Michael Goldman
email: goldman@math.univ-paris-diderot.fr

Benoît Merlet
email: benoit.merlet@math.univ-lille1.fr

Phase segregation

published or not. The documents may come

Introduction

In this paper, we investigate the asymptotic behavior as ε goes to zero of the Gross-Pitaievskii functional

F ε (η) = ε R 2 |∇η 1 | 2 + |∇η 2 | 2 + 1 ε R 2 1 2 η 4 1 + g 2 η 4 2 + Kη 2 1 η 2 2 + (η 2 1 + η 2 2 )V, (1)
under the mass constraint

R 2 η 2 1 = α 1 and R 2 η 2 2 = α 2 . (2)
This functional arises in the study of two component Bose-Einstein condensates. It has been widely studied, both in the physical and mathematical literature (see [START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF] and the references therein or the book [START_REF] Aftalion | Vortices in Bose-Einstein Condensates[END_REF]). The potential V is a trapping potential. For simplicity, we only consider here the harmonic potential V = |x| 2 . The constant g, measures the asymmetry between the intracomponent repulsive strengths of each component and K represents the intercomponent repulsive strength. Without loss of generality, we will take here g ≥ 1. The case K < √ g, where mixing of the two condensates occurs has recently been well understood in [START_REF] Aftalion | Thomas-Fermi approximation for coexisting two component Bose-Einstein condensates and nonexistence of vortices for small rotation[END_REF]. On the other hand, the case K > √ g, where it is expected both experimentally [START_REF] Mccarron | Dual-species Bose-Einstein condensate of 87 Rb and 133 Cs[END_REF][START_REF] Papp | Tunable miscibility in a dual-species Bose-Einstein condensate[END_REF], numerically [START_REF] Mason | Classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate[END_REF][START_REF] Öhberg | Hartree-Fock treatment of the two-component Bose-Einstein condensate[END_REF] and theoretically [START_REF] Kasamatsu | Vortices in multicomponent Bose-Einstein condensates[END_REF][START_REF] Ao | Binary Bose-Einstein condensate mixtures in weakly and strongly segregated phases[END_REF][START_REF] Timmermans | Phase separation of Bose-Einstein condensates[END_REF][START_REF] Barankov | Boundary of two mixed Bose-Einstein condensates[END_REF][START_REF] Van Schaeybroeck | Interface tension of Bose-Einstein condensates[END_REF] that segregation occurs, is maybe not yet so well understood.

In the symmetric case g = 1, it has been proved in [START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF][START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF] that, as expected from the physics literature, the Thomas-Fermi limit i.e. the limit of εF ε only imposes segregation but does not give any information about the actual shape of the minimizers. To gain such information, one has to go to the next order since F ε -min η F ε (η) converges in the sense of Γ-convergence [START_REF] Braides | Γ-convergence for beginners[END_REF] to a weighted isoperimetric problem. In this paper, we focus on the asymmetric case g > 1 and show that the situation is radically different.

Let us introduce the Thomas-Fermi energy:

E(ρ) = R 2 1 2 ρ 2 1 + g 2 ρ 2 2 + Kρ 1 ρ 2 + (ρ 1 + ρ 2 )V . (3)
Our first main theorem is the following:

Theorem 1.1. For every α 1 , α 2 , g, K > 0, with K ≥ √ g > 1, there exists a unique minimizer ρ 0 = (ρ 0 1 , ρ 0 2 ) of (3) under the volume constraints

R 2 ρ 1 = α 1 and R 2 ρ 2 = α 2 . ( 4
)
This minimizer ρ 0 is radially symmetric (see Lemma 2.1 for explicit formulas for ρ 0 and E 0 = E(ρ 0 )). Moreover, we have the following stability result: there exists C > 0 (which depends only on α 1 , α 2 and g) such that if ρ satisfies the constraints (4)

then 1 ρ -ρ 0 1 ≤ C E(ρ) -E 0 . (5)
Let us point out that the radial symmetry of the minimizer of the Thomas-Fermi energy was not completely expected (see [START_REF] Van Schaeybroeck | Interface tension of Bose-Einstein condensates[END_REF][START_REF] Mason | Classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate[END_REF] or the discussion in [START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF]Sec. 1.3.4]). As a consequence of this stability result, we prove that minimizers of F ε converge to ( ρ 0 1 , ρ 0 2 ).

Theorem 1.2. There exists C > 0 such that for ε ∈ (0, 1], any minimizer η ε of F ε under the constraints (2) satisfies

η ε - ρ 0 1 , ρ 0 2 2 ≤ Cε 1/4 . (6)
This theorem establishes that in the non-symmetric case, the Thomas-Fermi limit already provides full information on the limiting behavior of the minimizers of [START_REF] Acerbi | Minimality via second variation for a nonlocal isoperimetric problem[END_REF]. It is quite surprising that even without using isoperimetric effects, we are able to obtain strong convergence of the minimizers in the form of [START_REF] Alama | Domain walls in the coupled Gross-Pitaevskii equations[END_REF]. Let us point out that the idea of using stability inequalities such as [START_REF] Aftalion | Interface layer of a two-component Bose-Einstein condensate[END_REF] to get (quantitative) convergence results is far from new (see for instance [START_REF] Carlen | Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation[END_REF][START_REF] Bella | Study of island formation in epitaxially strained films on unbounded domains[END_REF]). One crucial point which explains the difference between the asymmetric case and the symmetric one is that here, there is a gap between the two Thomas-Fermi profiles ρ 0 1 and ρ 0 2 in the sense that there exists r 0 > 0 such that supp ρ 1 0 ⊂ B(0, r 0 ) and supp ρ 2 0 ⊂ R 2 \ B(0, r 0 ) with inf

r<r 0 ρ 0 1 (r) > sup r>r 0 ρ 0 2 (r).
We then study the crossover case where g = 1 + εξ for some ξ > 0. Let η ε be the minimizer of

G ε (η) = ε R 2 |∇η| 2 + 1 ε R 2 1 2 η 4 + V η 2
under the mass constraint

R 2 η 2 = α,
1 we denote by • p the L p norm where α = α 1 + α 2 . It is well known [START_REF] Ignat | The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate[END_REF][START_REF] Karali | The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit[END_REF] that η 2 ε converges when ε goes to zero to the Thomas-Fermi profile ρ = (R 2 -V )

+
where R is chosen such that R 2 ρ = α. Building on results obtained in the case g = 1 [START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF][START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF], we get Theorem 1.3. For K > 1, the functional (F ε -G ε (η ε )) Γ-converges as ε goes to zero for the strong L 1 topology to

G ξ (u 1 , u 2 ) =    σ K ∂E ρ 3/2 + ξ E c ρ 2 if u 1 = ρχ E , u 2 = ρχ E c and E ρ = α 1 , +∞ otherwise,
where σ K > 0 is defined by the one dimensional optimal transition problem

σ K = inf R |η ′ 1 | 2 + |η ′ 2 | 2 + 1 2 η 2 1 + η 2 2 -1 2 + (K -1)η 2 1 η 2 2 : lim -∞ η 1 = 0, lim +∞ η 1 = 1 .
We study the minimizers of the limiting functional G ξ and prove our second main result.

Theorem 1.4. The following holds:

• there exists α 0 ∈ (0, α/2] such that for every α 1 ∈ (α 0 , α -α 0 ) there exists ξ 1 α 1 such that the minimizer of G ξ is not radially symmetric for ξ ≤ ξ 1 α 1 ,

• for every α 1 ∈ (0, α), there exists ξ 2 α 1 such that the minimizer of G ξ is the centered ball for ξ ≥ ξ 2 α 1 .

The regime g = 1 + εξ corresponds to the numerical simulation of [START_REF] Mason | Classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate[END_REF]. In that paper, the observed numerical results (droplets) fit with the first point of Theorem 1.4 (see in particular [START_REF] Mason | Classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate[END_REF]Fig 1.c, Fig. 3.a]). The first part of Theorem 1.4 is a consequence of a symmetry breaking result from [START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF]. The second part follows from a combination of two results. The first is a stability result for the functional E c ρ 2 : Proposition 1.5. For every α ∈ (0, α), there exists C = C(α) > 0 such that for every measurable set E ⊂ R 2 with E ρ = α, we have,

E c ρ 2 - B c r ρ 2 ≥ C E∆Br ρ 2
where r is such that Br ρ = α.

The second is an estimate on the potential instability of the ball for the weighted isoperimetric problem.

Proposition 1.6. For every α ∈ (0, α), there exists c = c(α) > 0 such that for every set E ⊂ R 2 with locally finite perimeter and with E ρ = α, there holds

∂E ρ 3/2 - ∂Br ρ 3/2 ≥ -c E∆Br ρ 2 , ( 7
)
where r is such that Br ρ = α.

The rigidity result given by Theorem 1.4 is similar in spirit to several rigidity results obtained for variants of isoperimetric problems (see [START_REF] Knüpfer | On an isoperimetric problem with a competing nonlocal term II: The general case[END_REF][START_REF] Figalli | On the isoperimetric problem for radial log-convex densities[END_REF][START_REF] Goldman | Existence and Stability for a Non-Local Isoperimetric Model of Charged Liquid Drops[END_REF] for instance). However, the peculiar aspect here is that this rigidity does not come from the isoperimetric term but rather from the volume term. Nevertheless, the proof of [START_REF] Alikakos | On the connection problem for potentials with several global minima[END_REF] bounding the instability of the ball follows the strategy of [START_REF] Cicalese | A selection principle for the sharp quantitative isoperimetric inequality[END_REF] (see also [START_REF] Acerbi | Minimality via second variation for a nonlocal isoperimetric problem[END_REF][START_REF] Figalli | On the isoperimetric problem for radial log-convex densities[END_REF]) to prove quantitative stability estimates for isoperimetric problems. The idea is to show first the desired inequality for nearly spherical sets following ideas of [START_REF] Fuglede | Stability in the isoperimetric problem for convex or nearly spherical domains in R n[END_REF] and then use the regularity theory for minimal surfaces to reduce oneself to this situation. To the best of our knowledge, it is the first time that this strategy has been implemented to control the instability of the ball. Let us also point out that one of the ingredients in our proof is the following isoperimetric inequality: Lemma 1.7. There exists a constant c > 0 such that for every measurable set

E ⊂ R 2 satisfying E ρ ≤ α/2, there holds ∂E ρ 3/2 ≥ c E ρ 5/6 .
In recent years, there has been an increasing interest in studying isoperimetric problems with densities (see [START_REF] Morgan | Existence of isoperimetric regions in R n with density[END_REF][START_REF] Figalli | On the isoperimetric problem for radial log-convex densities[END_REF][START_REF] De Philippis | Existence of isoperimetric sets with densities[END_REF] for instance). However, most of these authors consider either problems where the same density is used for weighting the volume and the perimeter or weights which are increasing at infinity.

In the last part of the paper we come back to the situation g > 1 but consider an infinitely stiff trapping potential. That is, we assume that V is equal to zero inside some given open set Ω and is infinite outside. This is somehow the setting which is considered in [START_REF] Van Schaeybroeck | Interface tension of Bose-Einstein condensates[END_REF][START_REF] Van Schaeybroeck | Critical wetting, first-order wetting and prewetting phase transitions in binary mixtures of Bose-Einstein condensates[END_REF]. In this case, it is easier to work with slightly different parameters. After a new rescaling and some simple algebraic manipulations (see Section 4), the problem can be seen to be equivalent to minimizing

J ε (η) = ε Ω |∇η 1 | 2 + λ 2 |∇η 2 | 2 + 1 ε Ω 1 2 (η 2 1 + η 2 2 -1) 2 + (K -1)η 2 1 η 2 2 ( 8
)
with the volume constraint

Ω η 2 1 = α 1 and Ω η 2 2 = α 2 ,
for some λ ≤ 1, K > 1 and α 1 , α 2 ≥ 0 such that α 1 + α 2 = |Ω|. The main difficulty in studying the Γ-convergence of (8) to a sharp limit model is to obtain strong compactness for sequences of bounded energy. In the symmetric case λ = 1, which corresponds to g = 1 in (1), one can follow the strategy of [START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF][START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF] and use a nonlinear sigma model representation [START_REF] Kasamatsu | Vortices in multicomponent Bose-Einstein condensates[END_REF] to rewrite the problem in terms of an amplitude and a phase. In these unknowns, the functional takes a form similar to the celebrated Ambrosio-Tortorelli functional [START_REF] Ambrosio | On the approximation of free discontinuity problems[END_REF] from which one can get compactness. In the non-symmetric case, the methods of [START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF][START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF] do not apply. One has to find a different approach. Inspired by recent work on type-I superconductors [START_REF] Conti | A branched transport limit of the Ginzburg-Landau functional[END_REF], we prove instead that for every λ ≤ 1, the energy (8) directly controls a Modica-Mortola [START_REF] Modica | The gradient theory of phase transitions and the minimal interface criterion[END_REF] type energy where η 1 and η 2 are decoupled (see ( 51)). As a consequence, we recover the compactness of sequences of bounded energy. Besides extending results of [START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF][START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF] to the non-symmetric case, we believe that this more direct approach also has an intrinsic interest. In fact, it seems more natural and gives a better understanding of how the interaction between the two condensates via the term

Ω η 2 1 η 2 2
gives to the energy a structure of a double well potential. Using then classical arguments from Γ-convergence such as the slicing method [START_REF] Braides | Approximation of Free-Discontinuity Problems[END_REF], we can prove our last main theorem: Theorem 1.8. When ε → 0, J ε Γ-converges for the strong L 1 topology to

J (η 1 , η 2 ) = σ λ,K P (E, Ω) if η 1 = χ E = 1 -η 2 and |E| = α 1 +∞ otherwise,
where, for a set of finite perimeter E (see [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF][START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF]), P (E, Ω) denotes the relative perimeter of E inside Ω and where σ λ,K > 0 is defined by the one dimensional optimal transition problem

σ λ,K = inf R |η ′ 1 | 2 + λ 2 |η ′ 2 | 2 + 1 2 η 2 1 + η 2 2 -1 2 + (K -1)η 2 1 η 2 2 : lim -∞ η 1 = 0, lim +∞ η 1 = 1 .
At last, in the spirit of what was done in [START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF] in the case λ = 1 (see also the recent paper [START_REF] Aftalion | Interface layer of a two-component Bose-Einstein condensate[END_REF]), we study the asymptotic behavior of σ λ,K as K goes to one (mixing) or K goes to infinity (strong segregation) and recover good parts of what is expected from the physics literature [START_REF] Barankov | Boundary of two mixed Bose-Einstein condensates[END_REF][START_REF] Van Schaeybroeck | Interface tension of Bose-Einstein condensates[END_REF][START_REF] Ao | Binary Bose-Einstein condensate mixtures in weakly and strongly segregated phases[END_REF][START_REF] Timmermans | Phase separation of Bose-Einstein condensates[END_REF], namely lim

K→1 σ λ,K √ K -1 = 2 3 1 -λ 3 1 -λ 2 , σ λ,K -σ λ,∞ λ 2 K ↓0 ∼ - 1 K 1/2 + λ 1/2 K 1/4 ,
see Propositions 4.6 and 4.9.

Before closing this introduction, let us point out that most of the results, in particular Theorem 1.1, Theorem 1.3 and Theorem 1.8 can be easily generalized to arbitrary dimension and arbitrary radially symmetric strictly increasing confining potentials.

The structure of the paper is the following. In Section 2, we study the case of different intracomponent strengths in the presence of a confining potential. We then study in Section 3 the crossover case and finally in Section 4, we investigate the case of non-equal intracomponent strengths in the absence of confining potential and in a bounded domain.

Notation

For x ∈ R 2 and r > 0, we denote by B r (x) the ball of radius r centered at x and simply write B r when x = 0. Given a set E ⊂ R 2 , we let χ E be the characteristic function of E. For any integer k, we denote by H k the k-dimensional Hausdorff measure The letters, c, C denote universal constants which can vary from line to line. We also make use of the usual o and O notation. The symbols ∼, , indicate estimates that hold up to a positive constant. For instance, f g denotes the existence of a constant C > 0 such that f ≤ Cg. Throughout the paper, with a small abuse of language, we call sequence a family (u ε ) of functions labeled by a continuous parameter ε ∈ (0, 1]. A subsequence of (u ε ) is any sequence (u ε k ) such that ε k → 0 as k → +∞. For 1 ≤ p ≤ +∞ and a function f , we denote the L p norm of f by f p . In the sequel, when it is clear from the context, we omit to indicate the integrating measure. In particular, all integrals involving boundaries are with respect to H 1 and all integrals involving sets are with respect to the Lebesgue measure.

The case of different intracomponent strengths 2.1 The Thomas-Fermi profile

Let us consider the Thomas-Fermi approximation and prove Theorem 1.1. Let α = (α 1 , α 2 ) ∈ (0, +∞) 2 . We denote by X the set of pairs of measurable functions (ρ 1 , ρ 2 ) : R 2 → R 2 + and by X α the subset of pairs ρ ∈ X satisfying

R 2 ρ j = α j for j = 1, 2.
For ρ ∈ X we define the energy

E(ρ) = R 2 1 2 ρ 2 1 + g 2 ρ 2 2 + Kρ 1 ρ 2 + V (ρ 1 + ρ 2 ).
As stated in Theorem 1.1, we study the minimization of E in the class X α : we prove the existence of a unique minimizer for this problem and a stability result. We start by studying the minimization problem in the following subsets : given r > 0, let Lemma 2.1. For K ≥ √ g > 1 and r > 0, E admits a unique minimizer ρ 0,r = (ρ 0,r 1 , ρ 0,r 2 ) in X 0,r α . Let us set r = min(r, r 1 ) with

X 0,r α = ρ ∈ X α : supp(ρ 1 ) ⊂ B r , supp(ρ 2 ) ⊂ R 2 \ B r . and X 0 α = r>0 X 0,r α . ( 9
)
ρ 0 1 ρ 0 2 -r 0 r 0 σ + σ - T Figure
r 1 = 2α 1 π 1/4 and then R 2 1 = r 2 2 + r 4 1 2r 2 , R 2 2 = r 2 + 2gα 2 π 1/2 .
The unique minimizer is given by

ρ 0,r 1 (x) = (R 2 1 -|x| 2 ) + if |x| < r, 0 if |x| > r, ρ 0,r 2 (x) = 0 if |x| < r, 1 g (R 2 2 -|x| 2 ) + if |x| > r. ( 10
)
Optimizing in r > 0, the minimum of E in X 0 α is reached at ρ 0 = ρ 0,r 0 (see Figure 1) with

r 0 = r 1 1 + α 2 α 1 - α 2 α 1 1/2 ∈ (0, r 1 ). ( 11
)
In particular, there is a positive gap between ρ 0 2 and ρ 0 1 at the frontier ∂B r 0 :

σ + = inf Br 0 ρ 0 1 = √ g sup ρ 0 2 > sup ρ 0 2 = σ -. (12)
The minimal energy is

E 0 = E(ρ 0 ) = 2 3 2 π (α 1 + α 2 ) 3/2 + ( √ g -1) α 3/2 2 . ( 13
)
Proof.

Let r > 0. The optimization of E in X 0,r α splits into two independent optimization problems. From the associated Euler-Lagrange equations, we see that the minimizers have the form given by [START_REF] Ao | Binary Bose-Einstein condensate mixtures in weakly and strongly segregated phases[END_REF] with R 1 , R 2 fixed by the conditions R 2 ρ j = α j for j = 1, 2.

Let us now study the variation of r → E r = min X 0,r α E = E 1 (ρ 0,r 1 ) + E 2 (ρ 0,r 2 ). Using the notation t = (r/r 1 ) 2 , we obtain by direct computation,

E r = f (t) + 2 3 2gα 3 2 π 1/2 , with f (t) =          π 24 r 6 1 6t + 3 t -t 3 + α 2 r 2 1 t for t < 1, π 3 r 6 1 + α 2 r 2 1 t for t ≥ 1.
We have f ∈ C 2 (0, +∞) with f ′′ > 0 in (0, 1), f (t) → +∞ as t ↓ 0 and f ′ = f ′ (1) > 0 on (1, +∞). Therefore, f admits a unique minimiser t 0 = (r 0 /r 1 ) 2 ∈ (0, 1). Eventually, after some algebraic computations, we get that r 0 is given by [START_REF] Barankov | Boundary of two mixed Bose-Einstein condensates[END_REF], we also obtain [START_REF] Bella | Study of island formation in epitaxially strained films on unbounded domains[END_REF] and [START_REF] Braides | Approximation of Free-Discontinuity Problems[END_REF].

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1. Let ρ ∈ X α . We want to show that E(ρ) ≥ E(ρ 0 ). Since ρ 0 and E(ρ 0 ) do not depend on K and E(ρ) is a non decreasing function of K, we only consider the worst case K = √ g. Then, we write, ρ = ρ 0 + δρ, with δρ = (δρ 1 , δρ 2 ).

The energy expands as

E(ρ) = E(ρ 0 ) + L 1 (δρ 1 ) + L 2 (δρ 2 ) + Q(δρ), with L 1 (δρ 1 ) = R 2 (ρ 0 1 + √ gρ 0 2 + |x| 2 )δρ 1 , L 2 (δρ 2 ) = R 2 (gρ 0 2 + √ gρ 0 1 + |x| 2 )δρ 2 , Q(δρ) = 1 2 R 2 (δρ 1 + √ gδρ 2 ) 2 .
This last term is obviously non-negative. Let us study the terms L 1 (δρ 1 ) and L 2 (δρ 2 ). Let r 0 , R 1 , R 2 be as in the definition of ρ 0 in Lemma 2.1. We denote by A the annulus B R 2 \ B r 0 and by U the exterior domain

R 2 \ B R 2 .
Using the definition of ρ 0 1 and ρ 0 2 , we have

L 1 (δρ 1 ) = R 2 1 Br 0 δρ 1 + A 1 √ g R 2 2 + 1 - 1 √ g |x| 2 δρ 1 + U |x| 2 δρ 1 .
Since R 2 δρ 1 = 0, we have Br 0 δρ 1 = -A∪U δρ 1 . Using this, we get,

L 1 (δρ 1 ) = A∪U c 1 (|x|)δρ 1 , with c 1 (r) =    1 √ g (R 2 2 -r 2 ) -(R 2 1 -r 2 ) for r 0 < r < R 2 , r 2 -R 2 1 for r ≥ R 2 .
We notice that c 1 is continuous on [r 0 , +∞) and of the form a + br 2 with b > 0 on the two intervals (r 0 , R 2 ) and (R 2 , +∞). Moreover c 1 (r 0 ) = 0. Hence

c 1 > 0 in (r 0 , +∞), c 1 (r 0 ) = 0, c ′ 1 (r 0 ) > 0 and c 1 (r) r↑∞ ∼ r 2 . Now recall that ρ 0 1 + δρ 1 ≥ 0 and since ρ 0 1 ≡ 0 in A ∪ U , we have δρ 1 ≥ 0 in A ∪ U . Hence, L 1 (δρ 1 ) ≥ 0.
Similarly,

L 2 (δρ 2 ) = Br 0 ∪U c 2 (|x|)δρ 2 , with c 2 (r) = √ g(R 2 1 -r 2 ) -(R 2 2 -r 2 ) for r < r 0 , r 2 -R 2 2 for r > R 2 .
In the unbounded interval (R 2 , +∞), we have

c 2 (r) > 0 with c 2 (R 2 ) = 0, c ′ 2 (R 2 ) > 0 and c 2 (r) r↑∞ ∼ r 2 . In the interior interval (0, r 0 ), we have c 2 > 0, with c 2 (r 0 ) = 0, c ′ 2 (r) < 0. Hence c 2 > 0 in [0, r 0 ). Since δρ 2 ≥ 0 in B r 0 ∪ U , we have L 2 (δρ 2 ) ≥ 0.
To sum up, we have establish that for ρ = ρ 0 + δρ ∈ X α , there holds

E(ρ) = E(ρ 0 ) + H(δρ) ≥ E(ρ 0 ),
where we have set

H(δρ) = A∪U c 1 (|x|)δρ 1 + Br 0 ∪U c 2 (|x|)δρ 2 + 1 2 R 2 (δρ 1 + √ gδρ 2 ) 2 .
Now, for proving the stability estimate (5), we have to show that there exists C ≥ 0 such that for every function δρ = (δρ 1 , δρ 2 ) such that ρ 0 + δρ ∈ X α , there holds

δρ 2 L 1 ≤ CH(δρ). (14)
We prove [START_REF] Braides | Γ-convergence for beginners[END_REF] in several steps. First let us set

R = R 2 + 1, V = R 2 \ B R and à = A ∪ U \ V so that B r 0 ∪ à ∪ V = R 2 . We estimate successively δρ L 1 (V ) , δρ L 1 ( Ã) and δρ L 1 (Br 0 ) .
Step 1. Estimating δρ L 1 (V ) . In the set V , we have δρ 1 , δρ 2 ≥ 0. Using the fact that c 1 and c 2 are positive and increasing on [R, +∞), we deduce

V |δρ 1 | + |δρ 2 | ≤ 1 min(c 1 (R), c 2 (R)) V c 1 (|x|)δρ 1 + c 2 (|x|)δρ 2 ≤ CH(δρ).
Since the condition ρ

0 + δρ ∈ X α implies R 2 |δρ j | ≤ 2α j for j = 1, 2, we get δρ 2 L 1 (V ) ≤ CH(δρ). ( 15
)
Step 2. Estimating δρ L 1 ( Ã) . Notice that δρ 1 ≥ 0 in Ã. We split the annulus à into three

subsets à = Ã1 ∪ Ã2 ∪ Ã3 with Ã1 = x ∈ à : δρ 2 (x) ≥ 0 , Ã2 = x ∈ à : δρ 2 (x) < 0 and δρ 1 (x) > -2 √ gδρ 2 or - √ gδρ 2 (x) > 2δρ 1 (x) , Ã3 = {x ∈ à : δρ 2 (x) < 0 and δρ 1 (x)/2 ≤ - √ gδρ 2 (x) ≤ 2δρ 1 (x)}.
Step 2.1. In Ã1 , we have

|δρ 1 | + |δρ 2 | ≤ δρ 1 + √ gδρ 2 .
Hence, by Cauchy-Schwarz,

δρ 2 L 1 ( Ã1 ) ≤ | Ã1 | Ã1 (δρ 1 + √ gδρ 2 ) 2 ≤ CH(δρ). ( 16
)
Step 2.2. In Ã2 , there holds

|δρ 1 | + |δρ 2 | ≤ C|δρ 1 + √ gδρ 2 |
and as in the previous step,

δρ 2 L 1 ( Ã2 ) ≤ CH(δρ). (17)
Step.2.3. In Ã3 , we deduce from δρ 1 ≤ -2 √ gδρ 2 and the condition ρ 0 2 + δρ 2 ≥ 0 that

δρ 1 ≤ -2 √ gδρ 2 ≤ 2 √ g max ρ 0 2 = λ. Let us note m = Ã3 δρ 1 ≤ α 1 .
We have

H(δρ) ≥ Ã3 c 1 (|x|)δρ 1 ≥ inf à c 1 (|x|)v : v ∈ L 1 ( Ã, [0, λ]), à v = m .
Since c 1 is radial and increasing, the solution of the optimization problem is given by v

⋆ = λ1 Br ⋆ \Br 0 where r ⋆ > r 0 is such that à v ⋆ = m. Since c 1 (r) ≥ c ′ 1 (r 0 )(r -r 0 ) for r > r 0 with c ′ 1 (r 0 ) > 0, we deduce à c 1 (|x|)v ⋆ (x) ≥ c m 2 , for some c > 0 depending on λ, r 0 and c ′ 1 (r 0 ). This yields Ã3 δρ 1 2 = m 2 ≤ C à c 1 (|x|)v ⋆ (x) ≤ CH(δρ). Since |δρ 2 | ≤ 2/ √ g δρ 1 in Ã3 , we conclude that δρ 2 L 1 ( Ã3 ) ≤ CH(δρ). (18)
Gathering ( 16),( 17),( 18), we get

δρ 2 L 1 ( Ã) ≤ CH(δρ). ( 19
)
Step 3. Estimating δρ L 1 (Br 0 ) . Proceeding as in Step 2 and exchanging the roles of c 1 and c 2 and δρ 1 and √ gδρ 2 , we obtain

δρ 2 L 1 (Br 0 ) ≤ CH(δρ). (20)
Eventually, ( 15),( 19),( 20) yield the desired estimate ( 14).

Remark 2.2. One cannot hope for a stronger inequality of the form

ρ -ρ 0 1 E(ρ) -E(ρ 0 ) or ρ -ρ 0 2 2 E(ρ) -E(ρ 0 ).
since one can easily get a contradiction by exchanging ρ 0 1 and ρ 0 2 on small balls close to ∂B r 0 .

Approximation of the Thomas-Fermi limit by the Gross-Pitaievskii functional

We prove that minimizers of F ε converge to ( ρ 0 1 , ρ 0 2 ) as stated in Theorem 1.2.

Proof of Theorem 1.2. Let η ε be minimizers of F ε for ε ∈ (0, 1). Regularizing ( ρ 0 1 , ρ 0 2 ), one can easily construct a competitor ηε with F ε (η ε ) -E 0 /ε ≤ C. In particular, for minimizers η ε , there holds

E((η ε 1 ) 2 , (η ε 2 ) 2 ) -E 0 ε ≤ F ε (η ε ) - E 0 ε ≤ C so that (5) implies ((η ε 1 ) 2 , (η ε 2 ) 2 ) -(ρ 0 1 , ρ 0 2 ) L 1 ≤ Cε 1/2 . Now using that | √ a - √ b| 2 ≤ | √ a - √ b|| √ a + √ b| = |a -b|, we have indeed that (η ε 1 , η ε 2 ) - ρ 0 1 , ρ 0 2 L 2 ≤ ((η ε 1 ) 2 , (η ε 2 ) 2 ) -(ρ 0 1 , ρ 0 2 ) L 1 1/2 ≤ Cε 1/4 .

The crossover case

We now study what happens when g = 1 + ξε for some ξ > 0. For η satisfying (2), the energy then reads

F ε (η 1 , η 2 ) = ε R 2 |∇η 1 | 2 + |∇η 2 | 2 + 1 ε R 2 1 2 η 4 1 + 1 2 η 4 2 + Kη 2 1 η 2 2 + V (η 2 1 + η 2 2 ) + ξ 2 R 2 η 4 2 .
Let us first rewrite the energy in a more convenient way. For this, let η ε be the minimizer of

G ε (η) = ε R 2 |∇η| 2 + 1 ε R 2 1 2 η 4 + V η 2
under the constraint

R 2 η 2 = α,
where α = α 1 + α 2 . It is well known [START_REF] Ignat | The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate[END_REF][START_REF] Karali | The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit[END_REF] that η 2 ε converges to

ρ = (R 2 -V (x)) +
where R is such that R 2 ρ = 1. We denote by D the support of ρ (which is B R when V = |x| 2 ). We first rewrite the energy in a more convenient form.

Proposition 3.1. For u = (u 1 , u 2 ) a pair of non-negative functions, let

F ε (u) = ε R 2 η 2 ε (|∇u 1 | 2 + |∇u 2 | 2 ) + 1 ε R 2 1 2 η 4 ε 1 -(u 2 1 + u 2 2 ) 2 + (K -1)η 4 ε u 2 1 u 2 2 then if η = η ε u, there holds F ε (η) = G ε (η ε ) + F ε (u) + R 2 ξ 2 η 4 ε u 4 2
Proof. The proof follows as in [START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF]. We use the Lassoued-Mironescu trick [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF] and write

η 1 = η ε u 1 , η 2 = η ε u 2 to get |∇η 1 | 2 + |∇η 2 | 2 = (u 2 1 + u 2 2 )|∇η ε | 2 + η ε ∇η ε • ∇(u 2 1 + u 2 2 ) + η 2 ε (|∇u 1 | 2 + |∇u 2 | 2 ) ( 21
)
The function η ε solves the Euler-Lagrange equation

-ε∆η ε + 1 ε (η 3 ε + V (x)η ε ) = λ ε η ε (22)
where λ ε is some constant. Multiplying the equation ( 22) by η ε (u 2 1 + u 2 2 ), integrating and using integration by parts we get

ε R 2 (u 2 1 + u 2 2 )|∇η ε | 2 + η ε ∇η ε • ∇(u 2 1 + u 2 2 ) = λ ε α - 1 ε R 2 η 4 ε (u 2 1 + u 2 2 ) + V η 2 ε (u 2 1 + u 2 2 ).
On the other hand, multiplying [START_REF] Fuglede | Stability in the isoperimetric problem for convex or nearly spherical domains in R n[END_REF] by η ε and integrating, we find

λ ε α = G ε (η ε ) + 1 2ε R 2 η 4 ε so that (21) leads to ε R 2 |∇η 1 | 2 +|∇η 2 | 2 = R 2 εη 2 ε (|∇u 1 | 2 +|∇u 2 | 2 )+ 1 ε 1 2 η 4 ε (1 -2(u 2 1 + u 2 2 )) -V η 2 ε (u 2 1 + u 2 2 ) +G ε (η ε )
and therefore

F ε (η 1 , η 2 ) = G ε (η ε ) + R 2 ξ 2 η 4 ε u 4 1 + εη 2 ε (|∇u 1 | 2 + |∇u 2 | 2 ) + 1 ε 1 2 η 4 ε 1 -2(u 2 1 + u 2 2 ) + 2u 2 1 u 2 2 + (u 4 1 + u 2 ) 4 + (K -1)η 4 ε u 2 1 u 2 2 = G ε (η ε ) + R 2 ξ 2 η 4 ε u 4 1 + ε R 2 η 2 ε (|∇u 1 | 2 + |∇u 2 | 2 ) + 1 ε R 2 1 2 η 4 ε 1 -(u 2 1 + u 2 2 ) 2 + (K -1)η 4 ε u 2 1 u 2 2 = G ε (η ε ) + F ε (u 1 , u 2 ) + R 2 ξ 2 η 4 ε u 4 2 ,
which completes the proof.

For E a set of locally finite perimeter in D (see [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF][START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF]), let

F(E) = ∂E ρ 3/2 and V(E) = E ρ.
It is proved in [25, Th. 1.1] (see also [START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF]) that for all p < ∞, F ε L p -Γ converges to the functional

G 0 (u) = σ K F(E) if u 1 = χ E , u 2 = χ E c and V(E) = α 1 , +∞ otherwise.
where σ K > 0 is defined by the one dimensional optimal transition problem

σ K = inf R |η ′ 1 | 2 + |η ′ 2 | 2 + 1 2 η 2 1 + η 2 2 -1 2 + (K -1)η 2 1 η 2 2 : lim -∞ η 1 = 0, lim +∞ η 1 = 1 . Since ξ 2 η 4 ε u 4 2 is a continuous perturbation of F ε (u 1 , u 2 )
, we immediately obtain the following result.

Theorem 3.2. For every p < ∞, the functional

F ε (u 1 , u 2 ) + ξ 2 η 4 ε u 4 2 , L p -Γ converges to G ξ (u) =    σ K F(E) + ξ 2 E c ρ 2 if u 1 = χ E , u 2 = χ E c and V(E) = α 1 , +∞ otherwise.
Up to dividing G ξ by σ K and modifying ξ, we can assume that σ K = 1. If u 1 = χ E , u 2 = χ E c , we will, by a slight abuse of notation denote G ξ (u) by G ξ (E). We now want to study the minimizers of G ξ (whose existence follows from the Direct Method). As in [START_REF] Figalli | On the isoperimetric problem for radial log-convex densities[END_REF], by making a spherical symmetrization, we can restrict the analysis to spherically symmetric sets. For a given set E and a given half line ℓ starting from zero, such symmetrization is defined by replacing for every r > 0 the spherical slice E ∩ ∂B r by the spherical cap K(E, r) centered in ℓ ∩ ∂B r and such that

H 1 (K(E, r)) = H 1 (E ∩ ∂B r ). Proposition 3.3. For every minimizer E ξ of G ξ , ∂E ξ ∩ D (recall that D is the support of ρ) is a C ∞ hypersurface.
Moreover, for every half line ℓ starting from zero, there exists a minimizer E ξ such that for every r > 0, ∂B r ∩ E ξ is an arc of circle centered in ℓ.

Proof. Let E ξ be a minimizer of G ξ . The regularity of ∂E ξ is a consequence of the regularity theory for minimal surfaces. Indeed, since ρ is locally bounded away from zero in D, any minimizer of G ξ is locally a quasi-minimizer of the perimeter (see [START_REF] Figalli | On the isoperimetric problem for radial log-convex densities[END_REF]Th. 3.2] or [START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF][START_REF] Duzaar | Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals[END_REF] for instance). From this, one can infer C 1,α regularity of E ξ . Since ρ is smooth in D further regularity follows. The symmetry follows as in [START_REF] Figalli | On the isoperimetric problem for radial log-convex densities[END_REF]Th. 3.2] since spherical symmetrization reduces F and leaves

E c ξ ρ 2 invariant.
When ξ is small, the perimeter term is dominant. In this case, we are in a situation similar to the one studied in [START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF][START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF]. Proposition 3.4. There exists α 0 ∈ [0, 1/2) such that for every α 1 ∈ (α 0 , α/2 -α 0 ], there exists ξ 0 (α 1 ) such that for every ξ ≤ ξ 0 (α 1 ), the minimizer of G ξ is not radially symmetric.

Proof. It is proved in [START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF] that for every such α 1 , the minimizer of F under volume constraint is not radially symmetric and thus there exists E α 1 with

F(E α 1 ) < inf V(F )=α 1 {F(F ) : F radially symmetric } . but since G ξ (E α 1 ) = F(E α 1 ) + ξ E c α 1 ρ 2 it is clear that G ξ (E α 1 ) < inf V(F )=α 1 {F(F ) : F radially symmetric } < inf{G ξ (F ) : F radially symmetric }
for ξ small enough.

We now study the situation of large ξ. Our main result is a rigidity result stating that for large (but not infinite) ξ, the unique minimizer is the centered ball. Theorem 3.5. For every α 1 ∈ (0, α), there exists ξ 1 (α) such that for every ξ ≥ ξ 1 (α), the unique minimizer of G ξ is the centered ball B such that V( B) = α 1 .

In the rest of the section, α 1 is fixed. In order to ease notation, we assume that the unit ball is such that V(B) = α 1 (the general case follows by dilation). Theorem 3.5 follows from a combination of two results. The first one is a stability result for the volume term E c ρ 2 : Proposition 3.6. There exists c > 0 such that for every set E with V(E) = V(B),

E c ρ 2 - B c ρ 2 ≥ c E c ∆B c ρ 2 (23) Proof. Let E be such that V(E) = V(B) then E c ρ 2 - B c ρ 2 = E c ∩B ρ(1 -|x| 2 ) + B c ∩E ρ(|x| 2 -1) Let V = E c ∩B ρ
and δ be such that

B\B 1-δ ρ = V .
Letting

F δ = B\B 1-δ , since (E c ∩B)\F δ ρ = F δ \E c ρ, we have E c ∩B ρ(1 -|x| 2 ) - F δ ρ(1 -|x| 2 ) ≥ inf (E c ∩B)\F δ (1 -|x| 2 ) -sup F δ \E c (1 -|x| 2 ) F δ \E c ρ ≥ 0 and therefore, F δ minimizes G ρ(1 -|x| 2 ) among sets G ⊂ B with G ρ = V so that E c ∩B ρ(1 -|x| 2 ) ≥ F δ ρ(1 -|x| 2 ) ≥ cV 2 (24)
Similarly, letting

F δ = B 1+ δ ∩ B c with F δ ρ = V = B c ∩E ρ, then F δ minimizes G ρ(|x| 2 -1) among G ⊂ B c with G ρ = V and thus B c ∩E ρ(|x| 2 -1) ≥ F δ ρ(|x| 2 -1) ≥ CV 2 .
Together with [START_REF] Goldman | Existence and Stability for a Non-Local Isoperimetric Model of Charged Liquid Drops[END_REF], this gives [START_REF] Goldman | Volume-constrained minimizers for the prescribed curvature problem in periodic media[END_REF].

The second is an estimate on the possible instability of the ball for F.

Proposition 3.7. There exist ε > 0 and C > 0 such that for every set E with V(E) = V(B) and E∆B ρ ≤ ε

F(E) -F(B) ≥ -C E∆B ρ 2 . ( 25
)
Since the proof of Proposition 3.7 is long and involved, we postpone it. Let us show first how Proposition 3.6 and Proposition 3.7 yield together Theorem 3.5. Let E ξ be a minimizer of G ξ then using B as competitor, we obtain thanks to [START_REF] Goldman | Volume-constrained minimizers for the prescribed curvature problem in periodic media[END_REF] and ( 25),

E ξ ∆B ρ 2 ≤ 1 c E c ξ ρ 2 - B c ρ 2 = 1 cξ ([G ξ (E ξ ) -F(E ξ )] -[G ξ (B) -F(B)]) ≤ 1 cξ (F(B) -F(E ξ )) ≤ C cξ E ξ ∆B ρ 2 .
This implies E ξ ∆B ρ 2 = 0 for ξ > C/c and concludes the proof of Theorem 3.5.

Before going into the proof of Proposition 3.7, let us comment a bit on the statement and give the strategy for proving it. As explained below, in general, we do not expect the ball to be a local minimizer of F (and in particular, we cannot expect [START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF] to be true with a plus sign on the right-hand side). However, [START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF] shows that in some sense, the Hessian of F at the ball is bounded from below. The proof of (25) follows the strategy of [START_REF] Cicalese | A selection principle for the sharp quantitative isoperimetric inequality[END_REF] for proving the quantitative isoperimetric inequality (see also [START_REF] Acerbi | Minimality via second variation for a nonlocal isoperimetric problem[END_REF][START_REF] Figalli | On the isoperimetric problem for radial log-convex densities[END_REF]). Inequality ( 25) is first shown for nearly spherical sets, borrowing ideas from [START_REF] Fuglede | Stability in the isoperimetric problem for convex or nearly spherical domains in R n[END_REF]. The proof is then finished by arguing by contradiction, constructing a sequence (E n ) converging to the ball and contradicting the inequality. Using a Selection Principle and regularity theory for minimal surfaces, it is possible to replace (E n ) by a better sequence (F n ), still contradicting [START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF] but converging in a much stronger way to the ball. Since for n large enough F n are nearly spherical, we reach a contradiction and Proposition 3.7 is proved.

We thus start by proving [START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF] for nearly spherical sets (in the sense of Fuglede [START_REF] Fuglede | Stability in the isoperimetric problem for convex or nearly spherical domains in R n[END_REF]).

Proposition 3.8. There exist ε 0 > and

C 0 > 0 such that if ε ≤ ε 0 , every set ∂E = {(1+u(x))x : x ∈ ∂B} with u Lip ≤ ε and V(E) = V(B) satisfies F(E) -F(B) ≥ -C 0 E∆B ρ 2 . Proof. Recall that ρ(x) = (R 2 -|x| 2 ) + with R > 1.
The condition E ρ = B ρ can be written as

∂B 1 2 R 2 (1 + u) 2 -1 - 1 4 (1 + u) 4 -1 = 0 which leads to ∂B u = 1 2 3 -R 2 R 2 -1 ∂B u 2 + o u 2 L 2 (∂B) . (26)
We can now compute the energy

1 (R 2 -1) 3/2 (F(E) -F(B)) = ∂B 1 - 2u + u 2 R 2 -1 3/2 (1 + u) 1 + |∇u| 2 (1 + u) 2 1/2 -1 = ∂B u 1 - 3 R 2 -1 + 3 2 4 -3R 2 (R 2 -1) 2 u 2 + 1 2 ∇u 2 L 2 (∂B) + o u 2 L 2 (∂B) + ∇u 2 L 2 (∂B)
Using [START_REF] Ignat | The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate[END_REF], this turns into

1 (R 2 -1) 3/2 (F(E) -F(B)) = 1 2 ∂B - R 2 (2 + R 2 ) (R 2 -1) 2 u 2 + ∇u 2 L 2 (∂B) + o( u 2 L 2 (∂B) + ∇u 2 L 2 (∂B) ) ≥ 1 3 ∇u 2 L 2 (∂B) -C u 2 L 2 (∂B) . (27)
We now claim that for every δ > 0, there exists Λ δ > 0 such that

∂B u 2 ≤ δ ∂B |∇u| 2 + Λ δ ∂B |u| 2 . ( 28
)
Indeed, denoting by ū = 1 2π ∂B u and using Sobolev embedding and Young inequality, we compute

∂B u 2 = ∂B (u -ū)u + 2π(ū) 2 ≤ u -ū L ∞ (∂B) u L 1 (∂B) + (2π) -1 u 2 L 1 (∂B) ∇u L 2 (∂B) u L 1 (∂B) + u 2 L 1 (∂B) ≤ δ ∇u 2 L 2 (∂B) + Λ δ u 2 L 1 (∂B) .
This proves [START_REF] Kasamatsu | Vortices in multicomponent Bose-Einstein condensates[END_REF]. Using [START_REF] Kasamatsu | Vortices in multicomponent Bose-Einstein condensates[END_REF] in [START_REF] Karali | The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit[END_REF], we obtain,

F(E) -F(B) ≥ -C ∂B |u| 2 .
Now, since for nearly spherical sets, there holds

E∆B ρ ≃ |E∆B| ≃ ∂B |u|,
we can finally conclude the proof.

Remark 3.9. In the proof of Proposition 3.8, if we assume that E is centered, that is

E ρ(x)x = 0,
we could do as in [START_REF] Fuglede | Stability in the isoperimetric problem for convex or nearly spherical domains in R n[END_REF][START_REF] Cicalese | A selection principle for the sharp quantitative isoperimetric inequality[END_REF] and decompose u in Fourier series on ∂B to get

u = k u k Y k with Y 0 = 1 and Y 1 = x • ν for an appropriately chosen ν ∈ ∂B. So that ∂B R 2 3 (1 + u) 3 - 1 5 (1 + u) 5 - R 2 3 - 1 5 Y 1 = E ρ(x)x • ν - R 2 3 - 1 5 ∂B x • ν = 0.
We find thanks to ( 26)

u 0 = ∂B u = O u 2 L 2 (∂B) u 1 = ∂B uY 1 = O u 2 L 2 (∂B) .
From this, letting µ k = k 2 being the k-th eigenvalue of the Laplacian on ∂B,

∂B - R 2 (2 + R 2 ) (R 2 -1) 2 u 2 + ∂B |∇u| 2 = k≥2 u 2 k (µ k - R 2 (2 + R 2 ) (R 2 -1) 2 ) + o u 2 2 .
Since µ k ≥ 4 for k ≥ 2, and since

4 - R 2 (2 + R 2 ) (R 2 -1) 2 > 0 ⇐⇒ R < 1 √ 3 (5 - √ 13) 1/2 or R > 1 √ 3 (5 + √ 13) 1/2 ,
we expect the ball to be unstable for R ∈ (5 -√ 13) 1/2 / √ 3, (5 + √ 13) 1/2 / √ 3 and stable otherwise.

In order to go further, we need an isoperimetric inequality which is, we believe, of independent interest. Lemma 3.10. There exists c = c(α) > 0 such that for every set E with V(E) ≤ α/2, we have

F(E) ≥ c V(E) 5/6 . ( 29
)
Proof. Up to a dilation, we assume here that R = 1 that is ρ(x) = (1-|x| We now transform our problem in order to work on a periodic strip. Let S 1 be the unit torus and consider the diffeomorphism φ : B 1 \B 1/2 → S 1 × (0, 1/2) given by φ(θ, r) = θ 2π , 1 -r .

For F ⊂ S 1 × (0, 1/2), we let

V 1 (F ) = 1 0 1/2 0 χ F (x, y)y dydx and F 1 (F ) = ∂F y 3/2 then, since for E ⊂ B 1 \B 1/2 , V(E) ∼ V 1 (φ(E)) and F(E) ∼ F 1 (φ(E)), we are left to prove V 1 (F ) F 1 (F ) 6/5 ∀F ⊂ S 1 × (0, 1/2). ( 30
)
Notice also that since the sets E ⊂ B 1 \B 1/2 we started with were spherically symmetric, we can further assume that for every y ∈ (0, 1/2), F ∩ {y = y} is a segment centered in 0 × {y} for instance. We then make a convexification step. We define F as the smallest set which contains F and which is y-convex, that is for very x ∈ S 1 the set {y > 0 : (x, y) ∈ F } is a segment (see Figure 2). We have 000000000000000000 000000000000000000 000000000000000000 000000000000000000 000000000000000000 000000000000000000 000000000000000000 000000000000000000 000000000000000000 000000000000000000 000000000000000000 111111111111111111 111111111111111111 111111111111111111 111111111111111111 111111111111111111 111111111111111111 111111111111111111 111111111111111111 111111111111111111 111111111111111111 111111111111111111

-1/2 1/2 F x y F \ F Figure 2: Convexification step in the y-direction. V 1 ( F ) ≥ V 1 (F ) and F 1 ( F ) ≤ F 1 (F ).
Indeed, the first inequality comes from F ⊂ F and the second from the fact that for every x ∈ S 1 , the shortest (weighted) path between the points (x, y 1 ) and (x, y 2 ) is the straight segment. It is therefore enough to prove [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF] for the sets F . After these two symmetrizations, the set F is contained between two graphs y 1 : (-T, T ] → (0, 1/2) and y 2 : (-T, T ] → (0, 1/2) for some 0 < T ≤ 1/2 i.e. F = {(x, y) : -T < x ≤ T, y 1 (x) ≤ y ≤ y 2 (x)}.

Moreover, y 1 and y 2 are even, y 1 is non-decreasing and y 2 is non-increasing in [0, T ]. Using this parameterization, we have

V 1 ( F ) = 1 2 T -T [y 2 2 -y 2 1 ] and F 1 ( F ) = T -T y 3/2 1 1 + |Dy 1 | 2 + T -T y 3/2 2 1 + |Dy 2 | 2 ,
where for a function y of locally bounded variation,

F 1 (y ) = T -T y 3/2 1 + |Dy| 2 : = T -T y 3/2 1 + |y ′ | 2 + x∈Jy |y(x + ) 5/2 -y(x -) 5/2 |+ T -T y 3/2 d|D c y|,
see [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF]Th. 5.54]. Notice that we can assume that y 1 (±T ) = y 2 (±T ). Indeed, if T = 1/2, then F 1 ( F ) y 2 (1/2) 3/2 . In this case, we can add a vertical cut between y 2 (1/2) and y 1 (1/2). The additional contribution to the weighted perimeter is of order of y 2 (1/2) 5/2 -y 1 (1/2) 5/2 which is controlled by F 1 ( F ). Eventually, since F 1 is the relaxed functional of its restriction to C 1 paths with respect to L 1 convergence [8, Th. 5.54], we can further assume that y 1 and y 2 are smooth.

After, these symmetrization and regularization steps, we can write,

F 1 ( F ) ∼ T -T y 3/2 2 + T -T y 3/2 1 + y 2 (0) 5/2 -y 1 (0) 5/2 . ( 31
)
We then consider two cases. First, if y 1 (0) ≪ y 2 (0), then (31) yields

F 1 ( F ) T -T y 3/2 2 + y 2 (0) 5/2
from which we deduce

V 1 ( F ) T -T y 2 2 ≤ y 2 (0) 1/2 T -T y 3/2 2 F 1 ( F ) 1/5 F 1 ( F ) F 1 ( F ) 6/5 .
Then, in the case y 1 (0) ∼ y 2 (0), by convexity y 2 (0) 5/2 -y 1 (0) 5/2 ≥ 5 2 y 1 (0) 3/2 (y 2 (0) -y 1 (0)) so that [START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF] becomes

F 1 ( F ) T y 3/2 2 (0) + y 2 (0) 3/2 (y 2 (0) -y 1 (0)). We infer that V 1 ( F ) = T -T (y 2 -y 1 )(y 2 + y 1 ) ≤ 4T y 2 (0)(y 2 (0) -y 1 (0)) ≤ 4T y 2 (0)y 2 (0) 4/5 (y 2 (0) -y 1 (0)) 1/5 = 4T y 2 (0) 3/2 y 2 (0) 3/2 (y 2 (0) -y 1 (0)) 1/5 F 1 ( F )F 1 ( F ) 1/5 F 1 ( F ) 6/5 ,
which concludes the proof of (30).

Remark 3.11. The exponent 5/6 in ( 29) can be easily seen to be optimal by considering as competitor a small ball touching the boundary of B.

We can now prove the following ε-regularity result: Proposition 3.12. Let Λ > 0. Then, there exists ε 1 > 0 such that if E is a Λ-minimizer of F, i.e. for every G,

F(E) ≤ F(G) + Λ E∆G ρ and if E∆B ρ ≤ ε 1 then E is nearly spherical i.e. ∂E = {(1 + u(x))x : x ∈ ∂B} and u C 1,α ≤ ε 0 (
where ε 0 is the one defined in Proposition 3.8).

Proof. Fix δ > 0 then inside B 1+δ by classical ε-regularity results for quasi-minimizers of the perimeter (see [START_REF] Duzaar | Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals[END_REF]Th. 6.1] or [START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF]Th. II.6.3] for instance as well as [START_REF] Cicalese | A selection principle for the sharp quantitative isoperimetric inequality[END_REF]Lem. 3.6]), if

(E∩B 1+δ )∆B ρ ≤ ε 1 then ∂E ∩ B 1+δ is a small C 1,α perturbation of ∂B in particular, if E∆B ρ ≤ ε 1 , ∂E ∩ B 1+δ ⊂ B 1+ δ 2 and E can be written as E = E 1 ∪ E 2 where E 1 is nearly spherical, E 2 ⊂ B c
1+δ . By testing the Λ-minimality of E against E 1 , we find by ( 29)

cV(E 2 ) 5/6 ≤ F(E 2 ) ≤ ΛV(E 2 )
and thus if

V(E 2 ) = 0, c Λ 6 ≤ V(E 2 ) ≤ ε 1
which is absurd for ε 1 small enough.

We will also need the following simple lemma which is a weak version of (25).

Lemma 3.13. There exists Λ 1 > 0 such that for every set E,

F(E) -F(B) ≥ -Λ 1 E∆B ρ. ( 32
)
Proof. Let v be a vector field with supp

v ⊂ B R , |v| ≤ 1, v = x on ∂B and div v ∞ ≤ C and let w = 3 2 ρ -1/2 ∇ρ • v + ρ 1/2 div v,
so that div (ρ 3/2 v) = ρw. Then, for every set E, denoting by ν E the outward normal to E,

F(E) -F(B) ≥ ∂E ρ 3/2 v • ν E - ∂B ρ 3/2 v • ν B = E ρw - B ρw ≥ -w ∞ E∆B ρ.
We can finally prove [START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF].

Proof. We argue by contradiction. Assume that there exists a sequence of measurable sets (E n ) with En∆B ρ → 0 and

F(E n ) + 2C 0 En∆B ρ 2 ≤ F(B), (33)
where C 0 is the constant given by Proposition 3.8. Let ε n = En∆B ρ and for Λ 1 > 0 given by Lemma 3.13 and Λ 2 > 0, let F n be a minimizer of

F(F ) + 2Λ 1 ε n - F ∆B ρ 2 + ε n 1/2 + Λ 2 |V(F ) -V(B)|. ( 34
)
Step 1. We claim that if Λ 2 is large enough, V(F n ) = V(B) for all n. To prove this, we follow the approach of [START_REF] Esposito | A remark on a free interface problem with volume constraint[END_REF] (see also [START_REF] Goldman | Volume-constrained minimizers for the prescribed curvature problem in periodic media[END_REF] for another approach). Assume by contradiction that the claim does not hold, then there exist sequences of positive numbers ε ′ k → 0 and Λ 2,k → +∞, and a sequence of measurable sets

G k ⊂ R 2 such that G k minimizes F(G) + 2Λ 1 ε ′ k - G∆B ρ 2 + ε ′ k 1/2 + Λ 2,k |V(G) -V(B)| (35)
and for instance V(G k ) < V(B) (the other case is similar). In order to get a contradiction, we build a new sequence G k such that V( G k ) = V(B) and

F( G k ) + 2Λ 1 ε ′ k - G k ∆B ρ 2 + ε ′ k 1/2 |V(G k ) -V(B)|.
The construction is a bit delicate since we want all the constants to be uniform in ε ′ k . If this were not the case, one could have simply used Almgren's construction (see [START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF]). First, testing the energy [START_REF] Modica | Gradient theory of phase transitions with boundary contact energy[END_REF] with B, we find that F(G k ) ≤ C and therefore, up to extraction,

G k converges in L 1 loc (D) to some G ∞ . Moreover, since |V(G k ) -V(B)| ≤ C Λ 2,k , G ∞ satisfies V(G ∞ ) = V(B). The set G ∞ , minimizes F(G) + 2Λ 1 G∆B ρ under the constraint V(G) = V(B) (
this can be seen for instance by a Γ-convergence argument). Therefore, by [START_REF] Mason | Classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate[END_REF], G ∞ = B. Notice that arguing as in Proposition 3.3, we get that the sets G k ∩ D are C 1,α . Now, let us fix δ > 0 and r ≪ 1, let us choose x0 ∈ ∂B and let us set

x 0 = (1 + r 2 )x 0 and C = (1/2) B∩Br (x 0 ) ρ. Then since B r/2 (x 0 ) ∩ B = ∅, there hold G k ∩B r/2 (x 0 )
ρ ≤ δ and

G k ∩Br(x 0 ) ρ > C ( 36
)
for k large enough. Let 0 < σ k < 1/2 be a sequence to be fixed later and consider the bilipschitz maps:

Φ k (x 0 + x) -x 0 =      (1 -3σ k )x if |x| ≤ r 2 , x + σ k (1 -r 2 |x| 2 )x if r 2 ≤ |x| < r, x if |x| ≥ r,
and let G k = Φ k (E k ).
Step 1.1. We first prove that

F(G k ) -F( G k ) ≥ -Cσ k F(G k ). ( 37
)
Following the notation of [START_REF] Esposito | A remark on a free interface problem with volume constraint[END_REF], we let for x ∈ ∂G k , T k,x (τ ) = ∇Φ k (x) • τ for τ ∈ π k,x (where π k,x is the tangent space to ∂G k at x) and

J 1 T k,x = det(T * k,x • T k,x )
be the one Jacobian of T k,x so that

F( G k ) = ∂G k ρ 3/2 (Φ k (x))J 1 T k,x dH 1 .
In particular, it is proved in [START_REF] Esposito | A remark on a free interface problem with volume constraint[END_REF] that J 1 T k,x < 1 in B r/2 (x 0 ) and

J 1 T k,x ≤ 1 + 5σ k in C r (x 0 ) = B r (x 0 )\B r/2 (x 0 ). We can now decompose, F(G k ) -F( G k ) = ∂G k ∩Br(x 0 ) ρ 3/2 - ∂G k ∩Br(x 0 ) ρ 3/2 (Φ k (x))J 1 T k,x dH 1 = ∂G k ∩Cr(x 0 ) ρ 3/2 - ∂G k ∩Cr(x 0 ) ρ 3/2 (Φ k (x))J 1 T k,x dH 1 + ∂G k ∩B r/2 (x 0 ) ρ 3/2 - ∂G k ∩B r/2 (x 0 ) ρ 3/2 (Φ k (x))J 1 T k,x dH 1 . But since in B r (x 0 ), |ρ 3/2 (x) -ρ 3/2 (Φ k (x))| ≤ Cρ 3/2 (x)σ k , ∂G k ∩Cr(x 0 ) ρ 3/2 -ρ 3/2 (Φ k (x))J 1 T k,x dH 1 = ∂G k ∩Cr(x 0 ) (1 -J 1 T k,x )ρ 3/2 + J 1 T n,1 (ρ 3/2 (x) -ρ 3/2 (Φ k (x)))dH 1 ≥ -Cσ k ∂G k ∩Cr(x 0 ) ρ 3/2 .
Similarly,

∂G k ∩B r/2 (x 0 ) ρ 3/2 -ρ 3/2 (Φ k (x))J 1 T k,x dH 1 = ∂G k ∩B r/2 (x 0 ) (1 -J 1 T k,x )ρ 3/2 + J 1 T n,1 (ρ 3/2 (x) -ρ 3/2 (Φ k (x)))dH 1 ≥ -Cσ k ∂G k ∩B r/2 (x 0 ) ρ 3/2 .
Hence, (37) follows.

Step 1.2. We now prove that for some κ > 0,

V( G k ) -V(G k ) ≥ κσ k . (38)
If JΦ k denotes the Jacobian of Φ k , it is shown in [START_REF] Esposito | A remark on a free interface problem with volume constraint[END_REF] that

JΦ k ≤ 1 + 8σ k (39)
and that in C r (x 0 ),

JΦ k ≥ 1 + cσ k (40)
for some c > 0. We can decompose

G k ρ - G k ρ = G k ∩Br(x 0 ) ρ(Φ k (x))JΦ k (x) -ρ(x) = G k ∩Cr(x 0 ) ρ(Φ k (x))JΦ k (x) -ρ(x) + G k ∩B r/2 (x 0 ) ρ(Φ k (x))JΦ k (x) -ρ(x).
Thanks to [START_REF] Solci | Boundary contact energies for a variational model in phase separation[END_REF] and [START_REF] Sourdis | Remarks on the interface layer of a two-component Bose-Einstein condensate[END_REF], there holds

G k ∩Cr(x 0 ) ρ(Φ k (x))JΦ k (x) -ρ(x) = G k ∩Cr(x 0 ) (ρ(Φ k (x)) -ρ(x))JΦ k (x) + (JΦ k -1)ρ ≥ cσ k G k ∩Cr(x 0 ) ρ -2 G k ∩Cr(x 0 ) |ρ(Φ k (x)) -ρ(x))| = G k ∩Cr(x 0 ) ρ cσ k -2 ρ(Φ k (x)) -ρ(x) ρ(x) . Since ρ(x) = (R 2 -|x| 2 ) + , we see from the definition of Φ k that in C r (x 0 ), ρ(Φ k (x)) -ρ(x) ρ(x) ≤ Cσ k r
hence we can choose r small enough so that 2Cσ k r ≤ cσ k /2. So that

G k ∩Cr(x 0 ) ρ(Φ k (x))JΦ k (x) -ρ(x) ≥ κ 1 σ k G k ∩Cr(x 0 ) ρ, (41)
for some κ 1 > 0. Similarly, thanks to [START_REF] Sourdis | Remarks on the interface layer of a two-component Bose-Einstein condensate[END_REF],

G k ∩B r/2 (x 0 ) ρ(Φ k (x))JΦ k (x) -ρ(x) = G k ∩B r/2 (x 0 ) (ρ(Φ k (x)) -ρ(x))JΦ k (x) + (JΦ k -1)ρ ≥ -κ 2 σ k G k ∩B r/2 ρ,
for some κ 2 > 0. Combining this with ( 41) and [START_REF] Morgan | Existence of isoperimetric regions in R n with density[END_REF] gives

V( G k ) -V(G k ) ≥ σ k κ 1 G k ∩Br(x 0 ) ρ -κ 2 G k ∩B r/2 (x 0 ) ρ ≥ (κ 1 C -κ 2 δ)σ k ,
so that (38) holds if δ is small enough.

Step [START_REF] Papp | Tunable miscibility in a dual-species Bose-Einstein condensate[END_REF] and the continuity of the map

1.3. Since V(G k ) < V(B) and V(G k ) → V(B),
σ k → V( G k ), show that we can find σ k → 0 such that V( G k ) = V(B).
From this we get a contradiction. Indeed, by minimality of G k ,

F(G k ) + 2Λ 1 G k ∆B ρ -ε ′ k 2 + ε ′ k 1/2 + Λ 2,k |V(G k ) -V(B)| ≤ F( G k ) + 2Λ 1 G k ∆B ρ -ε ′ k 2 + ε ′ k 1/2 ≤ F(G k ) + Cσ k + 2Λ 1 G k ∆B ρ -ε ′ k 2 + ε ′ k 1/2
, where in the last line we used [START_REF] Öhberg | Hartree-Fock treatment of the two-component Bose-Einstein condensate[END_REF]. Since the function x → ((x -

ε ′ k ) 2 + ε ′ k ) 1/2 is 1-Lipschitz, we obtain, recalling (38), Λ 2,k σ k σ k + G k ∆B ρ - G k ∆B ρ .
Arguing as for [START_REF] Papp | Tunable miscibility in a dual-species Bose-Einstein condensate[END_REF], we can prove that

G k ∆B ρ - G k ∆B ρ σ k , from which we obtain Λ 2,k σ k σ k .
This is not possible since Λ 2,k → +∞.

Step 2. Going back to F n , minimizers of (34), we have that F n converge to some F ∞ minimizing

F(F ) + 2Λ 1 F ∆B ρ + Λ 2 |V(F ) -V(B)|
that is by [START_REF] Mason | Classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate[END_REF], F ∞ = B. The F n are Λ-minimizers of F. Indeed, for every E,

F(F n ) ≤ F(E) + 2Λ 1 E∆B ρ - Fn∆B ρ + Λ 2 Fn ρ - E ρ ≤ F(E) + (2Λ 1 + Λ 2 ) E∆Fn ρ.
Therefore, by Proposition 3.12 and Proposition 3.8, for n large enough,

F(F n ) -F(B) ≥ -C 0 Fn∆B ρ 2 . ( 42
)
Step 3. Let γ n = Fn∆B ρ. Then, by minimality of F n , there holds (using E n as a competitor and recalling (33)),

F(F n ) + 2Λ 1 (γ n -ε n ) 2 + ε n 1/2 ≤ F(E n ) + 2Λ 1 ε 1/2 n ≤ F(B) + 2Λ 1 ε 1/2 n -2C 0 ε 2 n which combined with (42) gives 2Λ 1 (γ n -ε n ) 2 + ε n 1/2 -C 0 γ 2 n ≤ -2C 0 ε 2 n + 2Λ 1 ε 1/2 n . (43)
From this we obtain

2Λ 1 (γ n -ε n ) 2 + ε n 1/2 ≤ C 0 γ 2 n + 2Λ 1 ε 1/2 n .
Dividing by 2Λ 1 and taking the square of both sides, we get

(γ n -ε n ) 2 + ε n ≤ C 2 0 4Λ 2 1 γ 4 n + C 0 Λ 1 γ 2 n ε 1/2 n + ε n subtracting ε n and dividing by γ 2 n we deduce 1 - ε n γ n 2 ≤ C 2 0 4Λ 2 1 γ 2 n + 2 C 0 Λ 1 ε 1/2
n hence 1 -εn γn → 0. Going back to [START_REF] Van Schaeybroeck | Critical wetting, first-order wetting and prewetting phase transitions in binary mixtures of Bose-Einstein condensates[END_REF], we obtain

2Λ 1 ε 1/2 n   ε n γ n ε n -1 2 + 1 1/2 -1   ≤ -C 0 ε 2 n 2 - γ 2 n ε 2
n from which we get a contradiction since for n large enough the left-hand side is positive while the right-hand side is negative. This ends the proof of Proposition 3.7.

4 The case without confining potential

The functional

We finally go back to the situation where g > 1 but where there is no trapping potential. We thus consider for a fixed bounded open set Ω ⊂ R 2

F ε (η) = ε Ω |∇η 1 | 2 + |∇η 2 | 2 + 1 ε Ω 1 2 η 4 1 + g 2 η 4 2 + Kη 2 1 η 2 2
with the constraints

Ω η 2 i = α i . ( 44
) Letting γ = (α 1 + α 2 g 1/2 ),
it can be easily seen that minimizers of the Thomas-Fermi energy

E(ρ) = Ω 1 2 ρ 2 1 + g 2 ρ 2 2 + Kρ 1 ρ 2
under the mass constraint

Ω ρ i = α i are given by ρ 1 = γ |Ω| χ E , ρ 2 = γ |Ω|g 1/2 χ Ω\E for any set E ⊂ Ω with |E| = |Ω|α 1 γ .
The minimal energy is then

E 0 = E(ρ) = γ 2 2|Ω|
This motivates the change of variables

η 1 = γ |Ω| -1/2 η 1 , η 2 = γ |Ω|g 1/2 -1/2 η 2 , x = γ |Ω| 1/2 x, J ε ( η) = |Ω| γ F ε (η)- γ 2ε yielding J ε ( η) = ε Ω |∇ η 1 | 2 + g -1/2 |∇ η 2 | 2 + 1 ε Ω 1 2 η 1 2 + η 2 2 -1 2 + ( K -1) η 1 2 η 2 2 where 1 < K = K g 1/2 , under the mass constraint Ω η 2 i = α i
where α 1 = α 1 and α 2 = g 1/2 α 2 so that α 1 + α 2 = γ = | Ω|. Forgetting the tildas and letting 1 > λ 2 = g -1/2 , we finally obtain that the original minimization problem is equivalent to minimizing

J ε (η) = ε Ω |∇η 1 | 2 + λ 2 |∇η 2 | 2 + 1 ε Ω 1 2 η 2 1 + η 2 2 -1 2 + (K -1)η 2 1 η 2 2
under the volume constraint (44).

The one dimensional transition problem

Let us introduce, the following energy defined for η

= (η 1 , η 2 ) ∈ W 1.2 loc (R, R 2 ), E λ,K (η) = R |η ′ 1 | 2 + λ 2 |η ′ 2 | 2 + W K (η 1 , η 2 ),
where for s, t ≥ 0, we introduced the potential

W K (s, t) = 1 2 (1 -s 2 -t 2 ) 2 + (K -1)s 2 t 2 .
We consider the minimization problem,

σ λ,K = inf E λ,K (η) : η = (η 1 , η 2 ) ∈ W 1,2 loc (R, R 2 + ), lim -∞ η 1 = 0, lim +∞ η 1 = 1 . (45)
Let us show that problem (45) admits a minimizer. The result also follows from [6, Th. 2.1] (see also [START_REF] Alikakos | On the connection problem for potentials with several global minima[END_REF]) with a different proof. Let us point out that uniqueness of the optimal profile has been recently shown [START_REF] Sourdis | Remarks on the interface layer of a two-component Bose-Einstein condensate[END_REF]. Proposition 4.1. There exist minimizing pairs to σ λ,K . Every such minimizing pair is smooth. Moreover, the following equipartition of energy holds:

|η ′ 1 | 2 + λ 2 |η ′ 2 | 2 = W K (η 1 , η 2 ).
Proof. We establish the existence of a minimizing pair η = (η 1 , η 2 ) by the Direct Method of the calculus of variations. The required compactness and semi-continuity result is stated in Lemma 4.2. The smoothness of η is the consequence of the Euler-Lagrange equations

-2η ′′ 1 -2λ 2 η ′′ 2 + ∇W K (η 1 , η 2 ) = 0, in R.
For the equipartition of energy, we take the dot product of the Euler-Lagrange equations with (η ′ 1 , η ′ 2 ) T . Integrating, we see that the quantity

-|η ′ 1 | 2 -λ 2 |η ′ 2 | 2 + W K (η 1 , η 2 )
does not depend on x. Using the conditions at infinity, we conclude that

|η ′ 1 | 2 + λ 2 |η ′ 2 | 2 = W K (η 1 , η 2 ) on R. Lemma 4.2. Let (η k ) k≥0 ⊂ W 1,2 loc (R, R 2
+ ) be such that there exists C 0 ≥ 0 such that for k ≥ 0,

lim inf x→-∞ η k 1 < 1/4, lim sup x→+∞ η k 1 > 3/4 and E λ,K (η k ) ≤ C 0 .
Then, there exist η = (η 1 , η 2 ) ∈ W 1.2 loc (R, R 2 + ), a subsequence (still denoted by (η k )) and a sequence (z k ) ⊂ R such that

η k (• -z k ) → η
uniformly on any bounded subset of R.

Moreover, lim

x→-∞ η = (0, 1), lim

x→+∞ η = (1, 0), (46)
and

E λ,K (η) ≤ lim inf k→+∞ E λ,K (η k ).
To prepare for the proof, we start by noticing that (similarly to the Ginzburg-Landau functional for Type-I superconductors [START_REF] Conti | A branched transport limit of the Ginzburg-Landau functional[END_REF]) the Gross-Pitaievskii energy controls a classical double well potential. This is the key ingredient in order to get compactness for sequences of bounded energy. Let us introduce the relaxed potential,

w K (s) = inf t∈R W K (s, t) =          1 2 (1 -s 2 ) 2 - 1 2 (1 -Ks 2 ) 2 if 0 ≤ s < K -1/2 , 1 2 (1 -s 2 ) 2 if s ≥ K -1/2 . (47)
The function w K is a standard double-well potential (see Figure 3). In particular, since there holds w

K (s) = w K (t) ≤ W K (s, t), we have for η = (η 1 , η 2 ) ∈ W 1,2 loc (R, R 2 + ) R |η ′ 1 | 2 + w K (η 1 ) ≤ E λ,K (η) and R λ 2 |η ′ 2 | 2 + w K (η 2 ) ≤ E λ,K (η). ( 48
)
Proof of Lemma 4.2. Let us consider a sequence η k = (η k 1 , η k 2 ) and C 0 ≥ 0 satisfying the hypotheses of the lemma. Let us fix k ≥ 0 and let us consider y < z such that either, η k 1 (y) = 1/4, η k 1 (z) = 3/4 or η k 1 (y) = 3/4, η k 1 (z) = 1/4. Using the Modica-Mortola trick, we see that Since δ > 0, taking into account (48) and the bound E λ,K (η k ) ≤ C 0 , we deduce that there exists an odd integer n k ∈ [1, C 0 /δ] and sequence of intervals

z y |(η k 1 ) ′ | 2 + w K η k 1 ≥ 2 z y w K η k 1 |(η k 1 ) ′ | ≥ 2 3/4 1/4 w K (s) ds = δ.
I k 0 = (-∞, x k 1 ), I k 1 = (x k 1 , x k 2 ), • • • , I k n k -1 = (x k n k -1 , x k n k ), I k n k = (x n k , +∞), such that η k 1 (x k j ) = 1/2 for j = 1, • • • , n k and there exists y k 0 , • • • , y k n k with y j ∈ I k j such that η k 1 (y k j ) = 1/4 and η k 1 < 3/4 in I k j if j is even, η k 1 (y k j ) = 3/4 and η k 1 > 1/4 in I k j if j is odd.
See the example of Figure 4. Now, up to extraction, we assume that n k = n does not depend on k. Now, if the sequence (x k 2 -x k 1 ) k is not bounded, we can extract a subsequence such that (x k 2 -x k 1 ) → ∞. Repeating the process, we can assume that the following property holds true: there exist R ≥ 0 and a partition of {1, .., n} = {a 0 , . .

. , b 0 } ∪ {a 1 , . . . , b 1 } ∪ • • • ∪ {a m , . . . , b m } with a 0 = 1, b m = n and a j+1 = b j + 1 for j = 0, . . . , m -1, such that, ∀ l ∈ {0, • • • , m} 0 < x k b l -x k a l ≤ R, ∀l ∈ {0, • • • , m -1} x k a l+1 -x k a l k→∞ -→ +∞.
Since a 0 = 1 and b m = n are odd there exists l ⋆ ∈ {0, • • • , m} such that a l⋆ and b l⋆ are odd. We set z k = x k a l⋆ and ηk

1 = η k 1 (• -z k ). By construction, there exists a sequence (R k ) ⊂ [R, +∞) with R k → +∞ such that for k ≥ 0, ηk 1 < 3/4 in [-R k , 0], ηk 1 (0) = 1/2 and ηk 1 > 1/4 in [R, R k ]. (49)
Now, from the energy bound E λ,K (η k ) ≤ C 0 , we see that (η k 1 ) is bounded in W 1,2 loc (R). Up to extraction, there exists η 1 ∈ W 1,2 loc (R) such that η k 1 (• -z k ) = ηk 1 → η 1 locally uniformly. Moreover, from (49), we have lim sup

-∞ η 1 ≤ 3/4, η 1 (0) = 1/2 and lim inf +∞ η 1 ≥ 1/4. (50)
Similarly, up to extraction, there exists η 2 ∈ W 1,2 loc (R) such that η k 2 (•-z k ) → η 2 locally uniformly. By lower semi-continuity of the Dirichlet energy, we have

E λ,K (η 1 , η 2 ) ≤ lim inf k→+∞ E λ,K (η k 1 , η k 2 ).
To end the proof of Lemma 4.2, we have to establish that η = (η 1 , η 2 ) satisfies the conditions at infinity (46). Since R |η ′ i | 2 is finite η admits limits η ± at ±∞. From the bound R W K (η 1 , η 2 ) ≤ C, we get η -, η + ∈ {(0, 1), (1, 0)}. Eventually (50) implies η -= (0, 1), η + = (1, 0), that is (46).

The Γ-convergence result

In this section we study the Γ-convergence of J ε as ε goes to zero. Theorem 4.3. When ε → 0, J ε Γ-converges for the strong L 1 topology to

J (η 1 , η 2 ) = σ λ,K P (E, Ω) if η 1 = χ E = 1 -η 2 and |E| = α 1 +∞ otherwise.
Proof. Since J ε (min(η 1 , 1 + δ), min(η 2 , 1 + δ)) ≤ J ε (η) for all δ > 0 and since the bound

Ω 1 ε (η ε 1 ) 2 + (η ε 2 ) 2 -1 2 ≤ C implies that if min(η ε i , 1 + δ) (
strongly) converges to some η then also η ε converges to the same η, we can always assume that sequences which are bounded in energy are bounded in L ∞ . Now, we use again that the Gross-Pitaievskii potential controls the double well-potential w K (see (47)). We have,

Ω w K (η 1 ) + Ω w K (η 2 ) ≤ 2 Ω W K (η 1 , η 2 ). (51)
Thanks to (51) and the usual Modica-Mortola argument, we then have that from every sequence (η ε 1 , η ε 2 ) of bounded energy, we can extract a subsequence converging strongly in L 1 to some pair (η 1 , η 2 ). Moreover, from the bound on the energy, we get that η 1 (x), η 2 (x) ∈ {0, 1} and η 1 (x)η 2 (x) = 0 for almost every x ∈ R. From the volume constraint and the strong convergence, we also deduce that Ω η 2 1 = α 1 . The lower bound inequality is then a standard application of the slicing technique (see [START_REF] Braides | Approximation of Free-Discontinuity Problems[END_REF][START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Γ-convergence[END_REF][START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF] for instance). The upper bound is also standard. By approximation it is enough doing the construction for a smooth set E. Let δ > 0 be fixed then we can find T > 0 and (η

δ 1 , η δ 2 ) with η δ 1 (-T ) = η δ 2 (T ) = 0 and η δ 1 (T ) = η δ 2 (-T ) = 1 with T -T |(η δ 1 ) ′ | 2 + λ 2 |(η δ 2 ) ′ | 2 + W K (η δ 1 , η δ 2 ) ≤ σ λ,K + δ
Let d E be the signed distance to ∂E.

We then let

η ε (x) = η δ d E (x) ε .
Using the coarea formula it can be seen that η ε converges strongly to (χ E , 1 -χ E ) and that lim sup

ε→0 J ε (η ε ) ≤ (σ λ,K + δ)P (E, Ω). Letting finally ηε = √ α 1 η ε 1 2 η ε 1 , √ α 2 η ε 2 2 η ε 2 ,
we have that by definition ηε satisfies the mass constraint. Moreover, using that η ε i 2 = α i +O(ε) (and therefore, ηε

i = η i (1 + O(ε))) we have lim sup ε→0 J ε (η ε ) ≤ (σ λ,K + δ)P (E, Ω). ( 52
)
Since δ is arbitrary in (52), this concludes the proof of the upper bound.

Remark 4.4. With a minor adaptation of this proof (see [START_REF] Modica | Gradient theory of phase transitions with boundary contact energy[END_REF][START_REF] Solci | Boundary contact energies for a variational model in phase separation[END_REF]), one could also deal with Dirichlet boundary conditions (that is impose η ε i = 0 on ∂Ω). Letting

γ i = inf +∞ 0 |η ′ 1 | 2 + λ 2 |η ′ 2 | 2 + 1 2 η 2 1 + η 2 2 -1 2 + (K -1)η 2 1 η 2 2 : η 1 (0) = η 2 (0) = 0, lim +∞ η i = 1 ,
we would obtain as Γ-limit (at least for ∂Ω of class C 2 ).

J Dir (η 1 , η 2 ) =      σ λ,K P (E, Ω) + γ 1 H 1 (∂Ω ∩ ∂E) + γ 2 H 1 (∂Ω ∩ ∂E c ) if η 1 = χ E = 1 -η 2 and |E| = α 1 +∞ otherwise.
Notice that in this case, by definition, γ i ≤ σ λ,K + γ j . Hence, on the macroscopic level, there is always a contact angle. Complete wetting is still possible in the form of an infinitely thin layer of one of the phases (this would show up in the γ i ). The behavior of γ i has been recently studied in the physics literature [START_REF] Van Schaeybroeck | Critical wetting, first-order wetting and prewetting phase transitions in binary mixtures of Bose-Einstein condensates[END_REF].

Remark 4.5. When K → +∞ as ε → 0 a refinement of the argument giving (51) gives also the optimal prefactor (see [START_REF] Conti | A branched transport limit of the Ginzburg-Landau functional[END_REF]Prop. 6.2]).

Study of the surface tension

We study the asymptotics of the surface tension σ λ,K in the limit regimes K → 1 and K → +∞. We first consider the case K → 1 (weak segregation).

Proposition 4.6. There holds:

lim K→1 σ λ,K √ K -1 = 2 3 1 -λ 3 1 -λ 2 .
Proof. Letting x = (K -1) -1/2 y in the definition of σ λ,K , we have

σ λ,K √ K -1 = inf R |η ′ 1 | 2 + λ 2 |η ′ 2 | 2 + η 2 1 η 2 2 + 1 2(K -1) η 2 1 + η 2 2 -1 2 : lim -∞ η 1 = 0, lim +∞ η 1 = 1 .
Hence, using similar Γ-convergence arguments as above (see also [START_REF] Goldman | Sharp interface limit for two components Bose-Einstein condensates[END_REF]), we get that,

lim K→1 σ λ,K √ K -1 = inf R |η ′ 1 | 2 + λ 2 |η ′ 2 | 2 + η 2 1 η 2 2 : lim -∞ η 1 = 0, lim +∞ η 1 = 1, η 2 2 = (1 -η 2 1 )
.

Letting η 1 = cos φ for lim -∞ φ = π 2 and lim +∞ φ = 0, and using that a 2 + b 2 ≥ 2ab, we find lim

K→1 σ λ,K √ K -1 ≥ 2 ∞ -∞ |φ ′ |(1 -(cos φ) 2 (1 -λ 2 )) 1/2 | cos φ sin φ| = 2 1 3(1 -λ 2 ) (cos 2 (x)λ 2 -cos 2 (x) + 1) 3/2 π 2 0 = 2 3 1 -λ 3 1 -λ 2
In order to reach equality it is enough to consider the unique solution of

φ ′ (1 -(cos φ) 2 (1 -λ 2 )) 1/2 = -cos φ sin φ, with φ(0) = π 4 .
The solution is indeed unique and defined on R since cos φ sin φ

(1-(cos φ) 2 (1-λ 2 )) 1/2 is Lipschitz continuous. Moreover, it is decreasing and has the right values at ±∞. Remark 4.7. Notice that the value of lim K→1 σ λ,K √ K-1 exactly coincides with the one found in [START_REF] Barankov | Boundary of two mixed Bose-Einstein condensates[END_REF] (see also [START_REF] Van Schaeybroeck | Interface tension of Bose-Einstein condensates[END_REF]). Using (51), it is moreover not hard to prove that there exists C > 0 (not depending on λ), such that

C -1 √ K -1 ≤ σ λ,K ≤ C √ K -1 when K → 1.
Remark 4.8. Proposition 4.6 also holds with the same proof for λ = 1. The limit is then equal to 1.

Eventually, we consider the strong segregation asymptotics, K → +∞ (notice that we recover the same scaling as the one predicted in the physics literature [START_REF] Van Schaeybroeck | Interface tension of Bose-Einstein condensates[END_REF]). Proposition 4.9. There exist 0 < c ≤ C such that for K > 1 and λ > 0, there holds

σ λ,∞ -C λ 1/2 (K -1) 1/4 + 1 K 1/2 ≤ σ λ,K ≤ σ λ,∞ -c 1 K 1/2 + λ 1/2 K 1/4 , (53)
with

σ λ,∞ = inf R |η ′ 1 | 2 + λ 2 |η ′ 2 | 2 + 1 2 η 2 1 + η 2 2 -1 2 : lim -∞ η 1 = 0, lim +∞ η 1 = 1, η 1 η 2 ≡ 0 , = (1 + λ) 2 √ 2 3 . ( 54
)
Proof. Let us establish the identity (54). Since any admissible pair (η 1 , η 2 ) with finite energy is continuous and satisfies η 1 (x) → 1 as x → +∞ and η 2 (x) → 1 as x → -∞, there exists x ∈ R such that (η 1 , η 2 )(x) = 0. Using translation and symmetry, we see that

σ λ,∞ = γ 1 + γ λ , with γ λ = inf +∞ 0 λ 2 |η ′ | 2 + 1 2 η 2 -1 2 : η ∈ W 1,2 loc ([0, +∞)), η(0) = 0, lim +∞ η = 1 .
The classical Modica-Mortola procedure applies to this minimization problem. For λ > 0 we have (see e.g. Chapter 6 in [START_REF] Braides | Γ-convergence for beginners[END_REF]),

γ λ = 2λ 2 1 0 (1 -s 2 ) 2 2λ 2 ds = 2 √ 2 3 λ.
Moreover, the minimizer is given by the formula

η(x) = tanh x √ 2λ for x ≥ 0. ( 55
)
Let us now establish the upper bound in (53) (right inequality). If λ K -1/2 (that is

λ 1/2 K 1/4
1 K 1/2 ), inspired by (55), we consider the profile

η 1 (x) =      0 if x < 0, tanh x √ 2 if x ≥ 0, η 2 (x) =      tanh δ -x √ 2λ if x ≤ δ, 0 if x > δ,
The parameter δ > 0 tunes the width of the overlap between the two species (see Figure 5). The energy f (δ) of this competitor is an upper bound for σ λ,K . We compute

f (δ) = E λ,K (η) = R |η ′ 1 | 2 + λ 2 |η ′ 2 | 2 + 1 2 η 2 1 + η 2 2 -1 2 + (K -1)η 2 1 η 2 2 = R |η ′ 1 | 2 + λ 2 |η ′ 2 | 2 + δ -∞ 1 2 (1 -η 2 2 ) 2 + +∞ 0 1 2 (1 -η 2 1 ) 2 + δ 0 (Kη 2 1 η 2 2 - 1 2 )
=

γ 1 + γ λ + δ 0 Kη 2 1 η 2 2 - 1 2 = σ λ,∞ + δ 0 Kη 2 1 η 2 2 - 1 2 .
Using the expressions of η 1 and η 2 and the concavity of t ∈ R + → tanh(t), we get

σ λ,K ≤ f (δ) ≤ σ λ,∞ - δ 2 + Kδ 5 4λ 2 1 0 s 2 (1 -s) 2 ds = σ λ,∞ - δ 2 + Kδ 5 120λ 2 .
Optimizing in δ, we obtain with δ = (12λ 2 /K) 1/4 , σ λ,K ≤ σ λ,∞ -4 5

12λ 2 K 1/4 = σ λ,∞ - 4(12) 1/4 5 √ λ K 1/4 .
This yields the right inequality of (53) with c = 4(12) 1/4 5 .

If on the contrary λ ≪ K -1/2 , we keep η 1 (x) = tanh(x/ √ 2) + as above and let

η 2 (x) =      1 x ≤ 0 1 -K 1/2 x x ∈ [0, K -1/2 ] 0 x ≥ K -1/2 .
The energy can then be estimated by

E λ,K (η 1 , η 2 ) ≤ γ 1 + λ 2 K 1/2 + K -1/2 0 1 2 (1 -η 2 2 ) 2 + (K -1)η 2 1 η 2 2 - 1 2 .
Notice now that λ 2 K 1/2 ≪ K -1/2 and that since tanh(x/ √ 2) ≤ x/ √ 2,

K -1/2 0 (1 -η 2 2 ) 2 + 2(K -1)η 2 1 η 2 2 -1 ≤ K -1/2 0 (1 -Kx 2 ) 2 + K 2 x 4 -1 ,
which implies,

K -1/2 0 1 2 (1 -η 2 2 ) 2 + (K -1)η 2 1 η 2 2 - 1 2 ≤ K -1/2 0 K 2 2 x 4 -Kx 2 = - 7 10 K -1/2 .
Putting all these estimates together gives the upper bound

σ λ,K ≤ σ λ,∞ -cK -1/2 .
We now turn to the proof of the lower bound (left inequality of (53)).

Let us consider an admissible pair η = (η 1 , η 2 ) ∈ W 1,2 loc (R, R 2 ) with finite energy. Without loss of generality, we may assume 0 < η 1 (x), η 2 (x) < 1 for x ∈ R. By continuity of η and from the conditions at infinity, there exists x 0 ∈ R such that η 1 (x 0 ) = λη 2 (x 0 ) = m and a maximal interval I 0 = [x 0 -δ -, x 0 +δ + ] such that m/2 ≤ η 1 , λη 2 ≤ 2m in I 0 . We define I -= (-∞, x 0 -δ -) and I + = (x 0 + δ + , +∞) and split the integration in three parts: E λ,K (η) = E -+ E 0 + E + , with

E i = I i |η ′ 1 | 2 + λ 2 |η ′ 2 | 2 + 1 2 η 2 1 + η 2 2 -1 2 + (K -1)η 2 1 η 2 2 ,
for i ∈ {-, 0, +}.

Using the Modica-Mortola trick, we have

E + ≥ +∞ x 0 +δ + |η ′ 1 | 2 + w K (η 1 ) ≥ 2 1 η 1 (x 0 +δ + )
w K (s) ds.

Using η 1 (x 0 + δ + ) ≤ 2m and 2 √ w K ≥ √ 2(1 -s 2 ) -√ 2χ [0,K -1/2 ] , we obtain,

E + ≥ √ 2 1 2m (1 -s 2 ) ds - √ 2K -1/2 ≥ 2 √ 2 3 -2 √ 2m - √ 2K -1/2 . (56)
Similarly,

E -≥ 2 √ 2λ 3 -2 √ 2m - √ 2K -1/2 . ( 57
)
Now we consider the middle part. Since min I 0 |η ′ 1 |, I 0 λ|η ′ 2 | ≥ m/2, we have by Cauchy-Schwarz inequality,

I 0 |η ′ 1 | 2 + λ 2 |η ′ 2 | 2 ≥ m 2 4(δ + -δ -)
.

On the other hand, since η 1 , λη 2 ≥ m/2 in I 0 , we have

I 0 1 2 η 2 1 + η 2 2 -1 2 + (K -1)η 2 1 η 2 2 ≥ (δ + -δ -)(K -1)m 4 16λ 2 .
Optimizing with respect to (δ + -δ -), we obtain,

E 0 = I 0 |η ′ 1 | 2 + λ 2 |η ′ 2 | 2 + I 0 1 2 η 2 1 + η 2 2 -1 2 + (K -1)η 2 1 η 2 2 ≥ √ K -1m 3 4λ . (58)

1 :

 1 Figure 1: Minimizer of the Thomas-Fermi energy.

Figure 3 :

 3 Figure 3: Relaxed potential w K (s) and limit potential w(s) = (1 -s 2 ) 2 (dashed line).

3 Figure 4 :

 34 Figure 4: Example of construction of the intervals I k j .

2 Figure 5 :

 25 Figure 5: Profile of the competitor for the upper bound

  (1 -s 2 ) = 29/192. Again, this gives[START_REF] Knüpfer | On an isoperimetric problem with a competing nonlocal term II: The general case[END_REF]. As in Proposition 3.3, we can make a spherical rearrangement and assume further that E is spherically symmetric.

	3/4
	1/2

2

) + . Since the estimate is sublinear, it is moreover enough to prove it for connected sets. Let us first show that we can assume that E ⊂ (B 1/2 ) c . Indeed, if this is not the case then there exists x ∈ ∂E ∩ B 1/2 . Then, there are two possibilities. Either E ⊂ B 3/4 and the classical isoperimetric inequality already gives

[START_REF] Knüpfer | On an isoperimetric problem with a competing nonlocal term II: The general case[END_REF]

or there is y ∈ ∂E ∩ B c 3/4 . Since ∂E is connected it contains a path from x ∈ B 1/2 to y ∈ B c 3/4 and F(E) is bounded from below by

Acknowledgment

We are grateful to an anonymous referee which pointed out a great simplification in the proof of Theorem 1.1.We thank G. De Philippis and V. Millot for helpful discussions about quantitative isoperimetric inequalities. B. Merlet was partially supported by the ANR research project GEOMETRYA, ANR-12-BS01-00 14-01

Gathering together (56),(57),(58), we get

Minimizing with respect to m, the minimum is reached for m = [16

This establishes the left inequality of (53) and ends the proof of Proposition 4.9.


F Perisse

P Venet

G Rojat

Reliability determination of aluminium electrolytic capacitors by the mean of various methods. Application to the protection system of the LHC

The lifetime of power electronic components is often calculated from reliability reports, but this method can be discussed. We compare in this article the results of various reliability reports to an accelerated ageing test of component and introduced the load-strength concept. Large aluminium electrolytic capacitors are taken here in example in the context of the protection system of LHC (Large Hadron Collider) in CERN where the level of reliability is essential. We notice important differences of MTBF (Mean Time Between Failure) according to the reliability report used. Accelerating ageing tests carried out prove that a Weibull law is more adapted to determinate failure rates of components. The load-strength concept associated with accelerated ageing tests can be a solution to determine the lifetime of power electronic components.

Introduction

The choice of the components in power electronics is guided by two principal parameters. First is commonplace, it relates to the rated values which is requested by the operation of the system. The second relates to reliability and is subject to discussion. Indeed, reliability depends on many parameters, it is difficult to evaluate in many cases. Currently the reliability of components is in the majority of the cases determined from reliability reports resulting from statistical results. We can quote for example the following reports: MILL HDBK 217F [START_REF] Specs | Military Handbook Reliability Prediction of Electronic Equipment, MIL-HDBK-217F[END_REF] Rdf 93, Rdf 2000 [START_REF]Recueil de données de fiabilité des composants électroniques RDF93[END_REF][START_REF]Modèle universel pour le calcul de la fiabilité prévisionnelle des composant, cartes et équipements électroniques[END_REF] Telcordia [4] 299B share Stress [5] HRD5 [6].

In this article, we propose to compare results from the reliability reports with results from accelerated ageing test. We describe also a righter approach of named load-strength concept.

We take as example aluminium electrolytic capacitors. These components are used by CERN (European Organization for Nuclear Research) in power supply "quench heaters" used for the protection of the superconductive magnets of the LHC (Large Hadron Collider) [START_REF] Evans | The LHC Project[END_REF]. As the electrolytic capacitors constitute the weak point of power supply [START_REF] Venet | Monitoring of power converter. Application to predictive maintenance[END_REF][START_REF] Mr | Life Prediction Model for Aluminium Electrolytic Capacitors[END_REF][START_REF] Perisse | Behavioral Study one the Reliability of Electrolytic Capacitors of Various Manufacturers[END_REF] their choice is of primary importance for the reliability of the LHC. The numbers of power supplies is 6000; the system thus comprises 36000 capacitors whose reliability is quite necessary.

Components and protection system studied

The components used in this study are large aluminium electrolytic capacitors of rated values C R = 4700µF, U R = 500V and temperature range -40°C, 85°C. We show in figure 1 the geometry of various components studied. Components produced by different manufacturers with the same rated values can have very different levels of reliability.

These differences are due to the technological parameters adopted by the manufacturers [START_REF] Perisse | Behavioral Study one the Reliability of Electrolytic Capacitors of Various Manufacturers[END_REF].

For example, we noted catastrophic failures related to:

• anode corrosion due to the intrusion of halogenous ions (D components) • internal overpressures due to the geometry of the components (E and F components)

•
breakdown due to the connection of the tabs on the terminals (A components)

The protection of the superconductive magnets is guaranteed when the energy which is accumulated in the capacitors reached 1.8kJ [START_REF] Perisse | Study and Analyses of Failure Modes of the Electrolytic Capacitors and Thyristors, Applied to the protection of the LHC (Large Hadron Collider)[END_REF]. This energy is transmitted to the protection system via two thyristors (1800V -80A) (see Fig. 2). The operating conditions are 450V DC and an ambient temperature of 20°C.

The six capacitors can store an maximum energy of 2.85kJ. Thus, the system can accept the failure of one or more capacitors according to the failure mode considered (short or open circuit).

We study two cases of failure:

• Case n°1: Failure of the capacitors by shortcircuit. • Case n°2: Failure of the capacitors by open circuit.

Normative approach

It consists of the use of reliability reports previously quoted. Calculations of failure rate are based on the characteristics of the components and on their way of use. For example, we show in table 1, principal parameters necessary to calculate failure rate of electrolytic capacitors for various reliability report.

Calculation of failure rates

The individual failure rates of capacitor and thyristor are defined in FIT (Failure In Time = 1 faillure/10 9 hours). The results are calculated for different reliability reports for 6000 power supplies and are defined by the use of MTBF (see. fig. 3).

We note significant differences between the reliability reports. These differences can partly be explained by the matter of fact that they privilege more or less some influence factors [START_REF] Jones | A Comparison of Electronic-Reliability Prediction Models[END_REF]. The diversity of these results confirms the importance of accelerated ageing test under the conditions which are close to the mission profile of the system.

Accelerating aging test, statistical approach

The principal cause of breakdown of electrolytic capacitors is a progressive failure. It is due to the timedependent evaporation of the electrolyte.

The lifetime of these components depends on the criterion chosen to define it. The parameter, which is varying more during ageing time of this component, is equivalent series resistance (ESR). The variation of this parameter is considered as the criterion of end of lifetime [START_REF] Venet | Monitoring of power converter. Application to predictive maintenance[END_REF][START_REF] Perisse | Behavioral Study one the Reliability of Electrolytic Capacitors of Various Manufacturers[END_REF][START_REF] Perisse | Study and Analyses of Failure Modes of the Electrolytic Capacitors and Thyristors, Applied to the protection of the LHC (Large Hadron Collider)[END_REF]. The increase allowed in the ESR depends on the application in which the component is used. We take a limit, which is defined by a multiplication by factor two of the ESR at 100Hz.

MIL-HDBK-217 FN2 X X X X X RDF 93 X X X X X Telcordia X X X X X 299 B Part Stress X X X X X HRD5 X X X RDF 2000 X X X X X 0
There are many failure laws possible to describe the lifetime of a system or a component. We use the exponential law, the Weibull law and normal law in order to determine their distribution functions of failure and to compare them. These various laws are characterized by their function of probability density f(t). From f(t) it is possible to determine the failure distribution function F(t). The relationship between these two functions and the failure rate λ versus time are defined by [START_REF] Klass B Klassen | System Reliability, Concept and Applications[END_REF]:

0 ( ) ( ) t F t f t = ∫ dt (1) ( ) ( ) 1 ( ) f t t F t λ = - (2)
These functions are described in equation 3 to 11 for different distribution laws.

Exponential law

Probability density function: ( ) .exp( . )

f t t λ λ = - (3)
Failure distribution function:

( ) 1 exp( . )

F t t λ = - - (4) Failure rate: ( ) t λ λ = (5)

Weibull law

Probability density function:

1 ( ) . .exp when t t f t t β β β γ γ γ η η η - ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ - - ⎜ ⎟ = - ≥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (6)
Failure distribution function:

( ) 1 exp when t F t t β γ γ η ⎛ ⎞ ⎛ ⎞ - ⎜ ⎟ = - - ≥ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (7)
Failure rate:

1 ( ) . when t t β β γ t λ γ η η - ⎛ ⎞ - = ⎜ ⎟ ⎝ ⎠ ≥ (8)
β, γ, η are coefficients dependent on the components studied.

Normal law

Probability density function:

2 2 ( ) 1 1 ( ) .exp . 2 . 2. t f t µ σ π σ ⎛ ⎞ - = - ⎜ ⎜ ⎝ ⎠ ⎟ ⎟ (9)
Failure distribution function: Failure rate:

2 2 ( ) 1 1 ( ) .exp . . 2 . 2. t t F t d µ σ π σ -∞ ⎛ ⎞ - = - ⎜ ⎜ ⎝ ⎠ ∫ t ⎟ ⎟ (
2 2 2 2 ( ) 1 exp . 2 ( ) ( ) 1 .exp . . 2 t t t t dt µ σ λ µ σ ∞ ⎛ ⎞ - - ⎜ ⎟ ⎜ ⎟ ⎝ = ⎛ ⎞ - - ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∫ ⎠ (11)
µ and σ are coefficients dependent on the components studied.

The failure distribution F(t) and the failure rate which is deduced from these various distribution laws are represented in figures 4 and 5.

The most precise failure distribution function is the normal law. However this normal law induces a reduction in the failure rate after a certain ageing time.

The real ageing of an aluminium electrolytic capacitor is not represented in this way. T he exponential law considers a constant failure rate. This failure rate does not represent the wear-out period. The most appropriate reliability law is the Weibull law.

Approach to load-strength

This approach is based on the energy storage concept originally introduced into reliability engineering by Haviland [START_REF] Haviland | Engineering Reliability and Long Life Design[END_REF][START_REF] Jensen | Electronic Component Reliability[END_REF]. We detail below the basis of this healthy components concept.

A component or a part stores energy when a load is applied. This energy can be electrical, thermal, mechanical one. The energy, which is stored in the component, is the stress, which is induced by the load and the environment. This stress can be compared to the maximum strength the component can support.

Capacitor example

We consider three cases. In the first case the load is represented by the voltage applied to the component, in the second case by the temperature, and in the last case by the energy stored in the component. In all cases we represent the load according to a normal distribution law.

-1 st case: the load of the component is defined by 450V±10% voltage. The manufacturers define the maximum strength as 115% of rated voltage i.e. 575V.

-2 nd case: the load is defined by environment temperature. The maximum strength is 85°C and load is a normal distribution whose average is equivalent to the applied ambient temperature of 20°C. Figure 6 to 8 shows the load and the maximum strength for these three cases.

A safety factor SF can be deduced. SF represents the separation of the load and the maximum strength [START_REF] Jensen | Electronic Component Reliability[END_REF]:

2 2 - = + S L S L SF σ σ (3)
Where:

-S : mean value of the strength = maximum strength in our case. -L : mean value of the applied load -σ L : standard deviation of the load distribution

Ageing effect

The component ageing induces the decrease of the maximum strength value. In order to guarantee reliability, make sure that there is no overlapping of the load diagram and strength curve (maximum strength in our cases). If there exists any overlapping the failure probability due to this one can be determined. The use of reliability reports can be discussed. Indeed the results are very different according to the reliability reports. Moreover they do not take account of the technological differences according to various manufacturers [START_REF] Perisse | Behavioral Study one the Reliability of Electrolytic Capacitors of Various Manufacturers[END_REF][START_REF] Perisse | Study and Analyses of Failure Modes of the Electrolytic Capacitors and Thyristors, Applied to the protection of the LHC (Large Hadron Collider)[END_REF]. We showed that the calculation of failure rate by the use of exponential law is too approximate. The use of Weibull law is better. The load-strength concept associated with accelerated ageing tests can be a solution to determine the lifetime of power electronic components.

Conclusion

Fig. 1 .

 1 Fig. 1. Capacitors studied

Fig. 2 .

 2 Fig. 2. General diagram of power supply

Fig. 3 MTBF

 3 Fig. 3 MTBF of 6000 power supplies for different reliability reports.

Fig. 5 .

 5 Fig. 5. Failure rate versus time

Fig. 4 .

 4 Fig. 4. Failure distribution versus time.

Fig. 7 .

 7 Fig. 7. Distribution of failure density versus temperature

Fig. 8 .

 8 Fig. 8. Distribution of failure density versus energy

Fig 6 .

 6 Fig 6. Distribution of failure density versus voltage

Table 1

 1 Principal parameters of various reliability reports

	Principal parameters

Acknowledgements

We thank the CERN represented by Mr. Felix Rodriguez-MATEOS and Mr. Reiner DENZ which allowed, by a profitable collaboration the realization of this work.