On the first critical field in the 3D Ginzburg-Landau model of superconductivity
Carlos Román


The Ginzburg-Landau model is a phenomenological description of superconductivity. A crucial feature of type-II superconductors is the occurrence of vortices, which appear above a certain value of the applied magnetic field called the first critical field. In this paper we estimate this value, when the Ginzburg-Landau parameter is large, and we characterize the behavior of the Meissner solution, the unique vortexless configuration that globally minimizes the Ginzburg-Landau energy below the first critical field. In addition, we show that beyond this value, for a certain range of the strength of the applied field, there exists a unique Meissner-type solution that locally minimizes the energy.

Detection of interstellar HCS and its metastable isomer HSC:

new pieces in the puzzle of sulfur chemistryBased on observations carried out with the IRAM 30m Telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain).
    
M. Agúndez1,
N. Marcelino1
J. Cernicharo1 M. Tafalla2

    Received; accepted
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§ INTRODUCTION



 §.§ Problem and background

Superconductors are certain metals and alloys, which, when cooled down below a critical (typically very low) temperature, lose their resistivity, which allows permanent currents to circulate without loss of energy. Superconductivity was discovered by Ohnes in 1911. As a phenomenological description of this phenomenon, Ginzburg and Landau <cit.> introduced in 1950 the Ginzburg-Landau model of superconductivity, which has been proven to effectively predict the behavior of superconductors and that was subsequently justified as a limit of the Bardeen-Cooper-Schrieffer (BCS) quantum theory <cit.>. It is a model of great importance in physics, with Nobel prizes awarded for it to Abrikosov, Ginzburg, and Landau.

The Ginzburg-Landau functional, which models the state of a superconducting sample in an applied magnetic field, assuming that the temperature is fixed and below the critical one, is

    GL_(u,A)=1/2∫_Ω |∇_A u|^2+1/2^2(1-|u|^2)^2+1/2∫_^3|H-H_|^2.

Here


  * Ω is a bounded domain of ^3, that we assume to be simply connected with C^2 boundary.

  * u:Ω→ℂ is called the order parameter. Its modulus squared (the density of Cooper pairs of superconducting electrons in the BCS quantum theory) indicates the local state of the superconductor: where |u|^2≈ 1 the material is in the superconducting phase, where |u|^2≈ 0 in the normal phase.

  * A:^3→^3 is the electromagnetic vector potential of the induced magnetic field H= A.

  * ∇_A denotes the covariant gradient ∇-iA.

  * H_:^3→^3 is a given external (or applied) magnetic field.

  * >0 is the inverse of the Ginzburg-Landau parameter usually denoted κ, a non-dimensional parameter depending only on the material. We will be interested in the regime of small , corresponding to extreme type-II superconductors. 


A key physical feature of this type of superconductors is the occurrence of co-dimension 2 topological singularities called vortices, which appear above a certain critical value of the strength of the applied field h_H__L^2(Ω,^3). There are three main critical values of h_ or critical fields H_c_1,H_c_2, and H_c_3, for which phase transitions occur.


  * Below H_c_1=O(|log|), the superconductor is everywhere in its superconducting phase, i.e. |u| is uniformly close to 1, and the applied field is expelled by the material due to the occurrence of supercurrents near ∂Ω. This phenomenon is known as the Meissner effect.

  * At H_c_1, the first vortice(s) appear and the applied field penetrates the superconductor through the vortice(s).

  * Between H_c_1 and H_c_2, the superconducting and normal phases coexist in the sample. As h_ increases, so does the number of vortices. The vortices repeal each other, while the external magnetic field confines them inside the sample. 

  * At H_c_2=O(1/^2), the superconductivity is lost in the bulk of the sample. 

  * Between H_c_2 and H_c_3, superconductivity persists only near the boundary. 

  * Above H_c_3=O(1/^2), the applied magnetic field completely penetrates the sample and the superconductivity is lost, i.e. u=0.

The Ginzburg-Landau model is known to be a 𝕌(1)-gauge theory. This means that all the meaningful physical quantities are invariant under the gauge transformations 

    u↦ ue^iϕ,   A↦ A+∇ϕ,

where ϕ is any real-valued function in H^2_(^3). The Ginzburg-Landau energy and its associated free energy 

    F_(u,A)= 1/2∫_Ω |∇_A u|^2+1/2^2(1-|u|^2)^2+| A|^2

are gauge invariant, as well as the density of superconducting Cooper pairs |u|^2, the induced magnetic field H, and the vorticity, defined, for any sufficiently regular configuration (u,A), as

    μ(u,A)= (iu,∇_A u)+ A,

where (·,·) denotes the scalar product in  identified with ^2 i.e. (a,b)=ab+ab/2. This quantity is the gauge-invariant version of the Jacobian determinant of u and is the analogue of the vorticity of a fluid. For further physics background on the model, we refer to Tin,DeG. 

The main purpose of this paper is to give a precise estimate of H_c_1 and to characterize the behavior of global minimizers of GL_ below this value in 3D. The analysis of H_c_2 or higher applied fields requires completely different techniques. The interested reader can refer to GioPhi,FouHel,FouKach,FouKachPer and references therein. 

The first critical field is (rigorously) defined by the fact that below H_c_1 global minimizers of the Ginzburg-Landau functional do not have vortices, while they do for applied fields whose strength is higher than H_c_1. In the 2D setting, Sandier and Serfaty (see Ser,SanSer1,SanSer2,SanSerBook) provided an expansion of the first critical field, up to an error o(1) as → 0, and rigorously characterized the behavior of global minimizers of the Ginzburg-Landau functional below and near this value. Conversely, in 3D much less is known. Very recently Baldo, Jerrard, Orlandi, and Soner <cit.>, via a Γ-convergence argument, provided the asymptotic leading order value of the first critical field as → 0 (see also <cit.> for related results). In short, in a uniform applied field, they proved that if (u_,A_) minimizes GL_(u_,A_) then there exists a measure μ_0 such that

    μ(u_,A_)/|log|→μ_0   as → 0

in weak sense (the precise type of convergence can be found in <cit.>*Proposition 1). Moreover, there exists a constant H^* such that if lim_→0h_/|log|<H^* then μ_0≡ 0, while μ_0≢0 if lim inf_→0h_/|log|>H^*. This result
gives H_c_1 up to an error o(|log|) as → 0 and agrees with previous work by Alama, Bronsard, and Montero <cit.> in the special when Ω is a ball. An intermediate situation, when the superconducting sample is a thin shell, was treated in <cit.>.

Before stating our results, let us recall the three-dimensional -level estimates for the Ginzburg-Landau functional provided by the author in <cit.>. These tools will play a crucial role in this paper.
 For any m,n,M>0 there exist C,_0>0 depending only on m,n,M, and ∂Ω, such that, for any <_0, if (u_,A_)∈ H^1(Ω,)× H^1(Ω,^3) is a configuration such that F_(u_,A_)≤ M|log|^m then there exists a polyhedral 1-dimensional current ν_ such that


  * ν_ /π is integer multiplicity,

  * ∂ν_=0 relative to Ω,

  * supp(ν_)⊂ S_ν_⊂Ω with |S_ν_|≤ C|log|^-q, where q(m,n)3/2 (m+n),

  * ∫_S_ν_|∇_A_ u_|^2+1/2^2(1-|u_|^2)^2+| A_|^2 ≥ |ν_|(Ω)(log1/-C loglog1/)-C/|log|^n,

  * and for any γ∈(0,1] there exists a constant C_γ depending only on γ and ∂Ω, such that 

    μ(u_,A_) -ν__C_T^0,γ(Ω)^*≤ C_γF_(u_,A_)+1/|log|^qγ.



Here and in the rest of the paper, C_T^0,γ(Ω) denotes the space of vector fields Φ∈ C^0,γ(Ω) such that Φ×ν=0 on ∂Ω, where ν is the outer unit normal to ∂Ω. The symbol ^* denotes its dual space.



 §.§ Main results

Throughout this article we assume that H_∈ L_^2(^3,^3) is such that H_=0 in ^3. In particular, we deduce that there exists a vector-potential A_∈ H^1_(^3,^3) such that 

    A_=H_,   A_=0 in ^3  and   A_·ν=0 on ∂Ω.

Let us define H_0,=h_^-1H_ and assume that this vector field is Hölder continuous in Ω with Hölder exponent β∈ (0,1] and Hölder norm bounded independently of . In particular, note that H_0,_L^2(Ω,^3)=1. It is also convenient to set A_0,=h_^-1A_.

We remark that the divergence-free assumption on the applied magnetic field is in accordance with the fact that magnetic monopoles do not exist in Maxwell's electromagnetism theory.

The natural space for the minimization of GL_ in 3D is H^1(Ω,)× [A_+H_], where

    H_{A∈ H^1_(^3,^3)  |   A∈ L^2(^3,^3)}.

Let us also introduce the homogeneous Sobolev space Ḣ^1(^3,^3), which is defined as the completion of C_0^∞ (^3,^3) with respect to the norm ∇ ( ·  ) _L^2(^3,^3). We observe that, by Sobolev embedding, there exists a constant C>0 such that 

    A_L^6(^3,^3)≤ C ∇ A_L^2(^3,^3)

for any A∈Ḣ^1(^3,^3). Moreover, by <cit.>*Proposition 2.4, we have 

    Ḣ^1(^3,^3)={A∈ L^6(^3,^3) |  ∇ A ∈ L^2(^3,^3)}.

It is also convenient to define the subspace 

    Ḣ^1_=0{A∈Ḣ^1(^3,^3)  |   A=0 in ^3}.

In this subspace, one has

    A_Ḣ^1_=0∇ A_L^2(^3,^3)=  A_L^2(^3,^3).


Let us now define a special vortexless configuration that turns out to be a good approximation of the so-called Meissner solution, i.e. the vortexless global minimizer of the Ginzburg-Landau energy below the first critical field, which, as we shall see, is unique up to a gauge transformation. By recalling that any vector field A∈ H^1(Ω,^3) can be decomposed as (see Lemma <ref>)

    {[          A          = B_A + ∇ϕ_A       in Ω;      B_A×ν          =          0      on ∂Ω;     ∇ϕ_A·ν          =        A·ν     on ∂Ω, ].

we consider the pair (u_0,h_A_0), where u_0=e^ih_ϕ_A_0 and A_0 is the unique minimizer (in a suitable space) of the functional 

    J(A)=1/2∫_Ω| B_A|^2+1/2∫_^3|(A-A_0,ex)|^2.

This special configuration satisfies the following properties:


  * GL_(u_0,h_A_0)=h_^2J(A_0) and |u_0|=1 in Ω.

  * H_0= A_0 satisfies the usually called London equation

    (H_0-H_0,)+H_0χ_Ω=0  in ^3,

where χ_Ω denotes the characteristic function of Ω.

  * The divergence-free vector field B_0=B_A_0∈ C_T^2,β(Ω,^3) satisfies 

    {[ -Δ B_0+B_0          =       H_0,       in Ω;      B_0×ν          =          0     on ∂Ω. ].

This vector field is the analog of the function ξ_0, considered by Sandier and Serfaty in the analysis of the first critical field in 2D (see Ser,Ser2,SanSer1,SanSer2). We shall see that B_0 plays an important role in our 3D analysis.

In addition, this pair allows us to split the Ginzburg-Landau energy of a given configuration (u,A). More precisely, by writing u'=u_0^-1u and A'=A-h_A_0, one can prove that (see Proposition <ref>)

    GL_(u,A)=h_^2J(A_0)+F_(u',A')+1/2∫_^3∖Ω| A'|^2-h_∫_Ωμ(u',A')∧ B_0+R_0,

where R_0=o(1), in particular, when h_ is bounded above by a negative power of |log|. Let me emphasize that one of the achievement of this paper is to find the right pair (u_0,h_A_0) to split the energy, which then allows to implement (almost) the same strategies as in 2D.

By combining this splitting with the optimal -level estimates of Theorem <ref>, we find

    GL_(u,A)≥ h_^2J(A_0)+1/2|ν_'|(Ω)(log1/-Cloglog1/)
    +1/2∫_^3∖Ω| A'|^2-h_∫_Ων_'∧ B_0+o(1),

where ν_' denotes the 1-current associated to (u',A') by Theorem <ref>. By construction of ν_' (see <cit.>*Section 5.2), we can write

    ν_'=∑_i∈ I_2πΓ_i^,

where the sum is understood in the sense of currents, I_ is a finite set of indices, and Γ_i^ is an oriented Lipschitz curve in Ω with multiplicity 1. Each of these curves, which are non-necessarily distinct, does not self intersect and is either a loop contained in Ω or has two different endpoints on ∂Ω. We will denote by X the class of Lipschitz curves, seen as 1-currents, described here.

Inserting this expression in the previous inequality, allows us to heuristically derive the leading order of the first critical field:

    H_c_1^01/2B_0_*|log|,

where

    B_0_*sup_Γ∈ X1/|Γ|(Ω)∫_ΩΓ∧ B_0[The notation used here is explained in the preliminaries (see Section <ref>).].

In Proposition <ref>, we compute this value in a special case. 

We may now state our first result, that characterizes the behavior of global minimizers of GL_ below H_c_1^0. In the 2D setting, an analogous result was proved by Sandier and Serfaty (see <cit.>*Theorem 1).

There exist constants _0,K_0>0 such that for any <_0 and h_≤ H_c_1^0-K_0log |log|, the global minimizers (u_,A_) of GL_ in H^1(Ω,)× [A_+H_] are vortexless configurations that satisfy

    h_^2J(A_0)+o(1)≤ GL_(u_,A_)≤ h_^2J(A_0)  and  1-|u_|_L^∞(Ω,)=o(1).


It is important to mention that in the proof of this theorem we use the fact that solutions of the Ginzburg-Landau equations (see Section <ref>), in the Coulomb gauge, satisfy a clearing-out result proved by Chiron <cit.>. Roughly speaking, this states that if the energy of a solution in a ball (with center in Ω) intersected with Ω is sufficiently small, then |u| is uniformly away from 0 in a ball of half radius intersected with Ω. The proof given by Chiron relies on monotonicity formulas, and is very much inspired by previous work by Bethuel, Orlandi, and Smets <cit.>. The interested reader can refer to Riv,LinRiv1,LinRiv2,BetBreOrl,SanSha for results in the same spirit.

Our second result provides bounds from above and below for the first critical field in 3D. 

There exist constants _0,K_0,K_1>0 such that for any <_0 we have 

    H_c_1^0-K_0log |log|≤ H_c_1≤ H_c_1^0+K_1.


In particular, these inequalities show that indeed H_c_1^0 is the leading order of H_c_1 as → 0. Of course this agrees with the previously mentioned result by Baldo, Jerrard, Orlandi, and Soner within their framework (i.e. when H_0, is taken to be a fixed unit vector). The author strongly believes that, as → 0,

    H_c_1=H_c_1^0+O(1).

To prove this result, one needs to avoid the uncertainty of order O(log|log|) in the lower bound for H_c_1 of Theorem <ref>. To accomplish this, it is crucially important to characterize, near the first critical field, the behavior of the vorticity μ(u,A) of global minimizers of GL_. We plan to address this question in future work.

Our next result shows that beyond the first critical field there exists a locally minimizing vortexless configuration. A similar result was proved by Serfaty in 2D (see <cit.>*Theorem 1).

 Let α∈(0,1/3). There exists _0>0 such that, for any <_0, if h_≤^-α then there exists a vortexless configuration (u_,A_)=(u_0u_',h_A_0+A_')∈ H^1(Ω,)× [A_+Ḣ^1_=0] with A_'·ν=0 on ∂Ω, which locally minimizes GL_ in H^1(Ω,)× [A_+H_]. In addition, it satisfies the following properties:


  * h_^2J(A_0)+o(1)≤ GL_(u_,A_)≤ h_^2J(A_0) and 1-|u_|_L^∞(Ω,)=o(1).

  * The configuration (u_',A_') satisfies

    inf_θ∈[0,2π]u_'-e^iθ_H^1(Ω,)+A_'_Ḣ^1_=0→ 0  as →0.


  * Up to a gauge transformation, (u_,A_) converges to (u_0,h_A_0). More precisely, we have

    inf_θ∈[0,2π]u_-e^iθu_0_H^1(Ω,)+A_-h_A_0_Ḣ^1_=0→ 0  as →0.



Let us point out that in Remark <ref> we explain why we require α<1/3.

Our last result concerns the uniqueness, up to a gauge transformation, of locally minimizing vortexless configurations.

Let α,c∈ (0,1). There exists _0>0 such that, for any <_0, if h_≤^-α then a configuration (u,A) which locally minimizes GL_ in H^1(Ω,)× [A_+H_] and satisfies |u|≥ c and F_(u',A')≤^1+ for some >0, is unique up to a gauge transformation.



The assumption that F_(u',A')≤^1+ for some >0 plays a crucial role in the proof of this result. In Proposition <ref>, we prove that if α∈(0,1/4) then this condition is implied by the other assumptions of this theorem provided that GL_(u,A)≤ GL_(u_0,h_A_0)=h_^2J(A_0), i.e. uniqueness holds without assuming that F_(u',A')≤^1+ for some >0 if the Ginzburg-Landau energy of the vortexless local minimizer is below the energy of (u_0,h_A_0). 
We observe that this condition is satisfied by the locally minimizing solution of Theorem <ref>.

Let us also note that if α≥1/4 then the strategy of the proof of Proposition <ref> fails. For this reason, we are able to guaranty the uniqueness of the locally minimizing vortexless configuration of Theorem <ref> only if α<1/4.

Finally, let us emphasize that this uniqueness result allows to conclude that the locally minimizing configuration of Theorem <ref> is, indeed, up to a gauge transformation, the unique global minimizer of the Ginzburg-Landau energy below the first critical field. Therefore Theorem <ref>, in particular, provides a detailed characterization of the behavior of the Meissner solution.

Thus, we prove that below the first critical field, up to a gauge transformation, the Meissner solution is the unique global minimizer of GL_. Beyond this value, at least up to h_=o(^-1/3), a Meissner-type solution continues to exists as a local minimizer of the Ginzburg-Landau energy. This solution is unique, up to a gauge transformation, at least up to h_=o(^-1/4). Since this branch of vortexless solutions remains stable, in the process of raising h_ vortices should not appear at H_c_1, but rather at a critical value of h_ called the superheating field H_sh, at which the Meissner-type solution becomes unstable. It is expected that H_sh=O(^-1). The interested reader can refer to <cit.> and references therein for further details.



 §.§ Outline of the paper
 The rest of the paper is organized as follows. In Section <ref> we introduce some basic quantities and notation, describe two Hodge-type decompositions, and present some classical results in Ginzburg-Landau theory. In Section <ref> we define the approximation of the Meissner solution, split the Ginzburg-Landau energy, and prove Theorem <ref>. In Section <ref> we present the proof of Theorem <ref> and compute B_0_* in a special case. Section <ref> contains the proof of Theorem <ref> and Section <ref> the proof of Theorem <ref>. Appendix <ref> is devoted to prove some improved estimates for locally minimizing configurations, that allow to obtain the uniqueness of the Meissner-type solution of Theorem <ref> for α<1/4, as a consequence of Theorem <ref>.



§ PRELIMINARIES



 §.§ Some definitions and notation

We define the superconducting current of a pair (u,A)∈ H^1(Ω,)× H^1(Ω,^3) as the 1-form

    j(u,A)=(iu,d_A u)=∑_k=1^3 (iu,∂_k u-iA_ku)dx_k.

It is related to the vorticity μ(u,A) of a configuration (u,A) through

    μ(u,A)=dj(u,A)+dA.

This quantity can be seen as a 1-current, which is defined through its action on 1-forms by the relation

    μ(u,A)(ϕ)=∫_Ωμ(u,A)∧ϕ.

We recall that the boundary of a 1-current T relative to a set Θ, is the 0-current ∂ T defined by

    ∂ T(ϕ)=T(dϕ)

for all smooth compactly supported 0-form ϕ defined in Θ. In particular, μ(u,A) has zero boundary relative to Ω. We denote by |T|(Θ) the mass of a 1-current T in Θ.



 §.§ Hodge-type decompositions

Next, we provide a decomposition of vector fields in H_.

Every vector field A∈ H_ can be decomposed as

    A= +∇Φ,

where ,∈Ḣ^1_=0 and Φ∈ H^2_(^3).


First, let us observe that there exists a function Φ_1∈ H^2_(^3,^3) such that

    ΔΦ_1 =  A ∈ L^2_(^3,^3).

Second, we consider the problem

    {[              B              = A ∈ L^2(^3,^3);              B              =             0. ].

By observing that B =-Δ B, <cit.>*Theorem 1 provides the existence of a solution ∈Ḣ^1_=0 to this problem such that ∈Ḣ^1_=0.

Finally, by noting that 

    (A-∇Φ_1-)= (A-∇Φ_1-)=0,

we deduce that 

    A-∇Φ_1-=∇Φ_2

for some harmonic function Φ_2∈ H^2_(^3,^3). By writing Φ=Φ_1+Φ_2, we obtain the result.


We now recall a decomposition of vector fields in H^1(Ω,^3). The proof of this result can be found in <cit.>*Appendix A.

There exists a constant C=C(Ω) such that for every A∈ H^1(Ω,^3) there exist a unique vector field B_A∈{B∈ H^2(Ω,^3) |   B=0 in Ω} and a unique function ϕ_A∈{ϕ∈ H^2(Ω) |  ∫_Ωϕ_A=0} satisfying 

    {[          A          = B_A + ∇ϕ_A       in Ω;      B_A×ν          =          0      on ∂Ω;     ∇ϕ_A·ν          =        A·ν      on ∂Ω ].

and

    B_A_H^2(Ω,^3)+ϕ_A_H^2(Ω)≤ CA_H^1(Ω,^3).





 §.§ Ginzburg-Landau equations

 We say that (u,A)∈ H^1(Ω,)× [A_+H_] is a critical point of GL_ if for every  smooth and compactly supported configuration (v,B) we have 

    d/dtGL_(u+tv,A+tB)|_t=0=0.


We now present the Euler-Lagrange equations satisfied by critical points of GL_. This is a well-known result, but for the sake of completeness we prove it here.

If (u,A)∈ H^1(Ω,)× [A_+H_] is a critical point of GL_ then (u,A) satisfies the system of equations

    GL{[      -(∇_A)^2u              = 1/^2u(1-|u|^2)           in Ω;         (H-H_)              =  (iu,∇_A u)χ_Ω          in ^3;        ∇_A u·ν              =              0          on ∂Ω;         H-H_×ν              =              0         on ∂Ω, ].

where χ_Ω is the characteristic function of Ω, [ ·  ] denotes the jump across ∂Ω, ∇_A u·ν=∑_j=1^3 (∂_ju-iA_ju)ν_j, and the covariant Laplacian (∇_A)^2is defined by

    (∇_A)^2u=( -iA· )∇_Au.



We have

    d/dtGL_(u+tv,A)|_t=0=∫_Ω (∇_A u,∇_A v)-1/^2∫_Ω (u,v)(1-|u|^2).

By noting that

    (∇_A u, ∇_A v)= (∇_A u,v)-((∇_A)^2u,v),

where (∇_A u,v)=((∂_1u-iA_1u,v),(∂_2u-iA_2u,v),(∂_3u-iA_3u,v)), and by integrating by parts, we obtain

    d/dtGL_(u+tv,A)|_t=0= ∫_∂Ω (∇_A u ·ν,v)-∫_Ω ((∇_A)^2u,v)-1/^2∫_Ω (u,v)(1-|u|^2).

Since this is true for any v, we find

    (∇_A)^2u=1/^2u(1-|u|^2) in Ω  and  ∇_A u·ν=0 on ∂Ω.

On the other hand, we have

    d/dtGL_(u,A+tB)|_t=0=-∫_Ω (i Bu,∇_A u)+∫_^3(H-H_)· B.

By integration by parts, we get

    d/dtGL_(u,A+tB)|_t=0=-∫_Ω (iu,∇_A u)· B+∫_^3(H-H_)· B.

We deduce that 

    (H-H_)=(iu,∇_A u)χ_Ω  in ^3.

By testing this equation against B and by integrating by parts over Ω, we find

    ∫_Ω (H-H_)· B-∫_∂Ω((H-H_)×ν)· B-∫_Ω (iu,∇_A u)· B=0.

Now, by integrating by parts over ^3∖Ω, we get

    ∫_^3 ∖Ω (H-H_)· B+∫_∂ (^3∖Ω)((H-H_)×ν)· B=0.

Thus

    ∫_∂Ω([H-H_]×ν)· B=0,

which implies that [H-H_]×ν=0 on ∂Ω.



By taking the curl of the second Ginzburg-Landau equation, we find

    (H-H_)+Hχ_Ω=μ(u,A)χ_Ω,

in the sense of currents.
We will come back to this equation later on.




 §.§ Minimization of the Ginzburg-Landau energy


The minimum of GL_ over H^1(Ω,)× [A_+H_] is achieved.


Let {(ũ_n,Ã_n)}_n be a minimizing sequence for GL_ in H^1(Ω,)× [A_+H_]. Lemma <ref> yields a gauge transformed sequence {(u_n,A_n)}_n such that A_n∈ [A_+Ḣ^1_=0]. In particular, we have that GL_(ũ_n,Ã_n)=GL_(u_n,A_n) and

    ∇ (A_n-A_)_L^2(^3,^3)= (A_n-A_)_L^2(^3,^3).

Using the bound GL_(u_n,A_n)≤ C, where C is independent of n, we find that 

    1-|u_n|^2_L^2(Ω,), ∇_A_nu_n_L^2(Ω,^3), and   (A_n-A_)_L^2(^3,^3)

are bounded independently of n. Therefore, by recalling (<ref>), we deduce that A_n-A_ is bounded in Ḣ^1(^3,^3). 
Because {u_n}_n is bounded in L^4(Ω) we find that {iA_nu_n}_n is bounded in L^2(Ω,^3). By noting that ∇ u_n=∇_A_nu_n+iA_nu_n, we conclude that u_n is bounded in H^1(Ω,).

We may then extract a subsequence, still denoted {(u_n,A_n)}_n, such that {(u_n,A_n-A_)}_n converges to some (u,A-A_) weakly in H^1(Ω,)×Ḣ_=0 and, by compact Sobolev embedding, strongly in every L^q(Ω,)× L^q(Ω,^3) for q<6. 

Let us now show that (u,A) is a minimizer of GL_. By strong L^4(Ω,) convergence, 

    lim inf_n 1-|u_n|^2_L^2(Ω,)=1-|u|^2_L^2(Ω,).

Also, by weak Ḣ^1(^3,^3) convergence, we have

    lim inf_n (A_n-A_)_L^2(^3,^3)   =lim inf_n∇ (A_n-A_)_L^2(^3,^3)
       ≥∇ (A-A_)_L^2(^3,^3)=(A-A_)_L^2(^3,^3).

Moreover, standard arguments show that 

    lim inf_n ∇_A_nu_n_L^2(Ω,^3)^2   =lim inf_n ∇ u_n^2_L^2(Ω,^3)-2∫_Ω (∇ u_n,iA_nu_n)+A_nu_n_L^2(Ω,^3)
       ≥∇_Au_L^2(Ω,^3).

Hence

    lim inf_n GL_(u_,A_)≥ GL_(u,A).




§ GLOBAL MINIMIZERS BELOW HC



 §.§ An approximation of the Meissner solution

Next, we find a configuration (u_0,h_A_0) with |u_0|=1 and which satisfies (<ref>) with zero right-hand side. As mentioned in the introduction, this turns out to be a good approximation of the Meissner solution, the vortexless configuration which minimizes GL_ below the first critical field. 

Let us consider a configuration of the form (e^i ϕ_0,h_A_0) with ϕ_0∈ H^2(Ω) and A_0∈ A_0,+Ḣ^1_=0. Observe that, by using Lemma <ref> and by letting u_0 e^i ϕ_0, we have

    GL_(u_0,h_A_0)=   1/2∫_Ω |∇ϕ_0-h_( B_A_0+∇ϕ_A_0)|^2+1/2∫_^3|h_ A_0-H_|^2
    
    =   1/2∫_Ω |∇ (ϕ_0-h_ϕ_A_0)|^2+ h_^2| B_A_0|^2+h_^2/2∫_^3| (A_0-A_0,)|^2.

By choosing ϕ_0=h_ϕ_A_0, we obtain

    GL_(u_0,h_A_0)=h_^2/2∫_Ω| B_A_0|^2+h_^2/2∫_^3| (A_0-A_0,)|^2 h_^2 J(A_0).

We let A_0 to be the minimizer of J in the space (A_0,+Ḣ^1_=0, · _Ḣ^1_=0), whose existence and uniqueness follows by noting that J is continuous, coercive, and strictly convex in this Hilbert space (recall (<ref>) and (<ref>)). We also let H_0= A_0 and here and in the rest of the paper we use the notation B_0 B_A_0. 

One can easily check that, for any A∈Ḣ^1_=0, we have 

    ∫_Ω B_0· B_A + ∫_^3 (H_0-H_0,)· A=0

Because

    ∫_Ω B_0·∇ϕ_A=∫_Ω B_0 ·∇ϕ_A-∫_∂Ω (B_0×ν)ϕ_A=0,

we have

    ∫_Ω B_0· A + ∫_^3 (H_0-H_0,)· A=0

Moreover, Lemma <ref> implies that this equality also holds for any A∈ H_.

Let us observe that, for any A∈ C_0^∞ (^3,^3), by integration by parts, we have

    ∫_Ω B_0· A+ ∫_^3(H_0-H_0,)· A=0.

Therefore A_0 satisfies the Euler-Lagrange equation

    (H_0-H_0,)+ B_0χ_Ω=0  in ^3.

In addition, it is easy to see that the boundary condition [H_0-H_0,]=0 on ∂Ω holds. 

By taking the curl of the previous equation, we find 

    (H_0-H_0,)+H_0χ_Ω=0  in ^3,

namely (up to multiplying by h_) (<ref>) with μ(u_0,A_0)=0. 

On the other hand, by integration by parts, for any vector field B∈ C_0^∞(Ω,^3), we have

    ∫_Ω B_0· B+∫_Ω (H_0-H_0,)· B=0.

Besides, for any function ϕ∈ C_0^∞(Ω), we have

    ∫_Ω (B_0+(H_0-H_0,))·∇ϕ =-∫_Ω (B_0+(H_0-H_0,)) ϕ =0

Then, given any vector field A∈ C_0^∞(Ω,^3), by taking B=B_A and ϕ=ϕ_A in the previous equalities, we find

    ∫_Ω (B_0+(H_0-H_0,))· ( B_A+∇ϕ_A)=∫_Ω (B_0+(H_0-H_0,))· A=0.

Hence, the free-divergence vector field B_0 weakly solves the problem

    {[ -Δ B_0+B_0          =       H_0,       in Ω;      B_0×ν          =          0     on ∂Ω. ].


Since we assume that H_0,_C^0,β(Ω,^3)<C, by standard elliptic regularity theory, we deduce that B_0∈ C_T^2,β(Ω,^3) with B_0_C_T^2,β(Ω,^3)≤ C for some constant independent of . In addition, if the applied field is taken to be uniform in Ω, i.e. if H_0, is a fixed unit vector in Ω, then B_0 depends on the domain Ω only.




 §.§ Energy-splitting

Next, by using the approximation of the Meissner solution, we present a splitting of GL_.
 For any (u,A)∈ H^1(Ω,)× [A_+H_], letting u=u_0u' and A=h_A_0+A', where (u_0,h_A_0) is the approximation of the Meissner solution, we have

    GL_(u,A)=h_^2J(A_0)+F_(u',A')+1/2∫_^3∖Ω| A'|^2-h_∫_Ωμ(u',A')∧ B_0+R_0,

where F_(u',A') is the free energy of the configuration (u',A')∈ H^1(Ω,)× H_, i.e.

    F_(u',A')=1/2∫_Ω |∇_A'u'|^2+1/2^2(1-|u'|^2)^2+| A'|^2

and

    R_0=h_^2/2∫_Ω (|u|^2-1)| B_0|^2.

In particular, |R_0|≤ C h_^2E_(|u|)^1/2 with E_(|u|)=1/2∫_Ω |∇ |u||^2+1/2^2(1-|u|^2)^2.


One immediately checks that A'∈ H_. Since u'=u_0^-1u=e^-ih_ϕ_0u and ϕ_0∈ H^2(Ω,), by Sobolev embedding we deduce that u'∈ H^1(Ω,). 

Writing u=u_0u' and A=h_ A_0+A' and plugging them into GL_(u,A), we obtain

    GL_(u,A)=
    1/2∫_Ω |∇_A'u'-ih_ B_0u'|^2+1/2^2(1-|u'|^2)^2
    +1/2∫_^3| A'+h_ (H_0-H_0,)|^2.

By expanding the square terms, we get

    GL_(u,A)=   1/2∫_Ω |∇_A'u'|^2+h_^2| B_0|^2|u'|^2-2h_(∇_A'u',iu')· B_0+1/2^2(1-|u'|^2)^2 
    
    +   1/2∫_^3 | A'|^2+h_^2 |H_0-H_0,|^2+2h_ A'· (H_0-H_0,).

Observe that, by (<ref>), we have

    ∫_^3 A'· (H_0-H_0,)=-∫_Ω A'· B_0.

Therefore, grouping terms and writing |u'|^2 as 1+(|u'|^2-1), we find

    GL_(u,A)=h_^2J(A_0)+F_(u',A')+1/2∫_^3∖Ω| A'|^2-h_∫_Ω (j(u',A')+A')· B_0 + R_0.

Then, an integration by parts yields

    ∫_Ω (j(u',A')+A')· B_0=∫_Ωμ(u',A')∧ B_0-∫_∂Ω(j(u',A')+A')· (B_0×ν).

By using the boundary condition B_0×ν=0 on ∂Ω, we find (<ref>). The inequality for R_0 follows directly from the Cauchy-Schwarz inequality. 


Let φ∈ C_T^0,1(Ω) be a 1-form. Observe that, by gauge invariance and by integration by parts, we have

    ∫_Ωμ(u,A)∧φ=∫_Ωμ(u',A'+h_ B_0)∧φ=∫_Ωμ(u',A')∧φ+h_(1-|u|^2) B_0·φ.

Moreover, the Cauchy-Schwarz inequality yields

    μ(u,A)-μ(u',A')_C_T^0,1(Ω)^*≤ C h_ E_(|u|)^1/2.




 §.§ Proof of Theorem <ref>
 

Proposition <ref> yields

    GL_(u_,A_)≥ h_^2J(A_0)+F_(u_',A_')-h_∫_Ωμ(u_',A_')∧ B_0+o(^1/2),

where (u_,A_)=(u_0u_',h_ A_0+A_').

Step 1. Estimating F_(u_',A_').
By minimality, we have

    inf_(u,A)∈ H^1(Ω,)× [A_+H_]GL_(u,A)≤ GL_ (u_0,h_ A_0)=h_^2 J(A_0).

On the other hand, by gauge invariance, we get

    F_(u_',A_')=F_(u_,A_-h_ B_0)    ≤ 2F_(u_,A_)+2F_(1,h_ B_0)
       ≤ 2F_(u_,A_)+Ch_^2,

which combined with (<ref>) implies that F_(u_',A_')≤ M |log|^2. We may then apply Theorem <ref> (with n large enough) to obtain

    F_(u_',A_')-h_∫_Ωμ(u_',A_') ∧ B_0 ≥
    1/2 |ν_'|(Ω)( log1/-C loglog1/) -h_∫_Ων_' ∧ B_0+o(|log|^-2),

where C>0 is a universal constant and ν_' denotes the polyhedral 1-dimensional current associated to the configuration (u_',A_') by Theorem <ref>.
By noting that 

    ∫_Ων_' ∧ B_0≤  |ν_'|(Ω)B_0_*,

we find

    F_(u_',A_')-h_∫_Ωμ(u_',A_') ∧ B_0 ≥
    1/2 |ν_'|(Ω)(log1/- 2B_0_*h_ -C loglog1/)+o(|log|^-2).

Writing h_=H_c_1^0-K_0log|log| with H_c_1^0=12B_0_*|log|, we get

    GL_(u_,A_)≥ h_^2J(A_0)+1/2 |ν_'|(Ω)(2B_0_*K_0-C)loglog1/+o(|log|^-2).

By using (<ref>), we deduce that

    o(|log|^-2)≥ |ν_'|(Ω)(2B_0_*K_0-C)loglog1/.

Therefore, by letting K_0 (2B_0_*)^-1C+1, we deduce that |ν_'|(Ω)=o(|log|^-2). In particular, from the vorticity estimate in Theorem <ref> and (<ref>), we deduce that h_∫_Ωμ(u_',A_')∧ B_0=o(|log|^-1). Therefore, by combining this with (<ref>) and (<ref>), we are led to

    F_ (u_',A_')+1/2∫_^3∖Ω| A_'|^2≤ o(|log|^-1).

In particular, we deduce that GL_(u_,A_)=h_^2J(A_0)+o(|log|^-1). 

Step 2. Applying a clearing out result.
To prove that (u_,A_) is a vortexless configuration we use a clearing out result. Let us define

    v_ e^-iφ_u_'  and   X_ A_'-∇φ_,

where φ_ satisfies

    {[    Δφ_      =    A_'   in Ω;  ∇φ_·ν      =  A_'·ν on ∂Ω. ].

This implies that X_ is in the Coulomb gauge, i.e. it satisfies

    {[     X_      =      0   in Ω;   X_·ν      =      0 on ∂Ω. ].

Since the configuration (u_,A_) minimizes GL_ in H^1(Ω,)× [A_+H_], it satisfies the Ginzburg-Landau equations (<ref>). By observing that the configurations (u_,A_) and (v_,X_+h_ B_0) are gauge equivalent in Ω, we deduce that v_ satisfies 

    {[ -(∇_X_+h_ B_0)^2v_                  =    1^2v_(1-|v_|^2)               in Ω;    ∇_X_+h_ B_0v_·ν                  =                  0             on ∂Ω. ].

Expanding the covariant Laplacian, and using (<ref>) and B_0·ν =0 on ∂Ω, which follows from B_0×ν=0 on ∂Ω, one can rewrite this problem in the form

    {[ -Δ v_ +i|log|c(x)·∇ v_+|log|^2d(x)v_                                    =                      1^2v_(1-|v_|^2)                                 in Ω;                               ∇ v_·ν                                    =                                    0                               on ∂Ω, ].

where

    c(x)2(X_+h_ B_0)/|log|  and   
    d(x)|X_+h_ B_0|^2/|log|^2.

By Remark <ref> and by standard elliptic regularity theory for solutions of the Ginzburg-Landau equations in the Coulomb gauge, we have

    c_L^∞(Ω,^3), ∇ c_L^∞(Ω,^3), d_L^∞(Ω), ∇ d_L^∞(Ω)≤Λ_0

for some constant Λ_0>0 independent of .

In addition, by gauge invariance, we have

    F(u_',A_')=F_(v_,X_).

Since (v_,X_) is in the Coulomb gauge, we have

    E_(v_) F_(v_,0)≤ C F_(v_,X_)

for some universal constant C>0. We define a_(x)=1-d(x)^2|log|^2 and observe that

    Ẽ_(v_)1/2∫_Ω |∇ v_|^2+1/2^2(a_(x)-|v_|^2)^2≤ E_(v_)+O(|log|^2).

This combined with (<ref>), implies that 

    Ẽ_(v_)=o(|log|^-1).

Finally, from (<ref>), (<ref>), (<ref>), and (<ref>), we conclude that all the hypotheses of <cit.>*Theorem 3 are fulfilled, and therefore

    1-|u_|_L^∞(Ω,)=1-|v_|_L^∞(Ω,)→ 0   as → 0.
 
This concludes the proof of the theorem.
 



§ THE FIRST CRITICAL FIELD

Let us recall that, given a fixed >0, the first critical field is defined as the value H_c_1=H_c_1() such that if h_<H_c_1 and (u_,A_) is a minimizer of GL_ then |u_|>0 in Ω, while if h_>H_c_1 and (u_,A_) minimizes GL_ then u_ must vanish in Ω. We now prove Theorem <ref>.

Theorem <ref> immediately implies that

    H_c_1^0-K_0log |log|≤ H_c_1.

It remains to prove that H_c_1≤ H_c_1^0+K_1, for some constant K_1 sufficiently large. Let us assume towards a contradiction that h_=H_c_1^0+K and (u_,A_) minimizes GL_ in H^1(Ω,)× [A_+H_][This in particular implies that (u_,A_) satisfies the Ginzburg-Landau equations (<ref>) and therefore u_ is continuous.] and |u_|>0. 

Step 1. Estimating GL_(u_,A_).
We write (u_,A_)=(u_0u_',h_A_0+A_'), where (u_0,h_A_0) is the approximation of the Meissner solution. Since |u_'|=|u_|>0, we deduce that the 1-dimensional current ν_' associated to (u_',A_') by Theorem <ref> vanishes identically, and therefore, by taking n large enough, we have

    μ(u_',A_')_C_T^0,1(Ω)^*≤C/|log|^2

The energy-splitting (<ref>) then yields

    GL_(u_,A_)   =h_^2J(A_0)+F_(u_',A_')+1/2∫_^3∖Ω| A_'|^2+o(|log|^-1)
       ≥ h_^2J(A_0)+o(|log|^-1).

But since (u_,A_) minimizes GL_, we have 

    GL_(u_,A_)≤ GL_(u_0,h_A_0)=h_^2J(A_0).

Combining these inequalities, we find

    GL_(u_,A_)=h_^2J(A_0)+o(|log|^-1).


Step 2. Definition of a vortex configuration.
To reach a contradiction, we will show that there exists a configuration (u^_1,A^_1), whose vorticity concentrates along a curve in X for which B_0_* is almost achieved, such that if h_≥ H_c_1^0+K then GL_(u^_1,A^_1)<GL_(u_,A_), provided K≥ K_1 for some constant K_1 independent of .

By definition of B_0_* (recall (<ref>)), there exists a Lipschitz curve _∈ X, seen as a 1-current, with multiplicity 1 such that ∂_=0 relative to Ω and 

    1/|_|(Ω)∫_Ω_∧ B_0=B_0_*+o(|log|^-1).

From the proof of <cit.>*Theorem 1.1 (ii), which in particular uses some results contained in <cit.>, we deduce that there exists v_∈ H^1(Ω,) such that

    F_(v_,0)≤π |_|(Ω)|log|+C_0

for some constant C_0>0 independent of , and

    μ(v_,0)-2π__C_0^0,1(Ω)^*=o(|log|^-1),

where C_0^0,1(Ω) denotes the space of Lipschitz vector fields with compact support in Ω. Arguing as in the proof of <cit.>*Proposition 3.2, we deduce that

    μ(v_,0)-2π__C_T^0,1(Ω)^*=o(|log|^-1).


Now we let (u_1^,A_1^) be define by

    u_1^=u_0v_,    A_1^=h_A_0.

Proposition <ref> yields

    GL_(u_1^,A_1^)=h_^2J(A_0)+F_(v_,0)-h_∫_Ωμ(v_,0)∧ B_0+R_0.

From (<ref>) and (<ref>), we get

    ∫_Ωμ(v_,0)∧ B_0=2πB_0_*|_|(Ω)+o(|log|^-1).

Inserting this and (<ref>) into (<ref>), we are led to

    GL_(u_1^,A_1^)≤ h_^2J(A_0)+π|_|(Ω)|log|+C_0-2πB_0_*h_|_|(Ω)+o(h_|log|^-1).


Step 3. Contradiction.
Writing h_=H_c_1^0+K with H_c_1^0=12B_0_*|log|, we get

    GL_(u_1^,A_1^)   ≤ h_^2J(A_0)+π|_|(Ω)|log|+C_0-π|_|(Ω)(|log|+2B_0_*K)+o(1)
       =h_^2J(A_0)+C_0-2πB_0_*K|_|(Ω)+o(1).


Clearly B_0_*>0, which implies that there exists a constant C_1∈ (0,1) such that |_|(Ω)≥ C_1, provided  is small enough. If this is not the case, there exists a sequence of Lipschitz curves _^n∈ X such that |_^n|(Ω)≤ 1/n and which satisfies (<ref>). If _^n is a loop contained in Ω then, by the Stokes' theorem, we have

    ∫_Ω_^n∧ B_0≤∫_S_n| B_0|,

where S_n denotes a surface with least area among those whose boundary is _^n, i.e. a solution to the associated Plateau's problem. By the isoperimetric inequality, we have

    ∫_S_n| B_0|≤ B_0_L^∞(Ω,^3)Area(S_n)≤ C |_^n|(Ω)^2.

On the other hand, if both different endpoints of _^n belong to ∂Ω, we consider the geodesic connecting the endpoints of _^n on ∂Ω, oriented accordingly to the orientation of _^n. We then denote by _^n the loop formed by the union of _ and this geodesic. Since B_0×ν=0 on ∂Ω, by Stokes' theorem, we have

    ∫_Ω^n∧ B_0≤∫_S_n | B_0|,

where S_n denotes a surface with least area among those whose boundary is _^n. Arguing as above, we conclude that 

    ∫_S_n| B_0|≤ B_0_L^∞(Ω,^3)Area(S_n)≤ C Length(_^n)^2≤ C(∂Ω) |_^n|^2.

Therefore 

    1/|_^n|(Ω)∫_Ω_^n∧ B_0 ≤ C|_^n|≤C/n,

but this is impossible, for n large enough, in view of (<ref>).

Hence

    GL_(u_1^,A_1^)=h_^2J(A_0)+C_0-2π C_1B_0_*K+o(1).

By choosing K_1 (2π C_1B_0_*)^-1C_0+1, we deduce that, for any K≥ K_1 

    GL_(u_1^,A_1^)≤ h_^2J(A_0)-1+o(1)<GL_(u_,A_).

This contradicts the fact that (u_,A_) globally minimizes GL_. Thus

    H_c_1≤ H_c_1^0+K_1.


The proof of the upper bound hints that, slightly above the first critical field, vortices will be located close to Lipschitz curves in X for which B_0_* is (almost) achieved. Let us now study this quantity in a special case.

Consider the special case Ω=B(0,R) and H_0,=ẑ in B(0,R). Then, if S_1 denotes the vertical diameter seen as a 1-current with multiplicity 1 and oriented in the direction of positive z axis, we have

    B_0_*=1/2R∫_Ω S_1∧ B_0=1/2R∫_-R^R B_0(0,0,z) ·ẑ dz=3/2(1-1/sinh R∫_0^Rsinh r/r dr).

Moreover, S_1 is the only curve in X achieving the maximum in (<ref>).


We follow some ideas from <cit.>. 

Step 1. Explicit computation of B_0.
When Ω=B(0,R) and H_0,=ẑ in B(0,R), the solution to (<ref>) can be explicitly computed (see <cit.>). By using spherical coordinates (r,θ,ϕ), where r is the Euclidean distance from the origin, θ is the azimuthal angle, and ϕ is the polar angle, we have

    B_0=-3R/r^2sinh R(cosh r-sinh r/r)cosϕr̂-3R/2r^2sinh R(cosh r-1+r^2/rsinh r)sinϕϕ̂-cẑ,

where c=32Rsinh R(cosh R-1+R^2R sinh R). In particular, we observe that B_0 does not depend on the azimuthal angle and therefore it is constant along θ̂. 

Step 2. Dimension reduction.
Let Γ∈ X with ∫_B(0,R)Γ∧ B_0>0. We will project it along the azimuthal angle onto B(0,R)^2,+{(x,z) ∈^2  |  x^2+z^2<R^2, x≥ 0}. For this, we consider the map q:B(0,R)⊂^3→ B(0,R)^2,+ defined by

    q(r,θ,ϕ)=(rsinϕ,rcosϕ),

and we let 

    Γ_2Γ∘ q.

It is easy to check that ∂Γ_2=0 relative to B(0,R)^2,

    ∫_B(0,R)Γ∧ B_0=∫_B(0,R)^2,+Γ_2∧ B_0,     |Γ_2|(B(0,R)^2,+)≤ |Γ|(B(0,R)).

Therefore

    1/|Γ|(Ω)∫_B(0,R)Γ∧ B_0≤1/|Γ_2|(Ω)∫_B(0,R)^2,+Γ_2∧ B_0.

Even though Γ_2 does not necessarily belong to X, we can decompose

    Γ_2=∑_i∈ IΓ_i,

where the sum is understood in the sense of currents, I is a finite set of indices, and Γ_i∈ X for all i∈ I. In particular,

    ∫_B(0,R)^2,+Γ_2∧ B_0≤∑_i∈ I |Γ_i|(B(0,R)^2,+)B_0_* =|Γ_2|(B(0,R)^2,+)B_0_*.

We deduce that in order to compute B_0_* it is enough to consider Lipschitz curves Γ∈ X contained in B(0,R)^2,+ with ∫_B(0,R)Γ∧ B_0>0. From now on we consider Γ of this form.

Step 3. Application of Stokes' theorem.
If Γ has both endpoints on ∂ B(0,R)∩∂ B(0,R)^2,+, we then define Γ̃ as the loop formed by the union of Γ and the curve lying on ∂ B(0,R)∩∂ B(0,R)^2,+ which connects the end points of Γ oriented accordingly to the orientation of Γ. Since B_0×ν =0 on ∂ B(0,R), the Stokes' theorem yields

    ∫_B(0,R)^2,+Γ∧ B_0=∫_B(0,R)^2,+Γ̃∧ B_0=∫_S_Γ B_0·ŷ,

where S_Γ is the surface enclosed by Γ̃. Of course if Γ is a loop contained in B(0,R)^2,+ then the Stokes' theorem gives

    ∫_B(0,R)^2,+Γ∧ B_0=∫_S_Γ B_0·ŷ,
 
where S_Γ is the surface enclosed by Γ. 

An explicit computation gives

    B_0·ŷ=3R/2sinh R(cosh r-sinh r/r)sinϕ/r≥ 0  in B(0,R)^2,+.

In what follows we use the notation

    f(r)3R/2sinh R(cosh r-sinh r/r).


Step 4. Estimate for curves with endpoints on  ∂ B(0,R)∩∂ B(0,R)^2,+. For a,b∈[0,π] with a<π-b let us define

    S_a,b{(r,ϕ) |  0≤ r≤ R, a≤ϕ≤π-b }.
 
We let ϕ_1,ϕ_2 be the maximum angles for which S_Γ⊂ S_ϕ_1,ϕ_2. From (<ref>) and (<ref>), we deduce that 

    ∫_B(0,R)^2,+Γ∧ B_0 ≤∫_S_ϕ_1,ϕ_2 B_0·ŷ   =∫_0^R ∫_ϕ_1^π-ϕ_2f(r)sinϕ dϕ dr
       =(cosϕ_1 +cosϕ_2)∫_0^R f(r)dr

On the other hand, by definition of ϕ_1,ϕ_2, S_Γ intersects the rays {(r,ϕ_1)  |  0≤ r≤ R } and {(r,ϕ_2)  |  0≤ r≤ R }.
Since the endpoints of Γ belong to ∂ B(0,R)∩∂ B(0,R)^2,+, a simple geometric argument shows that 

    |Γ|(B(0,R)^2,+)≥ d((R,ϕ_1),(R,ϕ_2)).

The law of cosines yields d((R,ϕ_1),(R,ϕ_2))=R√(2(1-cos(π-ϕ_1-ϕ_2))). Hence

    1/|Γ|(B(0,R)^2,+)∫_B(0,R)^2,+Γ∧ B_0≤cosϕ_1+cosϕ_2/√(2(1-cos(π-ϕ_1-ϕ_2)))∫_0^R f(r)dr/R.

We now estimate the right-hand side of this inequality. Let us observe that

    cosϕ_1+cosϕ_2=2cos(ϕ_1+ϕ_2/2) cos(ϕ_1-ϕ_2/2)

and

    cos(π-ϕ_1-ϕ_2)=
    cos(ϕ_1+ϕ_2)=cos^2(ϕ_1+ϕ_2/2)-sin^2(ϕ_1+ϕ_2/2)=2cos^2(ϕ_1+ϕ_2/2)-1.

Using 0≤ϕ_1+ϕ_2/2<π/2, we deduce that

    cosϕ_1+cosϕ_2/√(2(1-cos(π-ϕ_1-ϕ_2)))= cos(ϕ_1-ϕ_2/2)≤ 1,

with equality if and only if ϕ_1=ϕ_2. Therefore

    1/|Γ|(B(0,R)^2,+)∫_B(0,R)^2,+Γ∧ B_0≤∫_0^R f(r)dr/R   =3/2(1-1/sinh R∫_0^Rsinh r/r dr)
       =1/2R∫_B(0,R) S_1∧ B_0.

Besides, from the previous computations we easily deduce that the inequality is strict if Γ≠ S_1.

Step 5. Estimate for loops in B(0,R)^2,+.
Let us define 0<r_0<R as the minimum radius such that 

    S_Γ⊂ B(0,r_0)^2,+.

In particular, S_Γ∩ (∂ B(0,r_0)∩∂ B(0,r_0)^2,+)≠∅. We can then use the estimate provided in the previous step and conclude that

    1/|Γ|(B(0,R)^2,+)∫_B(0,R)^2,+Γ∧ B_0≤3/2(1-1/sinh r_0∫_0^r_0sinh r/r dr).

One can check that the function R→1/sinh R∫_0^Rsinh r/r dr is strictly decreasing in [0,∞) and therefore 

    1/|Γ|(B(0,R)^2,+)∫_B(0,R)^2,+Γ∧ B_0< 3/2(1-1/sinh R∫_0^Rsinh r/r dr).

This concludes the proof of the proposition.
  



§ A MEISSNER-TYPE SOLUTION BEYOND THE FIRST CRITICAL FIELD

In this section, we present the proof of Theorem <ref>.
 
Step 1. Existence of a locally minimizing vortexless configuration.
Let us introduce the set

    U={(u,A)∈ H^1(Ω,)× [A_+H_]  |  F_(u',A')< ^2/3},

where u'=u_0^-1u and A'=A-h_A_0.
Consider a minimizing sequence {(ũ_n,Ã_n)}_n∈ U. Lemma <ref> yields a gauge transformed sequence {u_n,A_n}_n∈ [A_+Ḣ^1_=0], that in particular satisfies F_(u_n',A_n')=F_(ũ_n',Ã_n')<^2/3. In addition, by gauge invariance, we can take this sequence to satisfy A_n'·ν=0 on ∂Ω. Then arguing as in Proposition <ref>, we deduce that (up to subsequence) {(u_n,A_n-A_)}_n converges to some (u,A-A_) weakly in H^1(Ω,)×Ḣ^1_=0. Arguing again as in Proposition <ref>, we find

    F_(u',A')≤lim inf_n F_(u_n',A_n')   and   GL_(u,A)≤lim inf_n GL_(u_n,A_n).

Hence (u,A)∈U∩ H^1(Ω,)× [A_+ Ḣ_=0^1] satisfies A'·ν=0 on ∂Ω and minimizes GL_ over U.

Let us now prove that (u,A)∈ U. We consider, for =()=c_1^1/3 and  sufficiently small, the grid (b_,R_0,) associated to (u',A') by <cit.>*Lemma 2.1[Though it is not explicitly written in the paper, if F_(u',A')≤^γ with γ∈ (0,1) then <cit.>*Lemma 2.1 holds for any c_1^1+γ/4≤≤_0. The proof of this is exactly the same.]. In particular, using the same notation as in this lemma, we have

    |u_|>5/8  on    _1((b_,R_0,δ)),
    
    I_^1∫__1((b_,R_0,δ))e_(u',A')d^̋1   ≤  C ^-2F_(u',A')≤ C^-2/3^2/3,
    
    I_^2∫__2((b_,R_0,δ))e_(u',A')d^̋2   ≤  C ^-1F_(u',A')≤ C^-1/3^2/3.

Then, the 2D ball construction method (see <cit.>*Theorem 4.1) applied in the 2D skeleton _2((b_,R_0,)) of the grid implies that for each face ω of a cube of the grid, the collection of connected components S_i,ω of {x∈ω |  |u(x)|<1/2} whose degree is different from zero is empty (see <cit.>*Section 4). We thus deduce that the 1-current ν_', which approximates well the vorticity μ(u',A'), vanishes identically in Ω. Then, from the proof of Theorem <ref> (see <cit.>*Section 8), we find

    μ(u',A')_C_T^0,1(Ω)^*≤ C F_(u',A')+C(1+I_^1+I_^2)≤ C( +^-2)F_(u',A').

Let us now use Proposition <ref>. From the previous inequality and since α<1/3, we have

    | h_∫_Ωμ(u',A')∧ B_0|≤ Ch_( +^-2)F_(u',A')≤ C^1/3-αF_(u',A')=o(^2/3).

On the other hand

    R_0≤ C h_^2E_(|u'|)^1/2≤ C h_^2F_(u',A')^1/2≤ C^1-2α^1/3=o(^2/3).

The energy-splitting (<ref>) then yields 

    GL_(u,A)=h_^2J(A_0)+F_(u',A')+1/2∫_^3∖Ω | A'|^2+o(^2/3).

But, since (u_0,h_A_0) belongs to U, we have

    GL_(u,A)≤ GL_(u_0,h_A_0)=h_^2J(A_0).

We thus deduce that

    F_(u',A')+1/2∫_^3∖Ω| A'|^2=o(^2/3),

and therefore (u,A)∈ U provided  is small enough. 

Now, since U is open in H^1(Ω,)× [A_+H_], the minimizer (u,A) must be a critical point of GL_ and therefore satisfies the Ginzburg-Landau equations (<ref>). Arguing as in the proof of Theorem <ref>, we deduce that (u,A) is a vortexless configuration and

    1-|u|_L^∞(Ω,)=1-|u'|_L^∞(Ω,)=o(1)  as → 0.

We note that we have omitted in our notation the dependence on  of the minimizer (u,A).

Step 2. Characterization of (u',A').
Since A'·ν =0 on ∂Ω, we have

    ∫_Ω |∇ u'|^2≤∫_Ω |∇_A' u'|^2+|A'|^2|u'|^2≤ CF_(u',A')≤ C^2/3

for some universal constant C>0.

Observe that, using the Poincaré-Wirtinger inequality, we have

    ∫_Ω |u'-u'|^2≤ C∫_Ω |∇ u'|^2,  where u'=1/|Ω|∫_Ω u'.

In addition, we have

    ∫_Ω| |u'|-|u'||^2≤∫_Ω |u'-u'|^2

and 

    ∫_Ω (1-|u'|)^2≤∫_Ω (1-|u'|^2)^2≤ 4^2 F_(u',A')≤ 4^2+2/3.

We deduce that 1-|u'|_L^2(Ω,)≤ C^1+1/3. But u' is a constant, thus u'=e^iθ_+O(^1+1/3) for some θ_∈[0,2π]. By combining with (<ref>) and (<ref>), we find

    ∫_Ω |u'-e^iθ_|^2≤ C^2/3.

Thus

    inf_θ∈[0,2π]u'-e^iθ_H^1(Ω,)→ 0  as → 0.

On the other hand, from (<ref>), we have A'^2_L^2(^3,^3)=o(^2/3), which combined with the fact that A'=A-h_A_0∈Ḣ^1_=0 implies that A'_Ḣ^1_=0^2=o(^2/3).

In particular, by noting that (e^iθ,h_ B_0) is gauge equivalent to (1,h_ B_0) in Ω for any θ∈[0,2π], we deduce that (up a gauge transformation) the configuration (u',A'+h_ B_0), which is gauge equivalent to (u,A) in Ω, gets closer and closer in the H^1(Ω,)× H^1(Ω,^3)-norm to (1,h_ B_0).

Step 3. (u,A) globally approaches (u_0,h_A_0).
Observe that, for any θ∈ [0,2π], we have

    ∫_Ω |u-e^iθu_0|^2=∫_Ω |u'u_0-e^iθu_0|^2=∫_Ω |u'-e^iθ|^2

and

    ∫_Ω |∇ (u-e^iθu_0)|^2≤∫_Ω |∇ u_0|^2|u'-e^iθ|^2+∫_Ω |∇ u'|^2.

From (<ref>), we deduce that 

    inf_θ∈[0,2π]u-e^iθu_0_L^2(Ω,)→ 0  as → 0.

Recall that u_0=e^ih_ϕ_0 and that A_0 satisfies the Euler-Lagrange equation (<ref>). Since (H_0-H_0,)= (A_0-A_0,)=-Δ(A_0-A_0,), standard elliptic regularity theory implies that ϕ_0=A_0- B_0∈ L^∞(Ω). Therefore

    ∫_Ω |∇ u_0|^2|u'-e^iθ|^2≤ h_^2∇ϕ_0_L^∞(Ω)^2 u'-e^iθ_L^2(Ω,)^2.

This combined with (<ref>) for θ=θ_, yields

    ∫_Ω |∇ u_0|^2|u'-e^iθ_|^2≤ C^-2α^2/3.

Since α<1/3, the right-hand side converges to 0 as →0. Using once again (<ref>), we obtain

    inf_θ∈[0,2π]∫_Ω |∇ (u-e^iθu_0)|^2→ 0  as → 0.

Hence

    inf_θ∈[0,2π]u-e^iθu_0_H^1(Ω,)→ 0  as → 0.

Moreover, we have

    A-h_A_0_Ḣ^1_=0=A'_Ḣ^1_=0→ 0  as → 0.


We have hence shown that, up to a gauge transformation in ^3, the solution (u,A) gets closer and closer in the H^1(Ω,)×Ḣ^1_=0-norm to (u_0,h_A_0). In addition, up to a (different) gauge transformation in Ω, the solution approaches in the H^1(Ω,)× H^1(Ω,^3)-norm the configuration (1,h_ B_0).



The assumption h_≤^-α for α<1/3 is needed to prove that 

    | h_∫_Ωμ(u',A')∧ B_0|≤ o(F_(u',A'));

see (<ref>). If α≥1/3, we are not able to show this, and our strategy to prove that (u,A)∈ U then fails.




§ UNIQUENESS OF LOCALLY MINIMIZING VORTEXLESS CONFIGURATIONS

In this section we prove Theorem <ref>. We follow the same strategy as in <cit.>*Section 2.

Since any pair (ũ,Ã)∈ H^1(Ω,)× [A_+ H_] is gauge equivalent to (u,A)∈ H^1(Ω,)× H^1_=0 with A_j'·ν=0 on ∂Ω, it is enough to prove the theorem in this gauge.

Let us assume towards a contradiction that there are two distinct locally minimizing vortexless solutions (u_j,A_j)=(u_0u_j',h_A_0+A_j') to (<ref>) with u_j∈ H^1(Ω,), |u_j|≥ c for some c∈(0,1), A_j'∈ H^1_=0, A_j'·ν=0 on ∂Ω, and

    F_(u_j',A_j')≤ C^1+  for j=1,2,

for some >0. As we shall see, this estimate is crucial to prove the theorem.

Since |u_j'|=|u_j|≥ c>0, we can write u_j'=η_je^iϕ_j in Ω for j=1,2. Note that the functions ϕ_0,ϕ_1,ϕ_2∈ H^2(Ω) can be extended to functions in H^2(^3). Therefore, for j=1,2, (u_j,A_j) is gauge equivalent to (η_j,Ã_j) with

    Ã_j=h_(A_0-∇ϕ_0)+A_j'-∇ϕ_j.


Step 1. Estimating Ã_j_L^∞(Ω,^3).
Let us show that, for j=1,2, we have

    Ã_j_L^∞(Ω,^3)≤ o(^-1).

By gauge equivalence, (u_j',h_(A_0-∇ϕ_0)+A_j') solves (<ref>). Then, by standard elliptic regularity theory for solutions of the Ginzburg-Landau equations in the Coulomb gauge, we have

    h_A_0 B_0+A_j'_L^∞(Ω,^3)≤ Ch_  and  ∇ u_j'_L^∞(Ω,^3)≤ C^-1.

Since

    ∇η_j_L^∞(Ω,^3)+∇ϕ_j_L^∞(Ω,^3)≤ 2∇ u_j'_L^∞(Ω,^3),

and h_=o(^-1), we find

    Ã_j_L^∞(Ω,^3)≤h_ B_0+A_j'_L^∞(Ω,^3)+∇ϕ_j_L^∞(Ω,^3)≤ C^-1.

We will now improve this estimate. By gauge equivalence, (η_j,Ã_j) solves (<ref>). In particular, the second Ginzburg-Landau equation in Ω reads

    ^2(Ã_j - A_)=-η_j^2Ã_j.

This implies that (η_j^2Ã_j)=0 in Ω. In addition, the boundary condition ∇_Ã_jη_j·ν=0 on ∂Ω, implies, in particular, that ∇ϕ_j ·ν =0 on ∂Ω. Therefore, ϕ_j satisfies the elliptic problem

    {[          Δϕ_j             = 2η_j∇η_j ·Ã_j          in Ω;        ∇ϕ_j·ν             =             0        on ∂Ω. ].

Because η_j≥ c>0, we deduce that, for any p>1,

    Δϕ_j _L^p(Ω)≤ CÃ_j_L^∞(Ω,^3)∇η_j_L^p(Ω,^3)≤ C^-1∇η_j_L^p(Ω,^3),

where the last inequality is obtained by using (<ref>).

On the other hand, since A_j'·ν =0 on ∂Ω, we have 

    ∇ u_j'^2_L^2(Ω,^3), A_j'^2_L^2(Ω,^3)≤ C F_(u_j',A_j')≤ C^1+.

This implies that

    ∫_Ω |∇η_j|^2+η_j^2|∇ϕ_j|^2=∫_Ω |∇ u_j'|^2≤ C^1+.

In addition, by interpolation, for any p>1, we have

    ∇η_j_L^p(Ω,^3)≤ C ∇η_j_L^∞(Ω,^3)^1-2/p∇η_j_L^2(Ω,^3)^2/p.

Combining the previous two inequalities with (<ref>), yields

    ∇η_j_L^p(Ω,^3)≤ C^-1+2/p^1+/p=C^3+-p/p.

Combining with (<ref>) for p=3+/2>3, we find

    Δϕ_j _L^p(Ω)≤ C^-1^/6+=o(^-1).

By an elliptic estimate and Sobolev embedding, we then obtain

    ∇ϕ_j _L^∞(Ω)≤ o(^-1).

Thus

    Ã_j_L^∞(Ω,^3)≤h_ B_0+A_j'_L^∞(Ω,^3)+∇ϕ_j_L^∞(Ω,^3)≤ Ch_ + o(^-1)=o(^-1).


Step 2. Energy estimate.
Let us prove that 

    YGL_(η_1,Ã_1)+GL_(η_2,Ã_2)/2-GL_(η_1+η_2/2,Ã_1+Ã_2/2)>0

First, observe that

    ∫_Ω |∇_Ã_jη_j|^2=∫_Ω |∇η_j|^2+η_j^2|Ã_j|^2.

We write Y=Y_0+Y_1+Y_2+Y_3 with

    Y_0   =1/2∫_Ω |∇η_1|^2+|∇η_2|^2-∫_Ω|∇(η_1+η_2/2)|^2,
    
    Y_1   =1/2∫_Ωη_1^2|Ã_1|^2+η_2^2|Ã_2|^2-∫_Ω(η_1+η_2/2)^2|Ã_1+Ã_2/2|^2,
    
    Y_2   =1/2(1/4^2∫_Ω (1-η_1^2)^2+(1-η_2^2)^2)-1/4^2∫_Ω(1-(η_1+η_2/2)^2)^2,
    
    Y_3   =1/2∫_Ω |Ã_1-H_|^2+|Ã_2-H_|^2-∫_Ω|(Ã_1+Ã_2/2)-H_|^2.

Note that, by convexity we have Y_0,Y_3≥ 0.

On the other hand, arguing exactly as in the proof of <cit.>*Lemma 2.5, we get

    Y_1=1/16∫_Ω |η_1-η_2|^2|Ã_1+Ã_2|^2+4η_1^2|Ã_1-Ã_2|^2
    
    -(η_1-η_2)(Ã_1-Ã_2)·(Ã_1(2η_1+4η_2)+Ã_2(6η_1+8η_2))

and

    Y_2≥3/64^2∫_Ω (η_1-η_2)^2.

Let us prove that Y_1+Y_2>0. We consider three cases.


  * If η_1=η_2 then 

    Y_1+Y_2≥∫_Ω 4η_1^2|Ã_1-Ã_2|^2>0.


  * If Ã_1=Ã_2 then Y_1≥ 0. Therefore

    Y_1+Y_2≥ Y_2≥3/64^2∫_Ω (η_1-η_2)^2>0.


  * If η_1≠η_2 and Ã_1≠Ã_2 then

    Y_1≥1/16∫_Ω |η_1-η_2|^2|Ã_1+Ã_2|^2+4η_1^2|Ã_1-Ã_2|^2-|η_1-η_2||Ã_1-Ã_2|(6|Ã_1|+14|Ã_2|).

By the Cauchy-Schwarz inequality, we have

    ∫_Ω|η_1-η_2||Ã_1-Ã_2|(6|Ã_1|+14|Ã_2|)
    ≤ 14(Ã_1_L^∞(Ω,^3)+Ã_2_L^∞(Ω,^3))η_1-η_2_L^2(Ω)Ã_1-Ã_2_L^2(Ω,^3),

which combined with (<ref>), yields

    ∫_Ω|η_1-η_2||Ã_1-Ã_2|(6|Ã_1|+14|Ã_2|)≤ o(^-1)η_1-η_2_L^2(Ω)Ã_1-Ã_2_L^2(Ω,^3).

On the other hand,

    ∫_Ω1/4η_1^2|Ã_1-Ã_2|^2+3/64^2(η_1-η_2)^2 ≥9/32η_1-η_2_L^2(Ω)Ã_1-Ã_2_L^2(Ω,^3).

Hence, if  is small enough then Y_1+Y_2>0.

We have thus proved that Y>0.

Step 3. Contradiction.
Assume without loss of generality that 

    GL_(η_1,Ã_1)≤ GL_(η_2,Ã_2).

From the previous step, we have

    GL_(η_1+η_2/2,Ã_1+Ã_2/2)<GL_(η_1,Ã_1)+GL_(η_2,Ã_2)/2≤ GL_(η_2,Ã_2).

A standard argument then shows that, for any t∈(0,1), 

    GL_(tη_1+(1-t)η_2,tÃ_1+(1-t)Ã_2)<GL_(η_2,Ã_2),

contradicting the fact that (η_2,Ã_2) is a local minimizer of the energy. Hence (η_1,Ã_1)=(η_2,Ã_2). This concludes the proof.




§ IMPROVED ESTIMATES FOR LOCALLY MINIMIZING VORTEXLESS CONFIGURATIONS


Let (u,A)∈ H^1(Ω,)× H^1(Ω,^3) with u continuous and |u|≥ c for some c∈(0,1). Then

    μ(u,A)_C_T^0,1(Ω,^3)^*≤ C F_(u,A).



Let φ∈ C_T^0,1(Ω,^3). By integration by parts, we have

    ∫_Ωμ(u,A)∧φ =∫_Ω (j(u,A)+A)·φ.

Since |u|≥ c>0, we can write u=|u|e^iϕ. A straightforward computation, shows that

    j(u,A)+A=|u|^2∇ϕ +(1-|u|^2)A=(1-|u|^2)(A-∇ϕ)+∇ϕ.

Observe that, by integration by parts, we have ∫_Ω∇ϕ·φ=0. Then, from the Cauchy-Schwarz inequality, we deduce that

    |∫_Ω (j(u,A)+A)·φ|≤∫_Ω (1-|u|^2)|A-∇ϕ||φ|≤ Cφ_L^∞(Ω,^3) F_(u,A).

Hence

    μ(u,A)_C_T^0,1(Ω,^3)^*≤ C F_(u,A).


With this estimate at hand, we prove the following result.
 
Denote (u_0,h_A_0) the approximation of the Meissner solution. Let (u,A)=(u_0u',h_A_0+A')∈ H^1(Ω,)× [A_+H_] with u continuous and |u|≥ c for some c∈(0,1). If h_≤^-α for some α∈(0,1/4) and GL_(u,A)≤ GL_(u_0,h_A_0) then, for any  sufficiently small, we have

    F_(u',A')+1/2∫_^3∖Ω | A'|^2≤ C^1+

for some ∈(0,1).


Let us first observe that, since GL_(u,A)≤ GL_(u_0,h_A_0)= h_^2J(A_0), we have

    F_(u',A')≤ Ch_^2 ≤ C ^-2α

for some constant C>0. We will now use Proposition (<ref>) to improve this estimate. By combining (<ref>) with the GL_(u,A)≤ GL_(u_0,h_A_0), we find

    F_(u',A')+1/2∫_^3∖Ω | A'|^2≤ h_∫_Ωμ(u',A')∧ B_0+C h_^2E_(|u'|)^1/2.

From Proposition <ref> and E_(|u'|)≤ F_(u',A'), we deduce that

    F_(u',A')+1/2∫_^3∖Ω | A'|^2≤ C h_F_(u',A')+C h_^2F_(u',A')^1/2≤ C h_^2F_(u',A')^1/2.

Inserting (<ref>) and h_≤^-α into (<ref>), we get 

    F_(u',A')+1/2∫_^3∖Ω | A'|^2≤ C^1-3α≤ C^(1-2α)(1+1/2).

By inserting this estimate into (<ref>), we obtain

    F_(u',A')+1/2∫_^3∖Ω | A'|^2≤ C^(1-2α)(1+1/2+1/4).

Repeating this process a finite number of times, we are led to

    F_(u',A')+1/2∫_^3∖Ω | A'|^2≤ C^(1-2α)(1+1/2+1/4+⋯+1/2k)

for some k∈. Since α<1/4 and ∑_i=0^∞1/2^i=2, we deduce that 

    (1-2α)∑_i=0^k 1/2^i>1
 
for any k sufficiently large. Hence

    F_(u',A')+1/2∫_^3∖Ω | A'|^2≤ C^1+

for some >0.

As a consequence, from Theorem <ref>, we obtain the uniqueness of the Meissner-type solution of Theorem <ref> for α<1/4.



§ ACKNOWLEDGMENTS

I am very grateful to my former Ph.D. advisors Etienne Sandier and Sylvia Serfaty for suggesting the problem and for useful comments.
Most of this work was done while I was a Ph.D. student at the Jacques-Louis Lions Laboratory of the Pierre and Marie Curie University, supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d'Avenir” program (reference: ANR-10-LABX-0098, LabEx SMP). Part of this work was supported by the German Science Foundation DFG in the context of the Emmy Noether junior research group BE 5922/1-1. 


