The ground state energy of the two dimensional Ginzburg–Landau functional with variable magnetic field

ScienceDirect

Ann. I. H. Poincaré – AN 32 (2015) 325–345

www.elsevier.com/locate/anihpc

The ground state energy of the two dimensional Ginzburg–Landau functional with variable magnetic field

K. Attar

aLaboratoire de Mathématiques, Université Paris-Sud, Rue G. Clémenceau, Bât 425, F-91405 Orsay Cedex, France bLaboratoire de Mathématiques, Université Libanaise, Hadath, Beyrouth, Liban

Received 24 May 2013; accepted 13 December 2013 Available online 7 January 2014

Abstract

We consider the Ginzburg–Landau functional with a variable applied magnetic field in a bounded and smooth two dimensional domain. We determine an accurate asymptotic formula for the minimizing energy when the Ginzburg–Landau parameter and the magnetic field are large and of the same order. As a consequence, it is shown how bulk superconductivity decreases in average as the applied magnetic field increases.

©2014 Elsevier Masson SAS. All rights reserved.

MSC:82D55

Keywords:Superconductivity; Ginzburg–Landau; Variable magnetic field

1. Introduction

1.1. The functional and main results

We consider a bounded open simply connected setΩ⊂R² with smooth boundary. We suppose thatΩ models a superconducting sample submitted to an applied external magnetic field. The energy of the sample is given by the Ginzburg–Landau functional,

Eκ,H(ψ,A)=

Ω

(∇ −iκHA)ψ²−κ²|ψ|²+κ² 2 |ψ|⁴

dx+κ²H²

Ω

|curlA−B₀|²dx. (1.1)

HereκandHare two positive parameters;κ(the Ginzburg–Landau constant) is a material parameter andHmeasures the intensity of the applied magnetic field. The wave function (order parameter)ψ∈H¹(Ω;C)describes the super- conducting properties of the material. The induced magnetic field is curlA, where the potentialA∈H_div¹ (Ω), with H_div¹ (Ω)is the space defined in(1.4)below. Finally,B₀∈C^∞(Ω)is the intensity of the external variable magnetic field and satisfies

|B0| + |∇B0|>0 inΩ. (1.2)

0294-1449/$ – see front matter ©2014 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2013.12.002

The assumption in(1.2)implies that for any open setωrelatively compact inΩ the set{x∈ω, B₀(x)=0}will be either empty, or consists of a union of smooth curves. LetF:Ω→R²be the unique vector field such that

divF=0 and curlF=B0 inΩ, ν·F=0 on∂Ω. (1.3)

The vectorνis the unit interior normal vector of∂Ω. The construction ofFis recalled inAppendix A. We define the space

H_div¹ (Ω)=

A=(A1,A2)∈H¹(Ω)²: divA=0 inΩ,A·ν=0 on∂Ω

. (1.4)

Critical points(ψ,A)∈H¹(Ω;C)×H_div¹ (Ω)ofEκ,H are weak solutions of the Ginzburg–Landau equations

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

−(∇ −iκHA)²ψ=κ²

1− |ψ|²

ψ inΩ,

−∇^⊥curl(A−F)= 1 κH Im

ψ(∇ −iκHA)ψ inΩ,

ν·(∇ −iκHA)ψ=0 on∂Ω,

curlA=curlF on∂Ω.

(1.5)

Here, curlA=∂_x1A2−∂_x2A1 and∇^⊥curlA=(∂_x2(curlA),−∂_x1(curlA)). If divA=0, then∇^⊥curlA=A. In this paper, we study the ground state energy defined as follows:

Eg(κ, H )=inf

Eκ,H(ψ,A): (ψ,A)∈H¹(Ω;C)×H_div¹ (Ω)

. (1.6)

More precisely, we give an asymptotic estimate which is valid in the simultaneous limitκ→ ∞andH→ ∞in such a way that ^H_κ remains asymptotically constant. The behavior of Eg(κ, H )involves an auxiliary functiong: [0,∞)→ [−¹₂,0]introduced in[9]whose definition will be recalled in(2.5)below. The functiongis increasing, continuous, g(b)=0 for allb1 andg(0)= −¹₂.

Theorem 1.1.Let0< Λ_min< Λ_max. Under assumption(1.2), there exist positive constantsC,κ₀andτ₀∈(1,2)such that if

κ₀κ, Λ_minH

κ Λ_max, then the ground state energy in(1.6)satisfies

Eg(κ, H )−κ²

Ω

g H

κ B₀(x) dx

Cκ^τ0. (1.7)

Theorem 1.1 was proved in[9] when the magnetic field is constant (B0(x)=1). However, the estimate of the remainder is not explicitly given in[9].

The approach used in the proof ofTheorem 1.1is slightly different from the one in[9], and is closer to that in[4]

which studies the same problem whenΩ⊂R³andB₀constant.

Corollary 1.2.Suppose that the assumptions ofTheorem1.1are satisfied. Then the magnetic energy of the minimizer satisfies, for some positive constantC

(κH )²

Ω

|curlA−B₀|²dxCκ^τ0. (1.8)

Remark 1.3. The value ofτ₀ depends on the properties ofB₀: we findτ₀=⁷₄ whenB₀does not vanish inΩ and τ0=¹⁵₈ in the general case.

Theorem 1.4. Suppose the assumptions of Theorem 1.1are satisfied. There exist positive constantsC,κ₀ and a negative constantτ₁∈(−1,0)such that, ifκκ₀, andDis regular set such thatD⊂Ω, then the following are true.

(1) If(ψ,A)∈H¹(Ω;C)×H_div¹ (Ω)is a solution of(1.5), then 1

2

D

|ψ|⁴dx−

D

g H

κ B₀(x)

dx+Cκ^τ1. (1.9)

(2) If(ψ,A)∈H¹(Ω;C)×H_div¹ (Ω)is a minimizer of(1.1), then

D

|ψ|⁴dx+2

D

g H

κ B₀(x) dx

Cκ^τ1. (1.10)

Remark 1.5.The value ofτ₁depends on the properties ofB₀: we findτ₁= −¹₄ whenB₀does not vanish inΩ and τ1= −¹₈ in the general case.

1.2. Discussion of main result

If {x ∈Ω: B0(x)=0} =∅ andH =bκ,b >0, theng(^H_κ|B0(x)|)=0 in D= {x∈Ω: ^H_κ|B0(x)|<1}, and

|D| =0. Consequently, forκ sufficiently large, the restriction ofψ onD is not zero inL²(Ω). This is a significant difference between our result and the one for constant magnetic field. When the magnetic field is a non-zero con- stant, then (see[3]), there is a universal constant0∈(¹₂,1)such that, ifH=bκ andb >⁻₀¹, thenψ=0 inΩ. Moreover, in the same situation, whenH=bκand 1< b <⁻₀¹, thenψis small everywhere except in a thin tubular neighborhood of∂Ω (see[6]). Our result goes in the same spirit as in[8], where the authors established under the assumption(1.2)that whenH=bκ²andb > b₀, thenψ=0 inΩ(b0is a constant).

1.3. Notation

Throughout the paper, we use the following notation:

• We writeEfor the functionalEκ,H in(1.1).

• The letterC denotes a positive constant that is independent of the parametersκ andH, and whose value may change from a formula to another.

• Ifa(κ)andb(κ)are two positive functions, we writea(κ) b(κ)ifa(κ)/b(κ)→0 asκ→ ∞.

• Ifa(κ)andb(κ)are two functions withb(κ)=0, we writea(κ)∼b(κ)ifa(κ)/b(κ)→1 asκ→ ∞.

• Ifa(κ)andb(κ)are two positive functions, we writea(κ)≈b(κ)if there exist positive constantsc₁,c₂andκ₀ such thatc₁b(κ)a(κ)c₂b(κ)for allκκ₀.

• Ifx∈R, we let[x]₊=max(x,0).

• GivenR >0 andx=(x₁, x₂)∈R², we denote byQ_R(x)=(−R/2+x₁, R/2+x₁)×(−R/2+x₂, R/2+x₂) the square of side lengthRcentered atx.

• We will use the standard Sobolev spacesW^s,p. For integer values ofsthese are given by W^n,p(Ω):=

u∈L^p(Ω): D^αu∈L^p(Ω)for all|α|n .

• Finally we use the standard symbolHⁿ(Ω)=W^n,2(Ω).

2. The limiting energy

2.1. Two-dimensional limiting energy

Given a constantb0 and an open setD⊂R², we define the following Ginzburg–Landau energy G^σ_b,D(u)=

D

b(∇ −iσA0)u²− |u|²+1 2|u|⁴

dx, ∀u∈H₀¹(D). (2.1)

Hereσ∈ {−1,+1}andA0is the canonical magnetic potential A0(x)=1

2(−x2, x1), ∀x=(x1, x2)∈R², (2.2)

that satisfies

curlA0=1 inR². We writeQ_R=Q_R(0)and let

m₀(b, R)= inf

u∈H₀¹(QR;C)

G⁺_b,Q¹

R(u). (2.3)

Remark 2.1.AsG⁺_b,¹_D(u)=G⁻_b,¹_D(u), it is immediate that inf

u∈H₀¹(Q_R;C)

G⁻_b,Q¹

R(u)= inf

u∈H₀¹(Q_R;C)

G⁺_b,Q¹

R(u). (2.4)

The main part of the next theorem was obtained by Sandier and Serfaty[9]and Aftalion and Serfaty[1, Lemma 2.4].

However, the estimate in(2.7)is obtained by Fournais and Kachmar[4].

Theorem 2.2.Letm0(b, R)be as defined in(2.3).

(1) For allb1andR >0, we havem0(b, R)=0.

(2) For anyb∈ [0,∞), there exists a constantg(b)0such that g(b)= lim

R→∞

m₀(b, R)

|QR| and g(0)= −1

2. (2.5)

(3) The function[0,+∞)b→g(b)is continuous, non-decreasing, concave and its range is the interval[−¹₂,0].

(4) There exists a constantα∈(0,¹₂)such that

∀b∈ [0,1], α(b−1)²g(b)1

2(b−1)². (2.6)

(5) There exist constantsCandR0such that

∀RR0, ∀b∈ [0,1], g(b)m₀(b, R)

R² g(b)+C

R. (2.7)

3. A priori estimates

The aim of this section is to givea prioriestimates for solutions of the Ginzburg–Landau equations(1.5). These estimates play an essential role in controlling the errors resulting from various approximations. The starting point is the followingL^∞-bound resulting from the maximum principle. Actually, if(ψ,A)∈H¹(Ω;C)×H_div¹ (Ω)is a solution of(1.5), then

ψL^∞(Ω)1. (3.1)

The set of estimates below is proved in[2, Theorem 3.3 and Eq. (3.35)](see also[7]for an earlier version).

Theorem 3.1.LetΩ⊂R²be bounded and smooth andB0∈C^∞(Ω).

(1) For allp∈(1,∞)there existsCp>0such that, if(ψ,A)∈H¹(Ω, C)×H_div¹ (Ω)is a solution of(1.5), then curlA−B₀W^1,p(Ω)C_p1+κH+κ²

κH ψL^∞(Ω)ψL²(Ω). (3.2)

(2) For allα∈(0,1)there existsC_α>0such that, if(ψ,A)∈H¹(Ω, C)×H_div¹ (Ω)is a solution of(1.5), then curlA−B0_C^0,α_(Ω)Cα

1+κH+κ²

κH ψL∞(Ω)ψL²(Ω). (3.3)

(3) For allp∈ [2,∞)there existsC >0such that, ifκ >0,H >0and(ψ,A)∈H¹(Ω, C)×H_div¹ (Ω)is a solution of(1.5), then

(∇ −iκHA)²ψ

pκ²ψp, (3.4)

(∇ −iκHA)ψ

2κψ2, (3.5)

curl(A−F)

W^1,p(Ω) C

κHψ∞(∇ −iκHA)ψ

p. (3.6)

Remark 3.2.

(1) Using theW^k,p-regularity of the curl–div system[3, Appendix A, Proposition A.5.1], we obtain from(3.2), A−FW^2,p(Ω)Cp

1+κH+κ²

κH ψL^∞(Ω)ψL²(Ω). (3.7)

The estimate is true for anyp∈ [2,∞).

(2) Using the Sobolev embedding theorem we get, for allα∈(0,1) A−F_C^1,α_(Ω)C_α1+κH+κ²

κH ψL^∞(Ω)ψL²(Ω). (3.8)

(3) Combining(3.5)and(3.6)(withp=2) yields curl(A−F)

L²(Ω)C

HψL^∞(Ω)ψL²(Ω). (3.9)

Theorem 3.1is needed in order to obtain the improveda prioriestimates of the next theorem. Similar estimates are given in[7].

Theorem 3.3.Suppose that0< Λ_minΛ_max. There exist constantsκ₀>1,C₁>0and for anyα∈(0,1),C_α>0 such that, if

κκ₀, Λ_minH

κ Λ_max, (3.10)

and(ψ,A)∈H¹(Ω;C)×H_div¹ (Ω)is a solution of(1.5), then (∇ −iκHA)ψ

C(Ω)C₁√

κHψL^∞(Ω), (3.11)

A−F_H²_(Ω)C₁curl(A−F)

L²(Ω)+ 1

√κHψL²(Ω)ψL^∞(Ω)

, (3.12)

A−F_C0,α(Ω)C_αcurl(A−F)

L²(Ω)+ 1

√κHψ_L²_(Ω)ψL^∞(Ω)

. (3.13)

Proof. Proof of (3.11): See[3, Proposition 12.4.4].

Proof of (3.12): Leta=A−F. Since diva=0 anda·ν=0 on∂Ω, we get by regularity of the curl–div system (seeAppendix A,Proposition A.1),

aH²(Ω)CcurlaH¹(Ω). (3.14)

The second equation in(1.5)reads as follows

−∇^⊥curla= 1 κH Im

ψ(∇ −iκHA)ψ .

The estimates in(3.11)and(3.14)now give aH²(Ω)C

curlaL²(Ω)+ 1

√κHψL²(Ω)ψL^∞(Ω)

.

Proof of (3.13): This is a consequence of the Sobolev embedding ofH²(Ω) into C^0,α(Ω) for any α∈(0,1) and(3.12). 2

4. Energy estimates in small squares

If(ψ,A)∈H¹(Ω;C)×H_div¹ (Ω), we introduce the energy density e(ψ,A)=(∇ −iκHA)ψ²−κ²|ψ|²+κ²

2 |ψ|⁴.

We also introduce the local energy of(ψ,A)in a domainD⊂Ω:

E0(u,A;D)=

D

e(ψ,A) dx. (4.1)

Furthermore, we define the Ginzburg–Landau energy of(ψ,A)in a domainD⊂Ω as follows, E(ψ,A;D)=E0(ψ,A;D)+(κH )²

Ω

curl(A−F)²dx. (4.2)

IfD=Ω, we sometimes omit the dependence on the domain and writeE0(ψ,A)forE0(ψ,A;Ω). We start with a lemma that will be useful in the proof ofProposition 4.2below. Before we start to state the lemma, we define for all (, x0)such thatQ(x0)⊂Ω,

B_Q(x0)= sup

x∈Q(x0)

B₀(x), (4.3)

whereB₀is introduced in(1.2). Laterx₀will be chosen in a lattice ofR².

Lemma 4.1.For anyα∈(0,1), there exist positive constantsCandκ₀such that if(3.10)holds,0< δ <1,0< <1, and (ψ,A)∈H¹(Ω;C)×H_div¹ (Ω)is a critical point of(1.1)(i.e. a solution of(1.5)), then, for any squareQ(x₀) relatively compact inΩ∩ {|B₀|>0}, there existsϕ∈H¹(Ω), such that

E0

ψ,A;Q(x0)

(1−δ)E0

e^{−iκH ϕ}ψ, σBQ(x₀)A0(x−x0), Q(x0)

−Cκ²

δ^−12α+δ⁻¹⁴κ²+δ

Q(x₀)

|ψ|²dx, (4.4)

whereσdenotes the sign ofB₀inQ(x₀).

Proof. Construction of ϕ: Letφ_x0(x)=(A(x0)−F(x0))·x, whereFis the magnetic potential introduced in(1.3).

Using the estimate in(3.13), we get for allx∈Q(x₀)andα∈(0,1), A(x)− ∇φ_x0−F(x)=(A−F)(x)−(A−F)(x0)

A−F_C^0,α· |x−x₀|^α

C

√λ

κH^α, (4.5)

where

λ=(κH )²curl(A−F)²

L²(Ω)+ 1

κHψ²_L2(Ω)

.

Using the boundψ_∞1 and the estimate in(3.9), we get

λCκ², (4.6)

which implies that

A(x)− ∇φx₀(x)−F(x)C^α

H. (4.7)

We estimate the energyE0(ψ,A;Q(x0))from below. We will need the functionϕ0introduced inLemma A.3and satisfying

F(x)−σB_Q(x0)A0(x−x₀)− ∇ϕ₀(x)C² inQ(x₀).

Let

u=e^{−iκH ϕ}ψ, (4.8)

whereϕ=ϕ₀+φ_x0.

Lower bound: We start with estimating the kinetic energy from below as follows. For anyδ∈(0,1), we write (∇ −iκHA)ψ²=∇ −iκH

σB_Q(x0)A0(x−x₀)+ ∇ϕ

ψ−iκH

A−σB_Q(x0)A0(x−x₀)− ∇ϕψ² (1−δ)∇ −iκH

σBQ(x₀)A0(x−x0)+ ∇ϕ ψ² +

1−δ⁻¹

(κH )²(A− ∇φ_x0−F)ψ+

F−σB_Q(x0)A0(x−x0)− ∇ϕ0

ψ². Using the estimates in(4.7),(A.3)and the assumption in(3.10), we get, for anyα∈(0,1)

(∇ −iκHA)ψ²(1−δ)∇ −iκH

σB_Q(x0)A0(x−x₀)+ ∇ϕψ²

−Cκ²

δ^−12α+δ⁻¹²²H2

|ψ|². (4.9)

Remembering the definition ofuin(4.8), then, we deduce the lower bound ofE0

E0

ψ,A;Q(x₀)

Q(x0)

(1−δ)∇ −iκH

σB_Q(x0)A0(x−x₀)

u²−κ²|u|²+κ² 2 |u|⁴

dx

−Cκ²

δ⁻¹²²κ+δ^−12α2

Q(x0)

|ψ|²dx

(1−δ)E0

u, σBQ(x₀)A0(x−x0);Q(x0)

−Cκ ²

δ⁻¹⁴κ²+δ^−12α+δ

Q(x0)

|ψ|²dx. (4.10)

This finishes the proof ofLemma 4.1. 2

Proposition 4.2.For allα∈(0,1), there exist positive constantsC,₀andκ₀such that, if(3.10)holds,κκ₀,∈ (0,¹₂),∈(0, 0),²κ² >1,(ψ,A)∈H¹(Ω;C)×H_div¹ (Ω)is a critical point of(1.1), andQ(x₀)⊂Ω∩{|B₀|> }, then

1

|Q(x₀)|E0

ψ,A;Q(x0)

g

H κ B_Q(x0)

κ²−C

³κ²+^2α−1+(κ)⁻¹+⁻¹ κ². Hereg(·)is the function introduced in(2.5), andBQ(x₀)is introduced in(4.3).

Proof. Using the inequalityψ∞1 and(4.4)to obtain E0

ψ,A;Q(x₀)

(1−δ)E0

u, σB_Q(x0)A0(x−x₀);Q(x₀)

−Cκ²

δ⁻¹⁴κ²+δ^−12α+δQ(x₀), (4.11)

whereuis defined in(4.8).

Let b=H

κ BQ(x₀), R=

κH BQ(x₀). (4.12)

Define the rescaled function v(x)=u

Rx+x₀

, ∀x∈Q_R. (4.13)

Remember thatσdenotes the sign ofB₀inQ(x₀). The change of variabley=^R(x−x₀)gives E0

u, σB_Q(x0)A0(x−x₀);Q(x₀)

=

Q_R

R

∇y−iσ

R A0(y)

v

²−κ²|v|²+κ² 2 |v|⁴

Rdy

=

QR

(∇y−iσA0)v²− κ

H B_Q(x0)|v|²+ κ 2H BQ(x0)

|v|⁴

dy

= κ

H BQ(x₀)

Q_R

b(∇y−iσA0)v²− |v|²+1 2|v|⁴

dy

=1 bG^σ_b,Q

R(v). (4.14)

We still need to estimate from below the reduced energyG^σ_b,Q

R(v). Sincevis not inH₀¹(Q_R), we introduce a cut-off functionχ_R∈C_c^∞(R²)such that

0χ_R1 inR², suppχ_R⊂Q_R, χ_R=1 inQ_R−1, and |∇χ_R|M inR². (4.15) The constantMis universal.

Let

u_R=χ_Rv. (4.16)

We have G^σ_b,Q

R(v)=

QR

b(∇ −iσA0)v²− |v|²+1 2|v|⁴

dx

QR

bχ_R(∇ −iσA0)v²− |χ_Rv|²+1 2|v|⁴+

χ_R²−1

|v|²

dx

G^σ_b,Q

R(χ_Rv)−

Q_R

1−χ_R²

|v|²dx−2

Q_R

(∇ −iσA0)χ_Rv,∇χ_Rvdy. (4.17) Having in mind(4.13)and(4.8), we get

∇y−iσA0(y)

v(y)=

R∇x−iκH σB_Q(x0)A0(x−x0) u(x). Using the estimate in(3.11),(4.7)and(A.3)we get

∇y−iσA0(y)

v(y)

R∇x−iκH σB_Q(x0)(A+ ∇ϕ) u(x) +κH

R A−σB_Q(x0)A0(x−x₀)− ∇ϕu(x)

C1

R

κ+κ^α+κ²²

. (4.18)

From the definition ofu_Rin(4.16)andχ_Rin(4.15)we get

|v|1. (4.19)

Using(4.19),(4.18)and the definition ofχRin(4.15), we get

QR

(∇ −iσA0)χ_Rv,∇χ_RvdyC1 R

κ+κ^α+κ²²

QR\QR−1

|∇χ_R|dx

C1

κ+κ^α+1+κ²³

, (4.20)

and

Q_R

1−χ_R²

|v|²dx|QR\QR−1|

R. (4.21)

Inserting(4.20)and(4.21)into(4.17), we get G^σ_b,Q

R(v)G^σ_b,Q

R(u_R)−C2

κ+κ^α+1+κ²³+κ√ G^σ_b,Q

R(u_R)−C₂ κ(√

+1)+κ²³ . There are two cases:

Case1:σ= +1, whenB₀>0, inQ(x₀).

Case2:σ= −1, whenB₀<0, inQ(x₀).

In Case 1, after recalling the definition ofm0(b, R)introduced in(2.3), wherebis introduced in(4.12)we get G⁺_b,Q¹

R(v)m0(b, R)−C2

κ(√

+1)+κ²³

. (4.22)

We get by collecting the estimates in(4.11)–(4.22):

1

|Q(x0)|E0

ψ,A;Q(x₀)

(1−δ) b²

m₀(b, R)−C₂

κ+κ²³(+1)

−C

δ⁻¹⁴κ²+δ^−12α+δ κ² (1−δ)

b² m₀(b, R)−r(κ), (4.23)

where

r(κ)=C₃

δ⁻¹⁴κ⁴+δ^−12ακ²+δκ²+ 1 b²

κ(√

+1)+κ²³

. (4.24)

Theorem 2.2tells us that m0(b, R)R²g(b) for allb∈ [0,1]andR sufficiently large. Hereg(b)is introduced in (2.5). Therefore, we get from(4.23)the estimate

1

|Q(x₀)|E0

ψ,A;Q(x₀)

(1−δ)R² b²

g(b)−r(κ), (4.25)

withbdefined in(4.12). By choosingδ=and using thatQ(x0)⊂ {|B0|> }, we get r(κ)=O

³κ²+^2α−1+1

(κ)⁻¹+

κ². (4.26)

This implies that 1

|Q(x₀)|E0

ψ,A;Q(x0)

g

H κ BQ(x₀)

κ²−C

³κ²+^2α−1+(κ)⁻¹+⁻¹ κ². Similarly, in Case 2, according toRemark 2.1, we get that

G⁻_b,Q¹

R(v)m₀(b, R)−C₂

κ+κ²³(+1) , and the rest of the proof is as for Case 1. 2

5. Proof ofTheorem 1.1 5.1. Upper bound

Proposition 5.1. There exist positive constants C and κ0 such that, if(3.10) holds, then the ground state energy Eg(κ, H )in(1.6)satisfies

Eg(κ, H )κ²

Ω

g H

κ B0(x)

dx+Cκ¹⁵⁸.

Proof. Let=(κ)and=(κ)be positive parameters such thatκ⁻¹ 1 andκ⁻¹ 1 asκ→ ∞. For someβ∈(0,1),μ∈(0,1)to be determined later, we will choose

=κ^−β, =κ^−μ. (5.1)

Consider the lattice Γ:=Z×Z and write forγ ∈Γ, Q_{γ ,}=Q(γ ). For anyγ ∈Γ such that Q_{γ ,}⊂Ω ∩ {|B₀|> }let

B_{γ ,}= inf

x∈Q_{γ ,}

B0(x). (5.2)

Let

I,=

γ: Q_{γ ,}⊂Ω∩

|B₀|>

, N=CardI,,

and

Ω,=int

γ∈I,

Qγ ,

.

It follows from(1.2)that N= |Ω|⁻²+O

⁻² +O

⁻¹

as→0 and→0.

Let

b=H

κ B_{γ ,}, R=

κH B_{γ ,}, (5.3)

andu_Rbe a minimizer of the functional in(2.1), i.e.

m₀(b, R)=

QR

b(∇ −iA0)u_R²− |u_R|²+1 2|u_R|⁴

dx.

We will need the functionϕ_γ introduced inLemma A.3which satisfies F(x)−σγ ,B_{γ ,}A0(x−γ )− ∇ϕγ(x)C² inQγ ,, whereσ_{γ ,}is the sign ofB0inQ_{γ ,}.

We define the function

v(x)=

⎧⎪

⎨

⎪⎩

e^{−iκH ϕγ}u_R(^R(x−γ )) ifx∈Q_{γ ,}⊂ {B₀> }, e^{−iκH ϕγ}uR(^R(x−γ )) ifx∈Qγ ,⊂ {B0<−},

0 ifx∈Ω\Ω_,.

Sinceu_R∈H₀¹(Q_R), thenv∈H¹(Ω). We compute the energy of the configuration(v,F). We get E(v,F)=

Ω

(∇ −iκHF)v²−κ²|v|²+κ² 2 |v|⁴

dx

=

γ∈I,

E0(v,F;Qγ ,). (5.4)

We estimate the termE0(v,F;Q_{γ ,})from above and we write E0(v,F;Q_{γ ,})=

Qγ ,

(∇ −iκHF)v²−κ²|v|²+κ² 2 |v|⁴dx

=

Q_{γ ,}

∇ −iκH

σγ ,B_{γ ,}A0(x−γ )+ ∇ϕγ(x) v

−iκH

F−σ_{γ ,}B_{γ ,}A0(x−γ )− ∇ϕ_γ(x)v²−κ²|v|²+κ² 2 |v|⁴dx

Qγ ,

(1+δ)∇ −iκH

σ_{γ ,}B_{γ ,}A0(x−γ )+ ∇ϕ_γ(x)v²−κ²|v|²+κ² 2 |v|⁴dx +C

1+δ⁻¹ (κH )²

Q_{γ ,}

F−σ_{γ ,}B_{γ ,}A0(x−γ )− ∇ϕ_γ(x) v²dx (1+δ)E0

e^{−iκH ϕγ}v, σ_{γ ,}B_{γ ,}A0(x−γ );Q_{γ ,} +C

δκ²+δ⁻¹κ⁴⁴

Qγ ,

|v|²dx. (5.5)

Having in mind thatu_Ris a minimizer of the functional in(2.1), and using the estimate in(3.1)we get

Q_{γ ,}

|v|²dx|Qγ ,|.

Remark 2.1and a change of variables give us

Qγ ,

∇ −iκH σ_{γ ,}

B_{γ ,}A0(x−γ )

e^{−iκH ϕγ}v²−κ²|v|²+κ² 2 |v|⁴

dx=m0(b, R) b . We insert this into(5.5)to obtain

E0(v,F;Q_{γ ,})(1+δ)m0(b, R) b +C

δκ²+δ⁻¹κ⁴⁴

². (5.6)

We know fromTheorem 2.2thatm0(b, R)g(b)R²+CRfor allb∈ [0,1]andRsufficiently large, wherebintro- duced in(5.3). We chooseδ=in(5.6). That way we get

E0(v,F;Q_{γ ,})g H

κ B_{γ ,}

²κ²+C 1

κ√

++κ²³

²κ². (5.7)

Summing(5.7)overγ inI_,, we recognize the lower Riemann sum ofx→g(^H_κ|B₀(x)|). By monotonicity ofg,g is Riemann-integrable and its integral is larger than any lower Riemann sum. Thus

E(v,F)

Ω,

g H

κ B₀(x) dx

κ²+C

1 κ√

++κ²³

κ². (5.8)

Notice that using the regularity of∂Ω and(1.2), there existsC >0 such that

|Ω\Ω_,| =O

|∂Ω| +C

, (5.9)

asandtend to 0.

Thus, we get by using the properties ofginTheorem 2.2,

Ω,

g H

κ B₀(x) dx

Ω

g H

κ B₀(x) dx+1

2|Ω\Ω_,|. This implies that

E(v,F)

Ω

g H

κ B₀(x) dx+C

1 κ√

+++κ²³

κ². (5.10)

We choose in(5.1) β=3

4 and μ=1

8. (5.11)

With this choice, we infer from(5.10), E(v,F)

Ω

g H

κ B₀(x)

dx+C₁κ¹⁵⁸, (5.12)

and

²κ²=κ³⁸ >1. (5.13)

This finishes the proof ofProposition 5.1. 2

Remark 5.2.In the case whenB₀does not vanish inΩ,disappears and{x∈Ω; |B₀(x)|>0} =Ω. Consequently, the Ginzburg–Landau energy of(v,F)in(4.2)satisfies

E(v,F)

Ω

g H

κ B₀(x) dx+C

1

κ++κ²³

κ².

We take the same choice ofβas in(5.11), then the ground state energy Eg(κ, H )in(1.6)satisfies Eg(κ, H )κ²

Ω

g H

κ B₀(x)

dx+Cκ⁷⁴.

5.2. Lower bound

We now establish a lower bound for the ground state energy Eg(κ, H ) in(1.6). The parametersandhave the same form as in(5.1).

Let

B_{γ ,}= sup

x∈Q_{γ ,}

B₀(x), (5.14)

and

b_{γ ,}=H

κ B_{γ ,}, R=

κH B_{γ ,}. (5.15)

If(ψ,A)is a minimizer of(1.1), we have

Eg(κ, H )=E0(ψ,A;Ω_,)+E0(ψ,A;Ω\Ω_,)+(κH )²

Ω

curl(A−F)²dx, (5.16)

where, for anyD⊂Ω, the energyE0(ψ,A;D)is introduced in(4.1). Since the magnetic energy term is positive, we may write

Eg(κ, H )E0(ψ,A;Ω_,)+E0(ψ,A;Ω\Ω_,). (5.17) Thus, we get by using(3.1),(3.11), and(5.9):

E0(ψ,A;Ω\Ω_,)

Ω\Ω,

(∇ −iκHA)ψ²+κ²|ψ|²+κ² 2 |ψ|⁴dx |Ω\Ω_,|

C₁κ²ψ²L^∞(Ω)+κ²ψ²L^∞(Ω)+κ²

2 ψ⁴L^∞(Ω)

C₂(+)κ². (5.18)

To estimateE0(ψ,A;Ω,), we notice that E0(ψ,A;Ω_,)=

γ∈I,

E0(ψ,A;Q_{γ ,}).

UsingProposition 4.2withα=²₃and(5.18)withβ=³₄andμ=¹₈in(5.1), we get E0(ψ,A;Ω,)

γ∈I,

g H

κ BQ(x₀)

²κ²−C

³κ²+^2α−1+(κ)⁻¹+⁻¹ κ²

κ²

γ∈I,

g H

κ B_Q(x0)

²−C1κ¹⁵⁸,

and

E0(ψ,A;Ω\Ω_,)−C₂κ¹⁵⁸. (5.19)

As for the upper bound, we can use the monotonicity ofgand recognize that the sum above is an upper Riemann sum ofg. In that way, we get

E0(ψ,A;Ω_,)κ²

Ω,

g H

κ B₀(x)

dx−C₁κ¹⁵⁸ .

Notice thatΩ_,⊂Ωand thatg0, we deduce that E0(ψ,A;Ω_,)κ²

Ω

g H

κ B₀(x)

dx−C₁κ¹⁵⁸ . (5.20)

Finally, putting(5.19)and(5.20)into(5.17), we obtain Eg(κ, H )κ²

Ω

g H

κ B₀(x)

dx−Cκ¹⁵⁸ . (5.21)

Remark 5.3.WhenB₀does not vanish, the local energy inQ(x₀)inProposition 4.2becomes 1

|Q(x₀)|E0

ψ,A;Q(x0)

g

H κ BQ(x₀)

κ²−C

³κ²+^2α−1+(κ)⁻¹ κ². Similarly, we chooseα=²₃and=κ⁻³⁴, we get

E0(ψ,A;Ω\Ω_,)−C₂κ⁷⁴, (5.22)