A rigorous derivation of a free-boundary problem arising in superconductivity

Ann. Scient. Éc. Norm. Sup., 4e série, t. 33, 2000, p. 561 à 592.

A RIGOROUS DERIVATION OF A FREE-BOUNDARY PROBLEM ARISING IN SUPERCONDUCTIVITY B Y É TIENNE SANDIER

AND

S YLVIA SERFATY

A BSTRACT . – We study the Ginzburg–Landau energy of superconductors submitted to a possibly nonuniform magnetic field, in the limit of a large Ginzburg–Landau parameter κ. We prove that the induced magnetic fields associated to minimizers of the energy-functional converge as κ → +∞ to the solution of a free-boundary problem. This free-boundary problem has a nontrivial solution only when the applied magnetic field is of the order of the “first critical field”, i.e. of the order of log κ. In other cases, our results are contained in those we had previously obtained [15, 16, 14]. We also derive a convergence result for the density of vortices.  2000 Éditions scientifiques et médicales Elsevier SAS R ÉSUMÉ . – On étudie l’énergie de Ginzburg–Landau des supraconducteurs soumis à un champ magnétique éventuellement non-uniforme, dans la limite d’un grand paramètre de Ginzburg–Landau κ. On montre que le champ magnétique induit associé aux minimiseurs de l’énergie converge lorsque κ → +∞ vers la solution d’un problème à frontière libre. Ce problème à frontière libre n’admet de solution non triviale que lorsque le champ magnétique appliqué est de l’ordre du “premier champ critique”, c’est-à-dire de l’ordre de log κ. Dans les autres cas, nos résultats sont inclus dans ceux que nous avions précédemment obtenus [15, 16, 14]. On obtient aussi un résultat de convergence de la densité de vortex.  2000 Éditions scientifiques et médicales Elsevier SAS

1. Introduction 1.1. The Ginzburg–Landau model of superconductivity The Ginzburg–Landau model was introduced in the fifties by Ginzburg and Landau as a phenomenological model of superconductivity. In this model, the Gibbs energy of a superconducting material, submitted to an external magnetic field is, in a suitable normalization, (1.1)

J(u, A) =

1 2

Z |∇A u|2 + Ω


2 κ2 1 − |u|2 + 2

Z |h − hex |2 . R3

Here, Ω is the domain occupied by the superconductor, κ is a dimensionless constant (the Ginzburg–Landau parameter) depending only on characteristic lengths of the material and on temperature. hex is the applied magnetic field, A : Ω 7→ R3 is the vector-potential, and the induced magnetic field in the material is h = curl A. ∇A = ∇ − iA is the associated covariant derivative. The complex-valued function u is called the “order-parameter”. It is a pseudo-wave function that indicates the local state of the material. There can be essentially two phases in a superconductor: |u(x)| ' 0 is the normal phase, |u(x)| ' 1, the superconducting phase. The ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE. – 0012-9593/00/04/ 2000 Éditions scientifiques et

médicales Elsevier SAS. All rights reserved

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Ginzburg–Landau model was based on Landau’s theory of phase-transitions. Since then, the model has been justified by the microscopic theory of Bardeen–Cooper–Schrieffer (BCS theory). |u(x)| is then understood as the local density of superconducting electron pairs, called “Cooper pairs”, responsible for the superconductivity phenomenon. A common simplification, that we make, is to restrict to the two-dimensional model corresponding to an infinite cylindrical domain of section Ω ⊂ R2 (smooth and simply connected), when the applied field is parallel to the axis of the cylinder, and all the quantities are translation-invariant. The energy-functional then reduces to (1.2)

1 J(u, A) = 2

Z |∇A u|2 + |h − hex |2 +


2 κ2 1 − |u|2 . 2

Ω

Then A : Ω 7→ R2 , h is real-valued, and hex is just a real parameter. The Ginzburg–Landau equations associated to this functional are  (G.L)

−∇2A u = κ2 u(1 − |u|2 ), −∇⊥ h = hiu, ∇A ui.

with the boundary conditions 

on ∂Ω , h = hex h∇u − iAu, ni = 0 on ∂Ω .

(Here ∇⊥ denotes (−∂x2 , ∂x1 ), and h. , .i denotes the scalar-product in R2 .) One can also notice that the problem is invariant under the gauge-transformations: 

u → ueiΦ , A → A + ∇Φ,

where Φ ∈ H 2 (Ω , R). Thus, the only quantities that are physically relevant are those that are gauge invariant, such as the energy J , the magnetic field h, the current j = hiu, ∇A ui, the zeros of u. We saw in [16,14] that, up to a gauge-transformation, the natural space over which to minimize J is {(u, A) ∈ H 1 (Ω , C) × H 1 (Ω , R2 )}. 1.2. Critical fields and vortices √ When κ > 1/ 2 the superconductor is said to be “of type-II”. As in our previous studies [14– 17], we shall only consider the case in which κ is large (i.e. we study the “London limit” κ → ∞ of “extreme type-II superconductors”), and for simplicity we set  = 1/κ. Superconductors in general, when cooled down below a critical temperature, become “superconducting”, which means in particular that there can be permanent currents without energy dissipation. But they also have a particular behaviour when a magnetic field is applied, which we now describe for type-II superconductors (for further physics reference, see for example [19]). When the applied field hex is small enough, the material is superconducting: |u| ∼ 1, and the magnetic field is “expelled” (this is called the Meissner effect): 

4e SÉRIE – TOME 33 – 2000 – N◦ 4

−∆h + h = 0 h = hex

in Ω , on ∂Ω .

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When hex is raised, one observes two main phase-transitions, for two critical fields Hc1 and Hc2 : for hex = Hc1 = O(|log |), there is a phase-transition from the superconducting state described above to the “mixed-phase” or “mixed-state”; for hex = Hc2 = O(1/2 ), from the “mixedphase” to the normal state (u ≡ 0, h ≡ hex ). The mixed-phase is defined by the coexistence of the normal and superconducting phases in the sample. The normal phase is localized in small regions of characteristic size  = 1/κ called “vortices”, surrounded by a superconducting region in which |u| ∼ 1. u vanishes at the center of a vortex, and if C is a small circle centered at this point and ϕ the phase of u on C, the degree of the vortex is defined as 1 2π

Z

∂ϕ =d∈Z ∂τ

C

or as the topological-degree of the map u/|u| : C 7→ S 1 . In stable stationary situations, the vortices are generally all of degree 1, and they repell one another according to a coulombian interaction. When hex is close to Hc1 , there are only a few vortices, and when hex is increased, their number increases, and, in order to minimize their repulsion, they tend to arrange in a triangular lattice, called the “Abrikosov lattice”. The induced magnetic field approximately satisfies  P −∆h + h = 2π i di δai in Ω , on ∂Ω , h = hex where the ai ’s are the centers of the vortices, and the di ’s their degrees. We are interested in describing rigorously the mixed-phase (which we have started to do in [15]). Several numerical or formal studies have been carried out to describe this mixed-phase, for example those of Chapman, Rubinstein and Chapman [8], and Berestycki, Bonnet and Chapman [3]. 1.3. Previous studies of vortices On a mathematical viewpoint, many papers have made clear the mathematical mechanisms of the apparition of vortices, and the definitions of what can be called a “vortex-structure”. The first and main work was the book of Bethuel, Brezis and Hélein [4], where the authors studied the functional Z
2 1 1 (1.3) |∇u|2 + 2 1 − |u|2 , F (u) = 2 2 Ω

in the limit  → 0. This corresponds to setting A and hex equal to zero. Then, the influence of the fields has to be replaced by a Dirichlet boundary condition u = g on ∂Ω , where g is an S 1 -valued map of winding-degree d > 0. This boundary condition triggers the apparition of vortices. They proved that minimizers of F have d vortices of degree one, and that the following expansion of the energy holds: (1.4)

F (u) ∼ πd|log | + W (a1 , . . . , ad ) as  → 0,

W , the “renormalized-energy” being a function depending only on the vortex-centers ai . Afterwards, Almeida and Bethuel [1], Sandier [13], Jerrard [10], independently developed methods to construct vortices (or define a vortex-structure) of energetical cost π|d||log | for arbitrary maps u (and not only for critical points of F ). There has also been a lot of research on the full Ginzburg–Landau functional J itself, for example the study of radial solutions by Berger and Chen. Bethuel and Rivière have proved in ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

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[6] results in the spirit of (1.4), replacing again the boundary conditions by Dirichlet boundary conditions (with hex = 0). In [16–18], the full functional was studied for hex not too high above Hc1 , and without Dirichlet boundary conditions. More precisely, local minimizers of J were found by minimizing it over the set of configurations such that F (u) < M |log |,

(1.5)

corresponding to configurations with a bounded number of vortices (as  → 0). This led to the following asymptotic expansion of Hc1 : Hc1 ∼ k1 (Ω )|log | + O(1) as  → 0,

(1.6) where


−1 , k1 (Ω ) = 2 max |ξ0 |

ξ0 being defined by



(1.7)

−∆ξ0 + ξ0 = −1 in Ω , on ∂Ω . ξ0 = 0

It also led to the existence of branches of stable n-vortices solutions, whose vortices were located. The estimate (1.5) allowed to use the method of construction of vortices of Almeida and Bethuel, with an a priori uniform bound on the number of vortices. Then, thanks to this construction and uniform bound, an expansion of J of the type (1.4), involving also a renormalized energy, was derived. The study of global minimizers of the energy (thus without the bound (1.5)) is more delicate in the sense that the number of vortices does not remain bounded as  → 0, and thus the type of analysis of [4] can no longer be reproduced. We have already studied two situations for the repartition of vortices. In the case hex < Hc1 , we proved in [14] that there are no vortices in the minimizing configurations. In the case Hc1  hex  1/2 , we proved in [15] that there is a uniform repartition of vortices, of density hex /2π. (Here, notice that hex has to be considered as a function of , and recall that Hc2 = O(1/2 ).) In both situations, vortices are defined through the methods of Sandier and Jerrard, i.e. in a weaker sense than as in [4] or [1], where their number remains bounded. 1.4. Purpose of this paper Our aim here is to describe the repartition of vortices in the minimizers for arbitrary applied fields, in particular for fields of the order of |log |, case which was left open. We show that in the limit  → 0, minimizers of J have a uniform vortex-distribution in a sub-region ωλ ⊂ Ω which is the solution of a free-boundary problem resembling the model of [8,3]. In [8], the authors formally derive the equation for the limiting magnetic field without however computing the number of vortices for a minimizing configuration. On the contrary, our approach, which consists in expanding min J as in (1.4) (except that the positions of vortices are replaced by a measure), allows it. For the sake of more generality, we consider in the sequel the functional (1.8)

1 J(u, A) = 2

Z |∇A u|2 + |h − phex |2 + Ω

4e SÉRIE – TOME 33 – 2000 – N◦ 4


2 κ2 1 − |u|2 , 2

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where p(x) is a smooth (C 2 is fine) positive weight. When p ≡ 1, it is the standard Ginzburg– Landau energy. Otherwise, we consider applied fields of intensity p(x)hex , which may be able to account for non-uniform applied fields. Note that in [14,15], only uniform applied fields were considered, but the analysis there could probably be adapted to the case of a weighted field. 1.5. The main result We consider hex as a function of , and assume throughout this paper that the limit |log | →0 hex ()

λ = lim

exists and is finite (this implies in particular that hex → ∞ as  → 0), and we also assume that (in the case λ = 0) 1 . 2 instead of hex () and J for the corresponding functional, the hex () 

From now on we will write hex -dependence being implicit. By testing J with (u ≡ 1, A ≡ 0) it is clear that, for minimizers, J(u, A)/h2ex remains bounded as  → 0. Here, we prove that J/h2ex converges in a sense similar to Γ -convergence to the limiting functional Z Z 1 λ ∇(f − p) 2 + |f − p|2 , −∆(f − p) + f + (1.9) E(f ) = 2 2 Ω

defined over

Ω

 V = f ∈ Hp1 (Ω ) | −∆(f − p) + f is a Radon measure ,

where Hp1 denotes the functions f in H 1 (Ω ) such that f (x) = p(x) on the boundary. More precisely, we prove that the induced magnetic fields of minimizers of J converge, after a renormalization, to the minimizer of E. The minimizer h∗ of E will be proved to be unique and to be the solution of the following variational inequality, usually called an “obstacle problem”:

(P)

 h∗ ∈ Hp1 (Ω ),      h∗ > p − λ in Ω , 2  λ  1  (Ω ) such that v > p − , ∀v ∈ H  p  2

Z


−∆(h∗ − p) + h∗ (v − h∗ ) > 0.

Ω

This obstacle problem is quite standard (one can refer to [12] for example). It is a free boundary problem in the sense that h∗ is determined by its coincidence set, defined by 

 λ . ωλ := x ∈ Ω | h∗ (x) = p(x) − 2 It is also a classical result that the free boundary ∂ωλ is not always smooth (there have been counterexamples by Schaeffer), yet, A. Bonnet and R. Monneau proved recently in [5] that it is smooth for almost every value of λ. When this is the case, ωλ is determined by the fact that there ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

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exists a solution h∗ to the over-determined system  −∆(h∗ − p) + h∗ = 0     λ    h∗ = p − 2  ∂(h∗ − p)   =0   ∂n   h∗ = p

in Ω \ωλ , in ωλ , on ∂ωλ , on ∂Ω ,

In addition, h∗ ∈ C 1,α (Ω ), ∀α < 1. The first theorem we prove is T HEOREM 1. – Assume λ = lim→0 |log |/hex exists and is finite and, if λ = 0, hex  1/2 . Consider, for every , (u , A ) minimizing J , and h = curl A the associated magnetic field. Then, as  → 0, h * h∗ hex →0

h → h∗ hex →0

weakly in H 1 (Ω ),

strongly in W 1,q (Ω ), ∀q < 2,

where h∗ is the unique minimizer of E, and the solution of the free-boundary problem (P). The lack of compactness in the above convergence is described by a defect measure:   2 ∇ h − h∗ → λµ, →0 hex in the sense of measures, where µ = −∆(h∗ − p) + h∗ . In addition, Z Z 2 λ 1 J(u , A ) ∇(h∗ − p) + |h∗ − p|2 , = E(h ) = |µ| + lim ∗ 2 →0 hex 2 2 Ω

Ω

where E is defined by (1.9). One can easily notice that if λ = 0 (i.e. if hex  |log |), the solution of (P) is h∗ = p, and E(h∗ ) = 0. In this case, the theorem asserts that h →p hex

strongly in H 1 ,

and

min J → 0. h2ex

This was already proved in [15] in the case p ≡ 1, and will be checked in other cases. In [15], the stronger result 1 1 min J ∼ vol(Ω )hex log √ 2  hex was proved (still for λ = 0). We conjecture that this stronger result holds when the applied field is phex , in the modified form:  Z 1 1 p(x) dx hex log √ . min J ∼ 2  hex Ω

We will thus treat separately the simple case λ = 0 in Corollary 2.2, and we now describe some stronger results for λ > 0. 4e SÉRIE – TOME 33 – 2000 – N◦ 4

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1.6. Results for the case λ > 0 In this case, hex satisfies the a priori bound hex 6 C|log |. Once we restrict ourselves to this case, the proof of Theorem 1 uses, as our previous papers, a construction of vortex-balls. Here, we use the following result, which is adjusted from [15]: P ROPOSITION 1.1. – If hex 6 C|log |, there exists 0 such that if  < 0 and (u , A ) is a minimizer of J , there exists a family of balls (depending on ) (Bi )i∈I = (B(ai , ri ))i∈I satisfying:  [  1 (1.10) B(ai , ri ). ⊂ x | |u | 6 2 i∈I

X

(1.11)

ri 6

i∈I

(1.12)

1 2

Z

1 , |log |6


∇(h − phex ) 2 > π|di ||log ε| 1 − o(1) ,

Bi

where h = curl A , and di = deg(u /|u |, ∂Bi ) if Bi ⊂ Ω and 0 otherwise. This proposition will be proved at the beginning of Section 3. Using it, Theorem 1 can be made more precise: T HEOREM 2. – Under the hypotheses of Theorem 1, assuming in addition that λ > 0, we have 2π X di δai → µ, →0 hex

(1.13)

i∈I

2π X |di |δai → µ, →0 hex

(1.14)

i∈I

in the sense of measures, where   λ 1 ωλ µ = −∆(h∗ − p) + h∗ = p − 2 and the (ai , di )’s are any “vortices” satisfying the results of Proposition 1.1. 1.7. Interpretations and consequences These results first indicate that hex h∗ can be seen as a good approximation of h as  → 0, and provide a new asymptotic expansion of the energy. Also, the vortex-density is approximately hex µ so that, when there are vortices, the domain is split in two regions given by problem (P): one central region ωλ in which the vortex-density is roughly equal to phex − 12 |log | and vortices have positive degrees, and a peripheral region in which there are no vortices. When hex becomes much higher than |log |, the central region tends to occupy the whole domain, and we are led to a vortex-density phex , which generalizes the results of [15]. Let us also point out that the defect measure for the H 1 -convergence of h/hex to h∗ in Theorem 1 exactly corresponds to the concentration of energy in the vortices, whereas S 1 2 kh − pk corresponds to the remaining energy outside of B ∗ i i . This appears clearly in H1 2 ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

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the proofs (see Section 1.8 for a sketch). This phenomenon is of the same type as the one that was described by Cioranescu and Murat in [7]. In this intermediate case hex ∼ C|log | (or 0 < λ < ∞), the energy that concentrates in the vortices (tending to the first term in E(h∗ )) is of the same order as the energy outside of the vortex-cores (which corresponds to the second term in E(h∗ )); whereas when hex  |log |, the outside energy becomes negligible as can be seen from the construction of [15]. This also explains why the analysis in this intermediate case is more delicate, the two contributions of energy having to be precisely taken into account. As already mentioned, these results, which describe a homogeneized behaviour of vortices, are reminiscent of existing free-boundary models for superconductivity of [8] and [3]. To our knowledge however, the range of hex for which our model is valid is somewhat clearer, and our approach, which consists in deriving the problem (P) rigorously from the minimization of J and using the vortices, seems to be new. Let us also mention that the behaviour of solutions of problems like (P) has been studied (see [12] for instance). The similar Berestycki–Bonnet–Chapman model has also been studied by Bonnet and Monneau in [5]. Now, problem (P) can be further analyzed to understand the apparition and location of vortices in link with the behaviour of the coincidence set. We proved in [16] and [14] that — in the case p(x) ≡ 1 — below the first critical field Hc1 , the minimizer of the energy is vortex-less, where Hc 1 =

(1.15)

1 |log | + O(1), 2 max|ξ0 |

and where ξ0 was defined in (1.7). Then, we naturally wish to compare this former result with the ones we present here. This is the content of the following proposition, easily derived from the maximum principle. Let ψ be the solution of  (1.16)

−∆ψ + ψ = −p ψ=0

in Ω , on ∂Ω .

ψ is the analogue of ξ0 with the weight p. P ROPOSITION 1.2. – The function ψ being defined in (1.16), and ωλ being the coincidence set corresponding to problem (P), (1) Ω \ωλ is connected. (2) 1 hex < . →0 |log | 2 max|ψ|

ωλ = ∅ ⇔ λ > 2 max |ψ| ⇔ lim

In this case h∗ = p + ψ = ∆ψ. In all cases, h∗ > p + ψ. (3) ∃C > 0,

dist(ωλ , ∂Ω ) > Cλ.

For p(x) ≡ 1, our results match with [14,16] since in this case ψ = ξ0 , giving the value (1.15) for Hc1 (actually, this only gives an equivalent as  → 0 of Hc1 , hence is less precise than our previous results). For λ > 2 max|ξ0 |, i.e. roughly for hex < Hc1 , Theorem 1 tells us that h/hex → ∆ξ0 , which was already proved in [18]: this corresponds to the Meissner (i.e. vortex-less) solution, for which the magnetic field was known to be approximately solution of 4e SÉRIE – TOME 33 – 2000 – N◦ 4

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the London equation



−∆h + h = 0 h = hex

569

in Ω , on ∂Ω ,

the solution of this equation being exactly hex ∆ξ0 . Therefore, when p ≡ 1, the results we get here are really new only for 0 < λ < 2 max|ξ0 |, which corresponds to the intermediate region Hc1 6 hex 6 O(|log |). On the other hand, if p(x) is not identically 1, they yield a new estimate of the corresponding “first critical field” Hc1 ∼

1 |log |. 2 max |ψ|

The last assertion in Proposition 1.2 allows us to control the growth of the vortex-region ωλ as it tends to occupy all of Ω . Proof of Proposition 1.2, assuming Theorem 2. – The proof of the first assertion follows [8]. If Ω \ ωλ = ∅, there is nothing to prove. If not, then λ 6= 0, therefore h∗ − p = −λ/2 < 0 on ωλ while h∗ − p = 0 on ∂Ω , thus ∂ωλ ∩ ∂Ω = ∅. Then from the simple connectedness of Ω , it follows that if Ω \ωλ was not connected, it would have a connected component Ω 0 such that ∂Ω 0 ⊂ Ω . Hence   −∆(h∗ − p) + h∗ = 0 in Ω 0 , λ  h∗ = p − on ∂Ω 0 . 2 Thus, h∗ < p − λ2 in Ω 0 by the maximum principle, contradicting (P). We turn to the second assertion. From Theorems 1 and 2, −∆(h∗ − p) + h∗ > 0 which can be written as −∆(h∗ − p) + h∗ − p > −p, while h∗ − p = 0 on ∂Ω . Subtracting (1.16) and using the maximum principle, we find that (1.17)

h∗ − p > ψ.

But h∗ − p = −λ/2 on ωλ . Thus, if ωλ 6= ∅, then |ψ| > λ/2 on ωλ hence λ 6 2 max |ψ|. On the other hand, if ωλ = ∅, h∗ − p = ψ. But from the second equation of (P), we have h∗ − p > −λ/2, this implies |ψ| < λ/2, finishing the proof of the second assertion. For the third assertion, we use (1.17) again, which yields ωλ ⊂ {x ∈ Ω | ψ(x) 6 −λ/2}. As ψ = 0 on ∂Ω , there exists a C > 0 such that ψ > −Cdist(. , ∂Ω ), hence the distance between ∂Ω and the set {x/ψ 6 −λ/2} remains bounded from below by Cλ for some other C > 0, proving the third assertion and the proposition. 2 1.8. Outline of the paper In Section 2, we prove the upper bound min J/h2ex 6 E(h) + o(1) — where E is defined in (1.9) — for any h ∈ Hp1 (Ω ) such that µ = −∆(h − p) + h is a positive Radon measure, absolutely continuous with respect to the Lebesgue measure. It will be proved in Section 3 that the measure corresponding to h∗ , the minimizer of E, indeed satisfies this condition. The upper bound is computed as follows: given h, and the corresponding measure µ, we construct a test-configuration having vortex-density close to µ. This construction is somewhat similar to that of the upper bound of [15] in thatPit starts from the idea of constructing first a magnetic field satisfying −∆(h − p) + h = 2π i δai , where the ai ’s denote the vortices. Again, this construction, and particularly that of [15], has some similarity with that of [7]. In [7] they constructed a periodic function on a domain with small holes, (corresponding to our vortex-cores) ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

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whose number diverges. The scale of the holes relatively to their distances were the same as in our construction. Yet, one of the main differences here is that the test-function is no longer periodic and thus we had to change the method of construction, as well as the method of evaluation of the energy of h. Once h is set, we construct a suitable u and evaluate the energy of the configuration, expressed in terms of an energy for the vorticity measure, similar to the interaction energy in potential theory. In Section 3, the matching lower bound is derived. Let us sketch the main steps of the proof. Considering (u , A ) any minimizer of the energy J , as a critical point, it satisfies the following Ginzburg–Landau equations :  1   −(∇ − iA)2 u = 2 u(1 − |u|2 ) in Ω ,  (G.L.) ⊥ 2   −∇ (h − phex ) = hiu, ∇u − iAui = ρ (∇ϕ − A) in Ω , on ∂Ω , h = hex where h = curl A is still the induced magnetic field, and u is written (where possible) u = ρeiϕ , with ρ 6 1 — a standard consequence of the maximum principle (see [16] for instance). On the other hand |∇A u|2 = |∇ρ|2 + ρ2 |∇ϕ − A|2 , hence from the second (G.L.) equation, we deduce that Z

Z |∇A u|2 >

(1.18) Ω

|∇ρ|2 +

|∇(h − phex )|2 > ρ2

Ω

Z

∇(h − phex ) 2 .

Ω

Next, we use the construction of vortices that we recalled in Proposition 1.1. Once these vortices of u are given, we can define the family of measures P 2π i∈I di δai (1.19) . µ = hex Moreover, as from (1.12) P each vortex carries at least an energy π|di ||log |, and J(u, A) 6 C|log |2 ; we have i |di | 6 C|log | 6 Chex , thus µ is a bounded family of measures. Therefore, up to extraction of a subsequence, we can find a measure µ0 such that (1.20)

µ → µ0

in the sense of measures. Similarly, from (1.18) and J(u, A) 6 Ch2ex , we can find h0 such that (1.21)

h * h0 hex

weakly in Hp1 .

Formally, when  = 0, the London-type equation holds: −∆(h − phex ) + h = 2π

X

di δai .

i

We can make this statement rigorous by proving the following identity: (1.22) 4e SÉRIE – TOME 33 – 2000 – N◦ 4

−∆(h0 − p) + h0 = µ0 .

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We can then derive the lower bound min J/h2ex > E(h0 ) − o(1). First, from (1.18), we have Z 2 1 ∇(h − phex ) + |h − hex |2 . (1.23) J(u , A ) > 2 Ω

S Then, we can split this energy between the vortex-energy contained in Bi , and the outer energy: Z 1X 1 J(u , A ) ∇(h − phex ) 2 > 2 2 hex 2 i hex Bi Z (1.24) 1 ∇(h − phex ) 2 + |h − phex |2 . + 2 2hex S Ω\

i

Bi

| R Using assertion (1.12) in Proposition 1.1, the first sum in (1.24) is larger than |log 2h2ex Ω |µ |, and R then from (1.20) and lower semi-continuity, larger than λ2R Ω |µ0 |. By (1.21) and lower semicontinuity again the second term in (1.24) is larger than 12 Ω |∇(h0 − p)|2 + |h0 − p|2 . Hence we obtain Z Z 2 1 J(u , A ) λ ∇(h0 − p) + |h0 − p|2 . (1.25) > |µ0 | + lim inf →0 h2ex 2 2 Ω

Ω

Using (1.22), the right-hand side in (1.25) is exactly E(h0 ), hence lim inf →0

J(u , A ) > E(h0 ) > min E = E(h∗ ). h2ex

The rest of Section 3 is then devoted to minimizing E and proving properties of its unique minimizer h∗ . Finally, in Section 4, we are able to derive the convergence of h /hex and of the measures µ from the fact that the upper and lower bounds of Sections 2 and 3 match. Remark on notations. – C always denotes a positive constant independent of . 2. Upper bound In this section, 0 6 λ < +∞. Recall the expression Z Z 1 λ ∇(f − p) 2 + |f − p|2 , (2.1) −∆(f − p) + f + E(f ) = 2 2 Ω

defined over

Ω

 V = f ∈ Hp1 (Ω ) |−∆(f − p) + f is a Radon measure .

The minimum of E is achieved, as will be shown in Section 3, by a function h∗ ∈ V for which µ = −∆(h∗ − p) + h∗ is in fact a positive absolutely continuous measure. Since trivially any f ∈ V is the solution of  −∆(f − p) + f − p = µ − p in Ω , (2.2) f −p=0 on ∂Ω , ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

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where µ = −∆(f − p) + f , we have Z G(x, y) d(µ − p)(y),

(f − p)(x) = Ω

where G(x, y) is the Green potential, solution of  (2.3)

−∆x G + G = δy G(x, y) = 0

in Ω , for x ∈ ∂Ω .

From (2.2), (2.3) it follows that (2.4)

E(f ) = I(µ) =

1 λ kµk + 2 2

Z G(x, y) d(µ − p)(x) d(µ − p)(y), Ω ×Ω

where kµk = |µ|(Ω ) is the total variation of µ. Note that I(µ) makes sense for any positive Radon measure if we admit the value +∞. We prove in this section P ROPOSITION 2.1. – Let hex be such that lim→0 |log |/hex = λ, with the additional condition, if λ = 0, that hex  1/2 , and µ be a positive Radon measure absolutely continuous with respect to the Lebesgue measure. Then, letting (u , A ) be a minimizer of J over H 1 (Ω , C) × H 1 (Ω , R2 ), lim sup

(2.5)

→0

1 J(u , A ) 6 I(µ). h2ex

As already mentioned, the measure µ corresponding to h∗ will be proved to be indeed positive and absolutely continuous (see Proposition 3.1), thus we have C OROLLARY 2.1. – Under the hypothesis of the proposition, (2.6)

lim sup →0

1 J(u , A ) 6 I(µ) = E(h∗ ) = min E(f ). f ∈V h2ex

C OROLLARY 2.2. – If λ = 0, and h is the induced magnetic field of a minimizing configuration, h →p hex →0 proving Theorem 1 in this particular case.

strongly in H 1 ,

Proof. – If λ = 0 then it is clear that the minimum of E over V is zero, and is uniquely achieved by h∗ = p. Hence from Corollary 2.1 lim sup →0

1 J(u , A ) = 0. h2ex

Therefore lim→0 J/h2ex = 0. But as pointed out in (1.17), Z 2 1 ∇(h − phex ) + |h − phex |2 , J(u , A ) > 2 Ω

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thus h/hex → p in H 1 .

573

2

The proof of the above proposition involves constructing a test configuration (u , A ) for J with normalized vortex density given by µ. But the size of a vortex, i.e. of the region where |u| < 1 is expected to be determined by the potential term in J . The factor 1/2 is scaled out from this term by letting v(x) = u(x) and expressing J(u, A) in terms of v. Thus we are led to believe that vortices have a size of the order of , and then that the vortex measure should be concentrated in balls of this size, each carrying a weight 2π. Our main task will therefore be to approximate µ by measures µ having this property, and to control I(µ ) as  → 0. In fact I(µ ) does not converge to I(µ) because there is a loss of energy due to the concentration at vortices. However we have P ROPOSITION 2.2. – Let µ, hex , λ be as in Proposition 2.1. Then for  > 0 small enough, there exist points ai , 1 6 i 6 n(), such that (1) hex µ(Ω ) , →0 2π

n() ∼

(2)

i

j

and, letting µi be the uniform measure on ∂B(ai , ) of mass 2π,

µ =

(3)

 a − a > 4,

1 X i µ →µ hex i 

in the sense of measures, as  → 0. Finally

lim sup →0

1 2

Z G(x, y) dµ (x) dµ (y) 6 Ω ×Ω

1 λ µ(Ω ) + 2 2

Z G(x, y) dµ(x) dµ(y). Ω ×Ω

This proposition shall be proved at the end of this section. Proposition 2.1 follows easily from the above result, once we know some easy properties of the function G(x, y) that we now list without proof. L EMMA 2.1. – The function G(x, y), solution to (2.3) has the following properties (∆ is the diagonal of R2 × R2 ). (1) G(x, y) is symmetric and positive. 1 log |x − y| is continuous on Ω × Ω . (2) G(x, y) + 2π (3) There exists C > 0 such that for all x, y ∈ Ω × Ω \ ∆   1 1 1 log − C 6 G(x, y) 6 C log +1 . 2π |x − y| |x − y| 2.1. Proof of Proposition 2.1 Step 1 Let µ be the sequence of measures given by Proposition 2.2, then (2.7)

lim sup →0

1 2

Z G(x, y) d(µ − p)(x) d(µ − p)(y) 6 I(µ). Ω ×Ω

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Indeed from Proposition 2.2, Z Z 1 λ 1 (2.8) lim sup G(x, y) dµ (x) dµ (y) 6 µ(Ω ) + 2 2 →0 2 Ω ×Ω

G(x, y) dµ(x) dµ(y).

Ω ×Ω

Moreover, from the weak convergence of µ to µ,   Z Z Z Z 1 1 (2.9) G(x, y)p(x) dx dµ (y) = G(x, y)p(x) dx dµ(y), lim sup 2 →0 2 Ω

Ω

Ω

Ω

the inner integral being a continuous function of y — an easy consequence of (2.3). Combining (2.8) and (2.9) yields (2.7). Step 2 Let h be the solution to  −∆(h − phex ) + h = hex µ (2.10) h = phex

in Ω , on ∂Ω .

This function will serve as the induced magnetic field of our test-configuration. For future reference, note that since Z Z 2 1 G(x, y) d(µ − p)(x) d(µ − p)(y), ∇(h − phex ) + |h − phex |2 dx = h2ex 2 Ω ×Ω

Ω

it follows from (2.7) that (2.11)

1 lim sup 2 h →0 ex

  Z 2 1 2 ∇(h − phex ) + |h − phex | dx 6 I(µ). 2 Ω

Step 3 We construct a test configuration (u , A ) such that lim sup

(2.12)

→0

1 J(u , A ) 6 I(µ). h2ex

We define A to be such thatPcurl A = h , and define u = ρ eiφ as follows. Recall from Proposition 2.2 that hex µ = i µi , with µi a uniform measure on ∂B(ai , ) of mass 2π, and for all i 6= j, |ai − aj | > 4. We let  0   (2.13)

ρ (x) =

 

|x − ai | 

  if |x − ai | 6  for some i, −1 if  < |x − ai | < 2 for some i,

1

otherwise.

The function S φ needs only to be defined modulo 2π, and where ρ 6= 0, i.e. on the set Ω = Ω \ i B(ai , ). Choosing a point x0 ∈ Ω , we let ∀x ∈ Ω I (2.14) φ (x) = A .τ − ∇(h − phex ).ν d`, (x0 ,x)

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where (x0 , x) is any curve joining x0 to x in Ω and (τ, ν) is the Frénet frame on the curve. By construction, ∇φ − A = −∇⊥ (h − phex ),

(2.15)

in Ω . Note that the function φ is well defined modulo 2π. Indeed if ω ⊂ Ω is such that ∂ω ⊂ Ω , Z Z A .τ − ∇(h − phex ).ν = −∆(h − phex ) + h , ω

∂ω

the right-hand side being equal to hex µ (ω) by (2.10). This quantity is in turn equal to 2πk, where k is the number of points ai in ω. Thus eiφ (x) defined by (2.14) does not depend on the particular curve (x0 , x) chosen. Step 4 We estimate J(u , A ). It follows easily from (2.13) — and the fact that from Proposition 2.2 there are n() ∼ hex µ(Ω )/2π points ai — that Z
2 1 1 |∇ρ |2 + 2 1 − ρ 2 6 Chex , 2 2 Ω

and therefore that (2.16)

1 lim sup 2 →0 hex

 Z 
1 1 2 2 2 |∇ρ | + 2 1 − ρ = 0. 2 2 Ω

Moreover, from (2.15), ρ 2 |∇φ − A |2 6 |∇(h − phex)|2 . Therefore, adding (2.11) and (2.16), lim sup →0

1 J (u , A ) 6 I(µ), h2ex

proving Proposition 2.1. 2.2. Proof of Proposition 2.2 Step 1 We claim that it suffices to prove the proposition for a measure µ with density u verifying 1/C < u(x) < C a.e. for some C > 0. Indeed if the density u of µ is an arbitrary positive function we may define truncated measures µn with densities un = (u ∧ n) ∨ 1/n. Applying the proposition to each µn we get approximating measures µn tending to µn as  → 0. Taking a diagonal sequence we may then construct a sequence µ → µ satisfying properties (1), (2) of Proposition 2.2 and such that Z 1 G(x, y) dµ (x) dµ (y) 6 lim inf I(µn ). lim sup n→+∞ →0 2 Ω ×Ω

That the right-hand side is less than or equal to I(µ) follows — as in Step 1 of the proof of Proposition 2.1 — from the weak convergence µn → µ and the fact that Z Z 1 1 G(x, y) dµn (x) dµn (y) 6 G(x, y) dµ(x) dµ(y). lim inf n→+∞ 2 2 Ω ×Ω

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Ω ×Ω

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This last inequality is proved by noting that since G is positive,     1 1 d µ+ , G(x, y) dµn dµn 6 G(x, y) d µ + n n integrating, and letting n → +∞. Step 2 From now on we assume the bound 1/C < u < C holds for the density of µ, and define the approximating measures µ . We choose a function δ() such that  δ  1. h−1/2 ex

(2.17)

Then for each  we define the family of squares {Ki }i∈I =



   kδ, (k + 1)δ × lδ, (l + 1)δ ⊂ Ω | k, l ∈ Z .

For each i ∈ I we let

 ni =

(2.18)

 hex µ(Ki ) , 2π

where [x] denotes the greatest integer less than or equal to x. Note that from our assumption on µ, δ 2 /C 6 µ(Ki ) 6 Cδ 2 .

(2.19)

It is possible to place ni points in each Ki evenly enough so that their mutual distance, as well √ as their distance to the boundary of Ki is of the order of δ/ ni . We call the resulting family of points (ai )i (we will specify more precisely how we place the points below). Note that in view of (2.17), (2.18), (2.19), if i 6= j then  ai − aj >

1 √  . C hex P Also, from (2.18), the total number of points n() = i ni is such that (2.20)

n() 6 But

P

i µ(Ki )

hex X µ(Ki ) 6 n() + Card(I). 2π i

∼ µ(Ω ) and Card(I) ∼ |Ω |/δ 2  hex , from (2.17). Therefore n() ∼

hex µ(Ω ), 2π

which, together with (2.20), shows that property (1) of Proposition 2.2 is verified. We define µi to be the uniform measure on Ci = ∂B(ai , ) of total mass 2π, and µ =

4e SÉRIE – TOME 33 – 2000 – N◦ 4

n() 1 X i µ. hex i=1 

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A FREE-BOUNDARY PROBLEM ARISING IN SUPERCONDUCTIVITY

Step 3 It remains to prove properties (2) and (3) for µ . The weak convergence of (2) is clear from the construction of µ , since 2π ni ≈ µ(Ki ) hex and since the size of the squares Ki tends to 0. The rest of the proof is devoted to proving property (3). Fix some η > 0 and let ∆η denote an η-neighbourhood of the diagonal in Ω × Ω . Since G(. , .) is continuous outside ∆η , it follows from the weak convergence µ → µ — which implies µ ⊗ µ → µ ⊗ µ — that Z Z 1 1 (2.21) G dµ dµ = G dµ dµ. lim →0 2 2 µ (Ki ) =

Ω ×Ω \∆η

Ω ×Ω \∆η

Therefore it suffices to prove that 1 lim sup →0 2

(2.22)

Z G dµ dµ 6

λ µ(Ω ) + C(η), 2

∆η

where C(η) is a function tending to 0 with η. Indeed if (2.22) is true, then adding it to (2.21) and letting η → 0 proves property (3) of Proposition 2.2. Step 4 We prove (2.22). Let Iη be the set of pairs of indices (i, j) such that Ci × Cj ∩ ∆η 6= ∅ (Iη depends in fact on , but we drop the  in our notation). We have (2.23)

1 2

ZZ ∆η

1 G dµ dµ 6 2 hex

X (i,j)∈Iη i6=j

1 2

ZZ

X 1 ZZ

!

n()

G dµi dµj +

i=1

2

G dµi dµi .

We first treat the second sum. From the definition of µi (2.24)

1 2

ZZ G dµi dµi

1 = 2

Z2πZ2π
G ai + eiθ1 , ai + eiθ2 dθ1 dθ2 . 0 0

We split the second sum in (2.23) in two by letting Jη = {i | d(ai , ∂Ω ) < η}. Then because of (2.19), #Jη < n()|{x | d(x, ∂Ω ) < η}| < Cηn() and then, using the upper bound (3) of Lemma 2.1 together with (2.24), X 1 ZZ
G dµi dµi 6 Cηπn() |log | + 1 . (2.25) 2 i∈Jη

On the other hand, from Lemma 2.1, (2), there exists a constant C(η) such that if x, y are at a distance at least η/2 from the boundary of Ω , then G(x, y) <

1 1 log + C(η), 2π |x − y|

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thus if  < η/2 and i ∈ / Jη , the same inequality holds true for any (x, y) ∈ Ci × Ci . Integrating i i over C × C yields, in view of (2.24), 1 2

ZZ G dµi dµi < π|log | + C(η).

Then, summing over indices i ∈ / Jη yields X 1 ZZ
G dµi dµi 6 n() π|log | + C(η) . 2

(2.26)

i∈J / η

Summing (2.26) and (2.25), and taking the limit as  → 0, we find n() Z Z 1 X1 G dµi dµi lim sup 2 h 2 →0 ex i=1

(2.27)

! 6

λ µ(Ω ) + Cη, 2

using the fact that lim→0 n()/hex () = µ(Ω )/2π and λ = lim→0 |log |/hex . It remains to treat the first sum in (2.23). First we get more specific as to how the points ai are placed. Each square Ki is partitioned into ni rectangles by induction as follows: (1) The first subdivision is S1 = {Ki }, it consists of one rectangle. (2) Given a subdivision Sn of Ki in n rectangles, Sn+1 is obtained by dividing the largest rectangle in Sn into two rectangles of same size, either vertically or horizontally according to which yields the rectangles with aspect ratio closest to 1. S This results in a partition of each Ki , and then a partition of i Ki , into a family of n() rectangles that we call S. A point is placed at the center of each rectangle in S. There are two fundamental facts that the reader may check easily for a rectangle R ∈ S with width w and height h: w ∈ {1/2, 1, 2}, h

(2.28)

1 C < |R| < , Cn() n()

where C does not depend on . From this we deduce (a) There exists a C > 0 independent of  such that for any pair of distinct rectangles Ri , Rj ∈ S with centers ai , aj ; any pair (x, y) ∈ Ri × Rj and any (θ1 , θ2 ) ∈ [0, 2π] × [0, 2π], (2.29)

|x − y| < C ai + eiθ1 − aj + eiθ2 .

This follows from the boundedness of the aspect ratio of the rectangles and the fact that the distance between the centers is at least 4. (b) For any rectangle R ∈ Sn , (2.30)

2π < Cµ(R). hex

This is because C|R| > 1/n() and n() ≈ µ(Ω )hex /2π. Using the lower bound u > 1/C on the density of µ yields (2.30). 4e SÉRIE – TOME 33 – 2000 – N◦ 4

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Now take (i, j) ∈ Jη a pair of distinct indices. Then, from the mean value theorem,

(2.31)

ZZ
1 G dµi dµj = G ai + eiθ1 , aj + eiθ2 , 2 4π Z Z 1 G dµ dµ = G(x, y), µ(Ri )µ(Rj ) Ri ×Rj

for some (x, y) ∈ Ri × Rj and (θ1 , θ2 ) ∈ [0, 2π] × [0, 2π]. But from Lemma 2.1, (3) and (2.29),
G ai + eiθ1 , aj + eiθ2 < CG(x, y).

(2.32)

Combining (2.30), (2.31) and (2.32) yields ZZ ZZ 1 i j dµ < C G dµ dµ, G dµ   h2ex Ri ×Rj

and then 1 h2ex

(2.33)

X (i,j)∈Iη i6=j

1 2

ZZ G dµi dµj 6 Cµ(∆2η ),

S since i,j Ri × Rj ⊂ ∆2η if  is small enough. But limη→0 µ(∆2η ) = 0, thus summing (2.33), (2.27) in view of (2.23) yields (2.22), completing this step and the proof of Proposition 2.2. 2 3. Lower bound We consider as usual (u , A ) minimizers of J , and h the associated magnetic fields. (We will often drop the subscripts .) We begin with the Proof of Proposition 1.1. – It is clear, as already said, that, by testing J with the configuration (u ≡ 1, A ≡ 0), the minimum of J is less than Ch2ex , for some constant C independent of . Then, writing ρ = |u |, Z
2 1 |∇ρ|2 + 2 1 − ρ2 6 C|log |2 . 2 Ω

For any t let Ωt = {x/ρ(x) < t} and let γt = ∂Ωt . Adapting Lemma 4.2 of [15] we find Z1 r(γt )

1 − t2 dt 6 

Z |∇ρ|2 +


2 1 1 − ρ2 , 2 2

Ω

C

where C above depends only on Ω and r(γt ) is the radius of γt , i.e. the infimum over all finite coverings of γt by balls B1 , . . . , Bk of the sum r1 + · · · + rk , where ri is the radius of Bi . Therefore, Z1 r(γt )

1 − t2 dt 6 C|log |2 . 

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From this we deduce, using the mean value theorem, that  ∃t ∈ 1 −

(3.1)

1 ,1 |log |

 such that r(γt ) < C|log |3 .

Now we apply Proposition 4.1 of [15] on Ω \ Ωt , taking v = eiϕ = u /|u |, σ = min(|log |−6 , 1 −1/2 ) and A = A . The conclusion is that there exist balls {Bi }i∈I such that (1.10), (1.11) 2 hex of the proposition hold and (3.2)

1 2

Z

1 |∇ϕ − A| + 2

Z

2

Bi \Ωt

 |h − hex | > π|di | log 2

σ −C C|log |3

Bi

 , +

where di is the winding number of u /|u | restricted to ∂Bi if Bi b Ω , and zero otherwise. We prove that this implies (1.12). First σ > |log |−p for some p > 0, thus the right-hand side above may be rewritten as
π|di ||log | 1 − o(1) .

(3.3)

Also, recall that the second Ginzburg–Landau equation is −∇⊥ (h − phex ) = ρ2 (∇ϕ − A).

(3.4)

Replacing the right-hand side of (3.2) by (3.3), and on the left using (3.4) and the fact that, from (3.1), ρ > 1 − |log |−1 on Ω \ Ωt , we find Z Z 1 ∇(h − phex ) 2 + 1 |h − hex |2 (3.5) 2 2 Bi \Ωt

1 > 2

Bi

Z

ρ4 |∇ϕ − A|2 + |h − hex |2 Bi \Ωt

  Z 1 C > 1− 2 |log |



Z |∇ϕ − A| +

|h − hex |

2

Bi \Ωt

2

Bi


> π|di ||log | 1 − o(1) .

On the other hand, we have the a priori estimate 1 2

Z Ω

∇(h − phex ) 2 6 1 2

Z |∇A u|2 6 Ch2ex , Ω

hence kh kH 1 (Ω ) 6 Chex 6 C|log |, therefore,

Z |h − hex |2 6 Cri kh − hex k2L4 6 C Bi

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where we have used (1.11). We conclude that (3.5) can be rewritten as (1.12): Z 2
1 ∇(h − phex ) > π|di ||log | 1 − o(1) . 2 Bi

2 From now on, we treat the case λ > 0 and we consider the vortices (ai , di ) of (u , A ) given by Proposition 1.1. We define P 2π i∈I di δai (3.6) . µ = hex In the sequel, M will denote the set of Radon measures on Ω . If we choose the Coulomb gauge  div A = 0 in Ω , A.n = 0 on ∂Ω , we have the a priori bounds: (3.7)

kAkL∞ 6 Chex ,

k∇ukL2 6 Chex ,

kukL∞ 6 1.

(See [16] or [14] for the proofs.) We also recall that, as in (1.23), Z 2
2 1 1 ∇(h − phex ) + |h − phex |2 + 2 1 − ρ2 . (3.8) J(u, A) > 2 2 Ω

We begin by extracting weakly convergent subsequences thanks to the a priori bounds. L EMMA 3.1. – From any sequence n → 0, we can extract a subsequence such that there exist h0 ∈ Hp1 (Ω ) and µ0 ∈ M such that

(3.9)

h n − p * h0 − p hex h n − p → h0 − p hex

(3.10)

µn → µ0

weakly in H01 (Ω ),

strongly in W01,q (Ω ), ∀q < 2, in the sense of measures,

with (3.11)

−∆(h0 − p) + h0 = µ0 .

Hence µ0 ∈ H −1 ∩ M. Proof. – We have the upper bound: J(u, A) 6 Ch2ex . Hence, (3.12)

and

J(u, A) 6

C hex |log |. λ

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Comparing (3.8) and (3.12), we deduce that subsequence,

h hex

− p is bounded in H01 (Ω ), hence, modulo a

h − p * h0 − p in H01 . hex On the other hand, {Bi }i being the family of balls constructed in Proposition 1.1, still from (3.8), Z XZ 2 1 ∇(h − phex ) 2 + |h − phex |2 + o(1). ∇(h − phex ) + (3.14) J(u, A) > 2 S i B (3.13)

Ω\

i

i

Bi

Hence, combining (3.12), (3.14) and (1.12), we get P XZ J(u, A) 1 C ∇(h − phex ) 2 > π i |di | + o(1). > > λ hex |log | 2hex |log | i hex Bi

Thus, µ is a bounded family of measures, we can then extract a subsequence converging to µ0 in the sense of measures. The next step is to compare µ0 and h0 . For that purpose we prove the following   h 2π X h (3.15) −p + − di δai → 0 in W −1,q (Ω ), ∀q < 2. −∆ →0 hex hex hex i First, the second Ginzburg–Landau equation holds: −∇⊥ (h − phex ) = (iu, ∇u) − |u|2 A, and taking the curl,


−∆(h − phex ) = curl(iu, ∇u) − curl |u|2 A   h curl(iu, ∇u) curl((1 − |u|2 )A) h −p + = + . −∆ hex hex hex hex

(3.16)

On the one hand, for any ξ ∈ H01 (Ω ), Z Z


ξcurl 1 − |u|2 A = A 1 − |u|2 .∇⊥ ξ. Ω

Ω

Then, using Cauchy–Schwartz and the a priori bounds (3.7), Z



ξcurl 1 − |u|2 A 6 Ck∇ξkL2 kAkL∞ 1 − |u|2 2 6 Ch2ex k∇ξkL2 , L Ω

the right-hand side tending to zero when  → 0. This means that

(3.17) curl 1 − |u|2 A → 0 in H −1 (Ω ). On the other hand, (3.18)

  X 1 di δai → 0 curl(iu, ∇u) − 2π hex i

4e SÉRIE – TOME 33 – 2000 – N◦ 4

in W −1,q (Ω ), ∀q < 2.

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This is a consequence of the proof of Lemma 2.3 of [14] (see [2] for a more detailed proof). Since h 1 −1 (Ω ), hence in view hex − p is bounded in H0 (Ω ), the left-hand side of (3.16) is bounded in H P 1 −1 of (3.17), hex curl(iu, ∇u) is bounded in H (Ω ). Therefore, from (3.18), µ = h2π i di δai ex −1,q is bounded in W (Ω ), ∀q < 2. It is also bounded in the sense of measures, hence, by a theorem of Murat (see [11]), it is compact in W −1,r (Ω ), ∀r < q < 2. Combining this with (3.17) and (3.18), we obtain that the right-hand side of (3.16) is compact in W −1,q (Ω ), ∀q < 2, and converges to µ0 . Thus,   h h −p + − µ0 → 0 −∆ →0 hex hex We deduce that satisfies

h hex

in W −1,q (Ω ), ∀q < 2.

− p → h0 − p strongly in W01,q (Ω ) ∀q < 2, and passing to the limit, h0 −∆(h0 − p) + h0 = µ0 .

2

L EMMA 3.2. – µ0 and h0 being defined in the previous lemma, lim inf n→∞

J(un , An ) λ > h2ex 2

Z

1 2

|µ0 | + Ω

Z

∇(h0 − p) 2 + |h0 − p|2 = E(h0 ),

Ω

where E(.) was defined in (1.2). Proof. – We start again from the lower bound (3.14): J(u, A) >

X1Z ∇(h − phex ) 2 + 1 2 2 i Bi

Z Ω\

∇(h − phex ) 2 + |h − phex |2 + o(1).

S i

Bi

Using (1.5), this can be rewritten (3.19)

J(u, A) π > h2ex

P

i |di ||log h2ex

|

+

h − phex 2 h − phex 2 ∇ + hex + o(1). hex

Z

1 2 Ω\

S i

Bi

On the other hand, we know from (1.11) that X

ri 6

i∈I

1 . |log |6

Therefore, one can extract from n a subsequence such that (3.20)

∞  X X n=0

 ri < ∞.

i∈In

Let us denote AN =

[ [ n>N i∈In

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

Bi .

584

E. SANDIER AND S. SERFATY

From (3.20), we have meas(AN ) → 0 as N → ∞.

(3.21) Now, we can write Z 1 lim inf 2 n→∞ hex S Ω\

> lim inf n→∞

∇(h − phex ) 2 + |h − phex |2

i

1 h2ex

Bi

Z

∇(h − phex ) 2 + |h − phex |2 >

Ω \AN

Z

∇(h0 − p) 2 + |h0 − p|2 ,

Ω \AN

by definition of AN and by lower-semicontinuity. Then, letting N → ∞, as we have (3.21), we deduce Z Z 1 ∇(h − phex ) 2 + |h − phex |2 > ∇(h0 − p) 2 + |h0 − p|2 . (3.22) lim inf 2 n→∞ hex S Ω Ω\

i

Bi

Returning to (3.19), we now have lim inf n→∞

π J(u, A) > lim inf n→∞ h2ex 1 > lim inf n→∞ 2

P

i |di || log n | h2ex

Z

+

1 2

Z Ω

| log n | 1 |µn | + hex 2

Ω

∇(h0 − p) 2 + |h0 − p|2

Z

∇(h0 − p) 2 + |h0 − p|2 ,

Ω

and by lower semi-continuity again, lim inf n→∞

J(u, A) λ > h2ex 2

Z |µ0 | + Ω

1 2

Z

∇(h0 − p) 2 + |h0 − p|2 = E(h0 ).

Ω

2 We will prove in the next section that h0 is in fact the unique minimizer h∗ of E, by using the upper bound of Proposition 2.1, but we first need some properties of h∗ . P ROPOSITION 3.1. – The minimum of E over V is uniquely achieved by h∗ ∈ C 1,α (Ω ) (∀α < 1) satisfying the following properties:

(3.23)

 λ   h∗ > p − ,   2    h∗ = p on ∂Ω ,   
λ   −∆(h∗ − p) + h∗ = 0 in Ω , h∗ − p −   2    µ = −∆(h∗ − p) + h∗ > 0,

problem (3.23) being equivalent to (P). We split the proof of this proposition into several lemmas. We denote by µ+ the positive part of µ, and µ− its negative part, so that µ = µ+ − µ− . 4e SÉRIE – TOME 33 – 2000 – N◦ 4

A FREE-BOUNDARY PROBLEM ARISING IN SUPERCONDUCTIVITY

585

L EMMA 3.3. – h∗ = p −

λ 2

µ+ -a.e.,

h∗ = p +

λ 2

µ− -a.e.,

λ λ 6 h∗ 6 p + 2 2

p−

on Ω .

Proof. – First, by lower semi-continuity, it is easy to see that the minimum of E is achieved. In addition, there is a unique minimizer by convexity of E. Also, recall that from Section 2, we may write (3.24)

E(h∗ ) = I(µ) =

λ 2

Z |µ| + Ω

1 2

Z G(x, y)d(µ − p)(x)d(µ − p)(y), Ω ×Ω

where G(x, y) is defined in (2.3). The functional I(.) is defined over M ∩ H −1 . Now, as µ minimizes I, we can make variations (1 + tf )µ, where f ∈ C 0 (Ω ); the first order in t → 0 has to vanish: Z Z 1

λ G(x, y)d (1 + tf )µ − p (x)d (1 + tf )µ − p (y) (1 + tf )µ + ∀t ∈ R 2 2 Ω

λ > 2

Z

1 |µ| + 2

Ω

Z

Ω ×Ω

G(x, y)d(µ − p)(x)d(µ − p)(y), Ω ×Ω

hence, by symmetry of G, looking at the first order, λ 2

Z

Z f |µ| +

G(x, y)d(f µ)(x)d(µ − p)(y) = 0.

Ω ×Ω

Ω

Using Fubini’s theorem, this means that λ 2

Z

Z f |µ| + Ω

(h∗ − p)f µ = 0. Ω

This is true for any f ∈ C 0 (Ω ), hence λ |µ| + (h∗ − p)µ = 0 2 We deduce the two assertions

(3.25)

 λ   h∗ = p − 2 λ  h = p + ∗ 2

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

in M.

µ+ -a.e., µ− -a.e.

586

E. SANDIER AND S. SERFATY

On the other hand, one can also consider variations µ + ν for ν ∈ M ∩ H −1 such that ν ⊥ µ (which means that µ and ν are mutually singular). Then, Z Z Z |µ + ν| = |µ| + |ν|, Ω

and λ 2

Z

1 |µ + ν| + 2

Ω

λ > 2

Z

Ω

Z G(x, y)d(µ + ν − p)(x)d(µ + ν − p)(y) Ω ×Ω

1 |µ| + 2

Ω

Ω

Z

G(x, y)d(µ − p)(x)d(µ − p)(y). Ω ×Ω

Consequently, as before, λ 2

(3.26)

Z

Z |ν| + Ω

(h∗ − p)ν > 0,

∀ν ⊥ µ.

Ω

(3.25) and (3.26) imply that λ λ 6 h∗ 6 p + 2 2

p− 2

a.e.

L EMMA 3.4. – A minimal µ is a positive measure, absolutely continuous with respect to the Lebesgue measure. As (h∗ − p) ∈ H01 (Ω ), (h∗ − p)+ ∈ H01 (Ω ), and, since µ ∈ H −1 , we can consider the integral Z Z Z µ(h∗ − p)+ = µ+ (h∗ − p)+ − µ− (h∗ − p)+ . Ω

Ω

Ω

But, from Lemma 3.3, (h∗ − p)+ = 0, µ+ -a.e., hence Z Z µ(h∗ − p)+ = − µ− (h∗ − p)+ 6 0. Ω

Ω

On the other hand, integrating by parts, Z Z Z
µ(h∗ − p)+ = −∆(h∗ − p) + h∗ (h∗ − p)+ = Ω

Ω

Hence, we deduce that

∇(h∗ − p) 2 +

h∗ >p

Z µ− (h∗ − p)+ = 0. Ω

But, as (h∗ − p)+ = 4e SÉRIE – TOME 33 – 2000 – N◦ 4

λ 2

µ− -a.e.,

Z

h∗ >p

h∗ (h∗ − p) > 0.

A FREE-BOUNDARY PROBLEM ARISING IN SUPERCONDUCTIVITY

we must have

Z µ− (h∗ − p)+ =

0= Therefore,

λ 2

Ω

R

587

Z |µ− |. Ω

|µ− | = 0 and µ− = 0. We conclude that µ = µ+ and µ is a positive measure. 2 Ω

We can then say that h∗ is solution of  λ  h∗ > p −    2    h∗ = p   
λ   −∆(h∗ − p) + h∗ = 0  h∗ − p −  2    µ = −∆(h∗ − p) + h∗ > 0.

in Ω , on ∂Ω , in Ω ,

This is equivalent to problem (P) (see [12] for Rexample, otherwise it can be checked directly). This equation characterizes a minimizer of Ω |∇l|2 + l2 − 2l∆p over the set of l ∈ Hp1 such that l > p − λ2 (obstacle problem). By convexity of the problem, there is a unique minimizer. In addition, the classical regularity theorem for such problems ([12] p. 141) applies and says that h∗ ∈ C 1,α (Ω ), ∀α < 1. Proposition 3.1 is then proved. For the convenience of the reader, we include an elementary proof of the fact that h∗ ∈ C 1,α , ∀α < 1 and that µ  dx (i.e. is absolutely continuous with respect to the Lebesgue measure). Proof. – We define ψα (t) over R by    1      ψα (t) = 0       1 λ   1 − t− α 2

λ if t 6 − , 2 λ if t > − + α, 2 λ λ if − 6 t 6 − + α. 2 2

Let ξ ∈ C0∞ (Ω ) such that ξ > 0. As ∆(h∗ − p) is in H −1 and ψα (h∗ − p) ∈ H01 , we can consider the following integral: Z Z Z
− ∆(h∗ − p)ξψα (h∗ − p) = ξ∇(h∗ − p).∇ ψα (h∗ − p) + ψα (h∗ − p)∇(h∗ − p).∇ξ. Ω

Ω

Ω

But, ∇(ψα (h∗ − p)) = − α1 1{− λ 6h∗ −p6α− λ} ∇(h∗ − p), hence 2

2

Z ∆(h∗ − p)ξψα (h∗ − p)

− (3.27)

Ω

1 =− α

Z

2 ξ ∇(h∗ − p) +

λ {− λ 2 6h∗ −p6α− 2 }

Z ψα (h∗ − p)∇(h∗ − p).∇ξ. Ω

We can write a Radon–Nikodym decomposition with respect to the Lebesgue measure dx of ∆(h∗ − p), which is a measure, as (∆(h∗ − p))R + (∆(h∗ − p))S where
R ∆(h∗ − p)  dx, ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE


S ∆(h∗ − p) ⊥ dx.

588

E. SANDIER AND S. SERFATY

On the one hand, we know that −∆(h∗ − p) + h∗ = 0 outside of ωλ = {h∗ − p = −λ/2}, hence the support of (∆(h∗ − p))S is included in ωλ , and ψα (h∗ − p) = 1 on the support of (∆(h∗ − p))S . On the other hand, by Lebesgue’s dominated convergence theorem, as ψα (h∗ − p) → 1ωλ

a.e.,

α→0

we have

Z

α→0 ωλ

Ω

and

Z


R −∆(h∗ − p) ξψα (h∗ − p) →


R −∆(h∗ − p) ξ,

Z

Z ∇(h∗ − p).(∇ξ)ψα (h∗ − p) →

α→0 ωλ

Ω

∇(h∗ − p).∇ξ = 0.

(Indeed ∇(h∗ − p) = 0 a.e. on ωλ .) Passing to the limit α → 0 in (3.27), we thus have Z


S −∆(h∗ − p) ξ +

Z


R −∆(h∗ − p) ξ 6 0.

ωλ

Ω

This is true for any ξ > 0 in C0∞ , therefore −∆(h∗ − p) 6 0 on ωλ . But −∆(h∗ − p) + h∗ > 0, hence (−∆(h∗ − p))S = (−∆(h∗ − p) + h∗ )S > 0, therefore (−∆(h∗ − p))S must be 0. Thus, we obtain that ∆(h∗ − p) is absolutely continuous (or L1 ), and that λ − p 6 −h∗ 6 −∆(h∗ − p) 6 0 2

on ωλ .

Hence, ∆h∗ is bounded on ωλ . It is also bounded on the complement since, there, ∆h∗ = h∗ + ∆p and h∗ is bounded. We conclude that ∆h∗ ∈ L∞ (Ω ), in particular µ is absolutely continuous, and h∗ ∈ C 1,α , ∀α < 1. 2

4. Completing the proofs of Theorems 1 and 2: the final convergence results We recall that the case λ = 0 has been totally treated in Corollary 2.2. For the case λ > 0, we start from the result of Lemma 3.2, stating that from any sequence n → 0, we can extract a subsequence, also denoted n such that hn /hex * h0 and µn → µ0 , for some h0 ∈ V , and µ0 = −∆(h0 − p) + h0 , with λ J(u, A) > E(h0 ) = lim inf n→∞ h2ex 2

Z Ω

1 |µ0 | + 2

Z |∇h0 − p|2 + |h0 − p|2 . Ω

But from Proposition 3.1 and Lemma 3.4, the minimum of E over V is achieved by a unique h∗ , and the measure µ = −∆(h∗ − p) + h∗ is positive and absolutely continuous. It then follows from the upper bound of Proposition 2.1 that lim sup n→∞

4e SÉRIE – TOME 33 – 2000 – N◦ 4

J(un , An ) 6 E(h∗ ), h2ex

A FREE-BOUNDARY PROBLEM ARISING IN SUPERCONDUCTIVITY

589

and then that h0 = h∗ , µ0 = µ. Also lim

n→∞

J(un , An ) = E(h∗ ). h2ex

Examining the proof of Lemma 3.2 and (3.18) especially, we have also proved that λ n→∞ 2

Z |µn | +

lim

Ω

λ 2

= E(h∗ ) =

(4.1)

Ω\

Z

hn − phex 2 hn − phex 2 ∇ + hex hex

Z

1 2

S i

|µ| +

Bi

1 2

Ω

Z

∇(h∗ − p) 2 + |h∗ − p|2 .

Ω

In addition, from Lemma 3.1, we have µn → µ in M, hence, by lower semi-continuity, λ lim inf n→∞ 2

(4.2)

Z

λ |µn | > 2

Ω

Z |µ|. Ω

From Lemma 3.1 too, h/hex * h∗ in Hp1 . By compact Sobolev embedding, and arguing as in (3.21), we have 2 Z h n 2 hex − p = |h∗ − p| .

Z lim

(4.3)

n→∞ Ω\

S i

Ω

Bi

As in (3.21) again, Z hn − phex 2 ∇ > ∇(h∗ − p) 2 . hex

Z (4.4)

lim inf n→∞

Ω\

S i

Ω

Bi

Comparing (4.2)–(4.4) to (4.1), we must have Z

Z |µn | =

lim

(4.5)

n→∞ Ω

Z |µ| =

Ω

µ, Ω

and Z hn − phex 2 ∇ = ∇(h∗ − p) 2 . hex

Z (4.6)

lim

n→∞ Ω\

S i

Ω

Bi

In addition, (3.14) also implies that (4.7)

 2 Z Z Z  2 hn − phex 6 λ µ + ∇(h∗ − p) . lim sup ∇ hex n→∞ Ω

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

Ω

Ω

590

E. SANDIER AND S. SERFATY

We deduce from (4.5) that, in addition to having µn → µ in M, we have |µn | → µ in M. Indeed, for any domain U ⊂ Ω Z

Z

Z

|µn | > lim inf

lim inf U

µn = U

and similarly

µ, U

Z

Z |µn | >

lim inf Ω \U

µ,

Ω \U

thus there must be equality. We can hence conclude that 2π

P

i di δai

hex 2π

P

i |di |δai

hex

  λ →µ= p− 1 ωλ , 2   λ →µ= p− 1 ωλ . 2

In order to complete the proof of Theorem 1, there remains to prove P ROPOSITION 4.1. –

  2 ∇ hn − h∗ → λµ in M. n→∞ hex

(4.8)

Proof. – From the Brezis–Lieb lemma, it suffices to prove that   2 ∇ hn − p → ∇(h∗ − p) 2 + λµ n→∞ hex

(4.9)

in M.

Considering any domain U ⊂ Ω , and arguing as in Lemma 3.2, we have  2 Z  h n − p lim inf ∇ n→∞ hex U

Z

= lim inf n→∞

(4.10)

Z >

U\

S i

Bi

  2 Z ∇ hn − p + hex S

∇(h∗ − p) 2 + lim inf n→∞

U

(

Z S i

(

i

  2 ∇ h n − p hex

Bi )∩U

  2 ∇ h n − p . hex

Bi )∩U

Now, as µ is a regular measure, there exists a family of compact sets Uη b U such that µ(U ) = limη→0 µ(Uη ). For any η, when  is sufficiently small, all the B(ai , ri ) such that ai ∈ Uη are included in U , hence    2  2 Z Z ∇ hn − p > ∇ hn − p hex hex S S (

i

Bi )∩U

4e SÉRIE – TOME 33 – 2000 – N◦ 4

i

B(ai ,ri )/ai ∈Uη

A FREE-BOUNDARY PROBLEM ARISING IN SUPERCONDUCTIVITY

>

2π h2ex

X

591


|di ||log | 1 − o(1) ,

i/ai ∈Uη

using (1.5). By definition of µ, this amounts to   2 ∇ h n − p > 1 hex h2ex

Z lim inf n→∞

(

S i

Bi )∩U

Z

! |µn | hex |log |,

Uη

hence by convergence of |µn |, we are led to   2 ∇ hn − p > λµ(Uη ). hex

Z (4.11)

lim inf n→∞

(

S i

Bi )∩U

Combining this to (4.10), we deduce lim inf n→∞

 2 Z Z  ∇ hn − p > ∇(h∗ − p) 2 + λµ(Uη ), hex U

U

and passing to the limit η → 0, we obtain lim inf n→∞

 2 Z Z  ∇ hn − p > ∇(h∗ − p) 2 + λµ(U ). hex U

U

Comparing this to (4.7), we obtain (4.9). 2 Now, all these limits are independent of the chosen subsequence; therefore, the whole sequence converges. Since this is true for any sequence n → 0, this is true for  → 0. This completes all the proofs.

Acknowledgements The authors are grateful to Régis Monneau for providing them information on obstacle problems.

REFERENCES [1] A LMEIDA L., B ETHUEL F., Topological methods for the Ginzburg–Landau equations, J. Math. Pures Appl. 77 (1998) 1–49. [2] AFTALION A., S ANDIER E., S ERFATY S., Pinning phenomena in the Ginzburg–Landau model of superconductivity, Preprint. [3] B ERESTYCKI H., B ONNET A., C HAPMAN J., A semi-elliptic system arising in the theory of type-II superconductivity, Comm. Appl. Nonlinear Anal. 1 (3) (1994) 1–21. [4] B ETHUEL F., B REZIS H., H ÉLEIN F., Ginzburg–Landau Vortices, Birkhäuser, 1994. [5] BONNET A., MONNEAU R., Existence of a smooth free-boundary in a superconductor with a Nash– Moser inverse function theorem argument, Interfaces and Free Boundaries (to appear). ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

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[6] B ETHUEL F., R IVIÈRE T., Vortices for a variational problem related to superconductivity, Annales IHP, Analyse non Linéaire 12 (1995) 243–303. [7] C IORANESCU D., M URAT F., Un terme étrange venu d’ailleurs, in: Nonlinear Partial Differential Equations and their Applications, Coll. de France Semin. Vol. II, Res. Notes Math., Vol. 60, 1982, pp. 98–138. [8] C HAPMAN S.J., RUBINSTEIN J., S CHATZMAN M., A mean-field model of superconducting vortices, Eur. J. Appl. Math. 7 (2) (1996) 97–111. [9] G IORGI T., P HILLIPS D., The breakdown of superconductivity due to strong fields for the Ginzburg– Landau model, SIAM J. Math. Anal. 30 (2) (1999) 341–359 (electronic). [10] J ERRARD R., Lower bounds for generalized Ginzburg–Landau functionals, SIAM J. Math. Anal. 30 (4) (1999) 721–746. [11] M URAT F., L’injection du cône positif de H −1 dans W −1,q est compacte pour tout q < 2, J. Math. Pures Appl. 60 (1981) 309–322. [12] RODRIGUES J.F., Obstacle Problems in Mathematical Physics, Mathematical Studies, North-Holland, 1987. [13] S ANDIER E., Lower bounds for the energy of unit vector fields and application, J. Functional Anal. 152 (2) (1998) 379–403. [14] S ANDIER E., S ERFATY S., Global minimizers for the Ginzburg–Landau functional below the first critical magnetic field, Annales IHP, Analyse non Linéaire 17 (1) (2000) 119–145. [15] S ANDIER E., S ERFATY S., On the energy of type-II superconductors in the mixed phase, Reviews in Math. Phys. (to appear). [16] S ERFATY S., Local minimizers for the Ginzburg–Landau energy near critical magnetic field, Part I, Comm. Contemp. Math. 1 (2) (1999) 213–254. [17] S ERFATY S., Local minimizers for the Ginzburg–Landau energy near critical magnetic field, Part II, Comm. Contemp. Math. 1 (3) (1999) 295–333. [18] S ERFATY S., Stable configurations in superconductivity: Uniqueness, multiplicity and vortexnucleation, Arch. Rat. Mech. Anal. 149 (1999) 329–365. [19] T INKHAM M., Introduction to Superconductivity, 2nd edn., McGraw-Hill, 1996. (Manuscript received September 10, 1999.)

Étienne S ANDIER Université François-Rabelais, Département de Mathématiques, Parc Grandmont, 37200 Tours, France E-mail: [email protected]

Sylvia S ERFATY CMLA, École Normale Supérieure de Cachan, 61, avenue du Président-Wilson, 94235 Cachan cedex, France E-mail: [email protected]

4e SÉRIE – TOME 33 – 2000 – N◦ 4