↓ 0 to deduce thatω ′ ×(ω ′ ) c |v k (x)v k (y)| 2 |x -y| n+1 dxdy -→ k→∞ ω ′ ×(ω ′ ) c |v(x)v(y)| 2|x -y| n+1 dxdy .

ω

  ′ ×(ω ′ ) c V (x, y) • F (x, y) |x -y| n+1 dxdy = lim k→∞ ω ′ ×(ω ′ ) c V k (x, y) • F (x, y) |x -y| n+1 dxdy = ω ′ ×(ω ′ ) c (v(x)v(y)) • F (x, y) |x -y| n+1 dxdy . Functions with compact support in ω ′ × ω ′ c being dense in L 2 (ω ′ × (ω ′ ) c , Λ), we conclude that ω ′ ×(ω ′ ) c V (x, y) • F (x, y) |x -y| n+1 dxdy = ω ′ ×(ω ′ ) c (v(x)v(y)) • F (x, y) |x -y| n+1 dxdy for all F ∈ L 2 (ω ′ × (ω ′ ) c , Λ). The Riesz Representation Theorem then yields V (x, y) = (v(x)v(y)). As a consequence, ω ′ ×(ω ′ ) c (v k (x)v k (y)) • (v(x)v(y)) |x -y| n+1 dxdy -→ k→∞ ω ′ ×(ω ′ ) c|v(x)v(y)| 2 |x -y| n+1 dxdy . (8.6)

Proof of Lemma 2 . 7 , 2 DD

 272 Part 2. Step 1. Let us first consider the case where v ∈ H 1/2 (ω;R m ) ∩ C ∞ (R n ). Then v e is smooth in R n+1 + . Let x0 ∈ ω and ρ > 0 such that D2ρ(x0) ⊆ ω. We consider a cut-off function χ ∈ C ∞ (R n+1 ; [0, 1]) satisfying χ = 1 in Bρ(x0), and χ = 0 in R n+1 \ B2ρ(x0). We set Φ := χ 2 v e and ϕ := Φ |R n = χ 2 v. From the harmonicity of v e in R n+1 + we infer that R n+1 + ∇v e • ∇Φ dx = R n ∂v e ∂ν • ϕ dx = lim δ↓0 R n v(x)v e (x, δ) • ϕ(x) δ dx .On the other hand, applying formula (2.9) to represent v e yieldsR n v(x)v e (x, δ) • ϕ(x) δ dx = γn R n ×R n (v(x)v(y)) • ϕ(x) (|x -y| 2 + δ 2 ) 2ρ (x 0 )×D 2ρ (x 0 ) (v(x)v(y)) • (ϕ(x)ϕ(y)) (|x -y| 2 + δ 2 ) 2ρ (x 0 )×D c 2ρ (x 0 ) (v(x)v(y)) • (ϕ(x)ϕ(y)) (|x -y| 2 + δ 2 ) n+12dxdy .

1 / 2 00 1 2+ 2 L 2 2 L 2 4 D

 12122224 (D2ρ(x0); R m ), we derive by dominated convergence thatlim δ↓0 R n v(x)v e (x, δ) • ϕ(x) δ dx = (-∆) v, ϕ D 2ρ (x 0 ) .As in (2.7), it then follows from Cauchy-Schwarz Inequality thatR n+1 + ∇v e • ∇Φ dx E(v, D2ρ(x0)) E(ϕ, D2ρ(x0)) 1 2 E(v, D2ρ(x0)) + E(ϕ, D2ρ(x0)).∇v e • ∇Φ dx B + 2ρ (x 0 ) χ 2 |∇v e | 2 dx -2 B + 2ρ (x 0 ) χ|v e | |∇v e | |∇χ| dx 1 2 B + 2ρ (x 0 ) χ 2 |∇v e | 2 dx -2 B + 2ρ (x 0 ) |v e | 2 |∇χ| 2 dx 1 2 B + ρ (x 0 ) |∇v e | 2 dx -Cρ v e (B + 2ρ (x 0 )) ,(8.8)for a constant Cρ > 0 independent of v. Then (8.7) and (8.8) yieldB + ρ (x 0 ) |∇v e | 2 dx E(v, D2ρ(x0)) + E(ϕ, D2ρ(x0)) + Cρ v e (B + 2ρ (x 0 )) ,(8.9)and it remains to estimate the second term in the right handside of this inequality. A straightforward computation yieldsE(ϕ, D2ρ(x0)) 2E(v, D2ρ(x0)) + 4ρ (x 0 ) D 2ρ (x 0 ) |χ 2 (x)χ 2 (y)| 2 |x -y| n+1 dx |v(y)| 2 dy + 4 D c 4ρ (x 0 ) D 2ρ (x 0 ) 1 |x -y| n+1 dx |v(y)| 2 dy .Noticing that for y ∈ D4ρ(x0),D 2ρ (x 0 ) |χ 2 (x)χ 2 (y)| 2 |x -y| n+1 dx Cρ D 2ρ (x 0 ) 1 |x -y| n-1 dx Cρ ,and that for y ∈ D c 4ρ (x0), D 2ρ (x 0 )

e k | 2 dx 1 2 1 2 1 2

 2111 Cρ E v, D2ρ(x0) + v 2 L 2 (D 2ρ (x 0 )) .Moreover, in view of the arbitrariness of x0, we conclude that v e ∈ H 1 (B + ρ (x); R m ) for all x ∈ ω and ρ > 0 such that D3ρ(x) ⊆ ω .(8.11) Finally, the conclusion v e ∈ H 1 loc (R n+1+∪ ω; R m ) follows from(8.11) together with a standard covering argument.Proof of Lemma 2.9. By the density of compactly supported smooth funtions in (2.2), we may assume without loss of generality that ϕ ∈ D(ω; R m ). Let Ω ⊆ R n+1 + be an admissible bounded open set such that supp ϕ ⊆ ∂ 0 Ω, and ∂ 0 Ω ⊆ ω. We then consider a smooth extension Φ of ϕ to R n+1+ which is compactly supported in Ω ∪ ∂ 0 Ω. Let {v k } ⊆ C ∞ (R n ; R m) be a sequence given by Lemma 8.2. Then,v k ∈ H 1/2 (∂ 0 Ω; R m ) for k large enough. Arguing as in the proof of Lemma 2.7 (Part 2, Step 1)v k , ϕ ∂ 0 Ω . By the proof of Lemma 2.7 (Part 2, Step 2), v e k ⇀ v e weakly in H 1 (Ω). Therefore, (v kv), ϕ ∂ 0 Ω ϕ H 1/2 00 (∂ 0 Ω) E(v kv, ∂ 0 Ω) v k , ϕ ∂ 0 Ω = (-∆) 1 2 v, ϕ ∂ 0 Ω = (-∆) 1 2 v, ϕ ω ,and the lemma is proved.

Remark 2.3. Consider an open set ω ′

							1 2 v
	to H ported in ω, i.e., 1/2 00 (ω; R m ) corresponds to the first variation of the energy E(v, ω) with respect to pertubations sup-
		(-∆)	1 2 v, ϕ ω =	d dt	E(v + tϕ, ω)	t=0	(2.8)
	for all ϕ ∈ H	1/2 00 (ω; R m ).

Harmonic extension and the Dirichlet-to-Neumann operator. Throughout

  the paper, for a measurable function v defined over R n , we shall denote by v e its extension to the half-space R n+1

	2.3. +	given by the
	convolution of v with the Poisson kernel, i.e.,
		v e (x) := γn	xn+1v(z)
			R n
						1 2 is the infinitesimal generator
	of the Poisson Kernel, i.e.,
	see e.g. [48].	e -t(-∆)	1 2 (x, y) =	γnt (|x -y| 2 + t 2 )	n+1 2	, t > 0 ,

Lemma 2.9. Let

  2 (ω; R m ) for some bounded open set ω ⊆ R n with Lipschitz boundary, we now observe that v e admits a distributional (exterior) normal derivative ∂ν v e on ω by its harmonicity in R n+1 + . More precisely, we define ∂ν v e through its action on ϕ ∈ D(ω; R m ) by ω ⊆ R n be a bounded open set with Lipschitz boundary. For every v ∈ H 1/2 (ω; R m ) we have

		∂v e ∂ν	, ϕ :=	R n+1 +	∇v e • ∇Φ dx ,	(2.14)
	where Φ is any smooth extension of ϕ compactly supported in R n+1 + of (2.14) is well defined by Lemma 2.7. Using the harmonicity of v e and the divergence theorem, it is ∪ ω. Note that the right hand side
	routine to check that the integral in (2.14) does not depend on the choice of the extension Φ, and that
	+ ∂ν v e coincides with the classical exterior normal derivative of v e on ω ⊆ ∂R n+1 It turns out that ∂ν v e coincides with the distribution (-∆) 1 2 v defined in the previous subsection. Here whenever v is smooth.
	again the proof is left to Appendix A.
	(-∆)	1 2 v, ϕ ω =	∂v e ∂ν	, ϕ	for all ϕ ∈ H	1/2 00 (ω; R m ) ,
	where v e is the harmonic extension of v to R n+1 +	given by (2.9).
	Up to now, we have not said anything about the local counterpart of Lemma 2.6 concerning the min-
	imality of v e for a function v in H 1/2 (ω; R m ). This is the purpose of the following lemma inspired from
	[18, Lemma 7.2].
	Corollary 2.10. Let ω ⊆ R n be a bounded open set, and let Ω ⊆ R n+1 +	be an admissible bounded open set such

  be a minimizer of Eε in ω, and let v e

		ε
	be its harmonic extension to R n+1 +	given by (2.9). Then v e ε is a minimizer of Eε in every admissible bounded open
	set

  According to Remark 4.24, Lemma 4.27 still holds if S 1 is replaced by a smooth compact manifold N ⊆ R m without boundary, i.e., if v ∈ Ḣ1/2 (R; R m ) is a nontrivial entire 1/2harmonic map into N , then v e is conformal or anti-conformal. If m = 3 and N is a surface, it shows that the image of R 2 + by v e is a minimal surface in R 3 whose boundary lies in N , and meets N orthogonaly, see[START_REF] Struwe | On a free boundary problem for minimal surfaces[END_REF]. If N is a closed boundary curve, then the image of R 2 + by v e is a minimal surface spanned by N .Proof of Theorem 4.25. By Lemma 4.27 we may assume that v e is conformal (otherwise we simply consider the complex conjugate of v). We introduce the so-called Caley transform C : R

2 + ).

Remark 4.28 (Minimal surfaces).

2 + → D defined by

  |∇T ( v e + tΦ)| 2 dH 2 .

									+ ), we can argue as in Step 1 to obtain
					1 2 D	|∇(wg + twϕ)| 2 dx =	1 2 S 2 +
	Consequently,
	d dt	E	g + tϕ |g + tϕ|	, S 1	t=0	=	d dt	1 2 S 2 +	|∇T ( v e + tΦ)| 2 dH 2	t=0	=	S 2 +	∇T v e • ∇T Φ dH 2 . (4.20)

  2 eε Cη0 .

	Choosing σ = 1/4 in (5.21) yields
	sup B + 1/4	|∇uε| 2 Cη0 ,
	which is the announced estimate.
		Rε	7	1 √ eε	.	(5.23)
	Indeed, assume that (5.23) does not hold. Then, arguing as in Case 1), we obtain
	eε	C ε R 2	η0 Ceεη0 .
	Hence 1 Cη0, which is impossible whenever η0 is small enough.
	Next we assume that η0 is sufficiently small so that (5.23) holds. Then,
	|xε -xε| min	2	1 √ eε	,	ρε 2	.
	Noticing that B + ρε (xε) ⊆ B + σε+ρε , we deduce that
	sup B + ρε (xε)	|∇uε| 2		sup B + σε +ρε	|∇uε| 2 .	(5.24)
	Since σε + ρε ∈ (0, 1/2), we infer from (5.21) and (5.24) that
		sup
	B + ρε (xε)

Case 2)

. We now assume that Rε ρε/7, and we claim that if η0 is small enough, only depending on the dimensions n and m, then

  2 ε eε 4 . We are now going to prove (5.26) by contradiction, assuming that rε > 1. The following proposition is the key point of the argument. If η0 > 0 is small enough (depending only on nand m), then there exists a constant ς0 > 0 (depending only on n and m) such that for each ε > 0 and each map vε ∈ C 2 (B

	Choosing σ = 1/4 in the inequality above, we obtain
	sup B + 1/4	|∇uε| 2 64 Cη0 ,
	as desired.
	5.3. Proof of Theorem 5.1, Step 2. Proposition 5.5.

+ 1

  the limiting 1/2-harmonic map v * and the defect measure µsing satisfy From the assumptions |v k | 1 and sup k Eε k (v k , ω) < ∞, we first deduce from Lemma 2.1 that the sequence {v k } is bounded in L 2 (R n , m), where the measure m is defined in(2.10). Therefore, we can find a subsequence and v * ∈ L 2

	d dt	E v * • φt, ω	t=0	=	1 2 Σ	divΣX dµsing
	for all vector fields X ∈ C 1 (R Proof. Step 1.

n ; R n ) compactly supported in ω, where {φt} t∈R denotes the flow on R n generated by X;

(vi) if n = 1, the set Σ is locally finite in ω and v * ∈ C ∞ (ω).

  y)| 2 |x -y| n+1 dxdy ω ′ ×ω ′ Dε k ̺ k (z) |v(x + z)v(y + z)| 2 |x -y| n+1 dxdy dz Hence {v k } is bounded in H 1/2 (ω ′ ; R m ), and since v k → v in L 2 (ω ′ ) we deduce that v k ⇀ v weakly in H 1/2 (ω ′ ; R m). Now we infer from (8.4) and Fatou's lemma thatω ′ ×ω ′ |v(x)v(y)| 2 |x -y| n+1 dxdy lim inf k→+∞ ω ′ ×ω ′ |v k (x)v k (y)| 2 |x -y| n+1 dxdy lim sup k→+∞ ω ′ ×ω ′ |v k (x)v k (y)| 2 |x -y| n+1 dxdy ω δ ×ω δ |v(x)v(y)| 2 |x -y| n+1 dxdy . Since [v] 2 H 1/2 (ω δ ) → [v] 2 H 1/2 (ω ′ )as δ ↓ 0 by monotone convergence, we conclude from this last inequality and the arbitrariness ofδ that [v k ] 2 H 1/2 (ω ′ ) → [v] 2 H 1/2 (ω ′ ). Therefore v k → v strongly in H 1/2 (ω ′ ). Similarly to (8.4), we estimate

	Dε k	̺ k (z)	(z+ω ′ )×(z+ω ′ )	|v(x) -v(y)| 2 |x -y| n+1 dxdy dz
	ω δ ×ω δ	|v(x) -v(y)| 2 |x -y| n+1 dxdy .	(8.4)

  In view of Lemma 2.1, we have thus proved thatB + ρ (x 0 ) |∇v e | 2 dx Cρ E v, D2ρ(x0) + v 2 L 2 (D 2ρ (x 0 )) .Step 2. Let us now consider an arbitrary v ∈ H 1/2 (ω; R m ), and a sequence{v k } ⊆ C ∞ (R n ; R m )given by Lemma 8.2. Let x0 ∈ ω and ρ > 0 such that D3ρ(x0) ⊆ ω. Then, for k large enough,v k ∈ H 1/2 (D2ρ(x0); R m ). AppyingStep 1 to v k we infer that for k large enough,B + ρ (x 0 ) |∇v e k | 2 dx Cρ E(v k , D2ρ(x0)) + v k 2 L 2 (D 2ρ (x 0 )) .By Lemma 8.2, the right hand side of this inequality is uniformly bounded with respect to k. Since v e k → v e in L 2 (B + ρ (x0)) (still by Lemma 8.2), we deduce that v e k ⇀ v e weakly in H 1 (B + ρ (x0)). By lower semicontinuity and (ii) in Lemma 8.2, we obtain

			1 |x -y| n+1 dx	Cρ (|y -x0| + 1) n+1 ,
	we deduce that
	E ϕ, D2ρ(x0)	2E(v, D2ρ(x0)) + Cρ	R n	|v(z)| 2 (|z -x0| + 1) n+1 dz .	(8.10)
	Finally, gathering (8.9) with (8.3) and (8.10), we infer that
	B + ρ (x 0 )	|∇v e | 2 dx 3E(v, D2ρ(x0)) + Cρ	R n	|v(z)| 2 (|z -x0| + 1) n+1 dz .

B + ρ (x 0 ) |∇v e | 2 dx lim inf

APPENDIX B

This purpose of this last appendix is to provide the elliptic regularity results we have been using during the proof of Lemma 5.6. Lemma 9.1. For ε > 0, let uε ∈ H 1 (B + 1 ) be the unique (variational) solution of

, and

for some constant C > 0 which only depends on the dimension.

Proof. The H ölder continuity of uε (up to the boundary) follows from the general results of STAMPAC-CHIA [START_REF] Stampacchia | Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie h ölderiane[END_REF][START_REF] Stampacchia | Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus[END_REF] (see also [START_REF] Chicco | H ölder regularity for solutions of mixed boundary value problems containing boundary terms[END_REF]), while higher order regularity away from ∂D1 follows from standard elliptic regularity theory. The fact that uε > 0 in B + 1 is an easy consequence of the maximum principle and the Hopf boundary Lemma. Indeed, assume that x0 be a minimum point of uε such that uε(x0) 0. By the maximum principle x0 ∈ ∂B + 1 , and thus x0 ∈ D1 since uε = 1 on ∂ + B1 and uε is continuous on B + 1 . Since uε is smooth up to D1 (away from ∂D1), we can apply the Hopf boundary Lemma to infer that ∂ν uε(x0) < 0. Then ε∂ν uε(x0) + uε(x0) < 0 which contradicts the equation on D1.

Let us now consider the unique solution

It is well known that w ∈ C 0,1 (B + 1 ) ∩ C ∞ B + 1 \ ∂D1 by convexity of B + 1 (see e.g. [START_REF] Fromm | Potential space estimates for Green potentials in convex domains[END_REF]). We set vε := uεw, so that vε

Setting

we observe that κ only depends on the dimension. Next we define ūε ∈ H 1 (B + 1 ) as the unique (variational) solution of

As previously, ūε ∈ C 0,β (B + 1 ) ∩ C ∞ B + 1 \ ∂D1 for some 0 < β < 1, and by the maximum principle and the Hopf boundary Lemma, we have

Observe that

Indeed, consider the function h

, still by the maximum principle and the Hopf boundary Lemma. Finally, in view of (9.1) and according to [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]Theorem 6.26],

C , for a constant C > 0 which only depends on the dimension. In particular |∂ν ūε| C on D 1/2 , and thus

Since vε = uε on D1, the conclusion now follows from (9.2) and (9.3).

We

Then,

) ) for every 0 < r < 1 and a constant Cr,α > 0 which only depends on n, α, r.