analysis of minimizers of the lawrence-doniach energy for ...baumanp/paper-bau... ·...

Analysis of Minimizers of the Lawrence-Doniach Energy for

Superconductors in Applied Fields

Patricia Bauman∗ and Guanying Peng∗

Department of Mathematics, Purdue UniversityWest Lafayette, IN 47907, USA

E-mail address: [email protected] address: [email protected]

Abstract

We analyze minimizers of the Lawrence-Doniach energy for layered superconductors occupyinga bounded generalized cylinder, Ω×[0, L], in R3, where Ω is a bounded simply connected Lipschitzdomain in R2. For an applied magnetic field ~Hex = hex~e3 that is perpendicular to the layers with|ln ε| hex ε−2 as ε→ 0, where ε is the reciprocal of the Ginzburg-Landau parameter, we provean asymptotic formula for the minimum Lawrence-Doniach energy as ε and the interlayer distances tend to zero. Under appropriate assumptions on s versus ε, we establish comparison resultsbetween the minimum Lawrence-Doniach energy and the minimum three-dimensional anisotropicGinzburg-Landau energy. As a consequence, our asymptotic formula also describes the minimumthree-dimensional anisotropic energy as ε tends to zero.

1 Introduction

The Lawrence-Doniach model was formulated by Lawrence and Doniach in 1971 as a macroscopicmodel for layered superconductors. While the standard Ginzburg-Landau model has been well ac-cepted as a macroscopic model for isotropic superconductors, it does not account for the anisotropyin three-dimensional high temperature superconducting materials. For these materials, dependingon the nature of the anisotropy in the material, physicists have used the Lawrence-Doniach model(which treats the superconducting material as a stack of parallel superconducting layers with nonlin-ear Josephson coupling between them) or the three-dimensional anisotropic Ginzburg-Landau model(which is a slight modification of the standard three-dimensional Ginzburg-Landau model).

The standard two-dimensional Ginzburg-Landau energy model (with energy given by (1.6)) hasbeen intensively investigated. In this case, an analysis of the behavior of energy minimizers and theirvortex structure in a perpendicular applied magnetic field with modulus hex in different regimes (e.g.,hex ∼ | ln ε|, | ln ε| hex ε−2, or hex ≥ C

ε2 ) is now well understood. (See [9].) However, for theLawrence-Doniach energy (see (1.1)), an analysis for hex in the first two regimes has been done onlyfor the gauge-periodic problem, in which the superconductor is assumed to occupy all of R3 and thegauge invariant quantities are assumed to be periodic with respect to a given parallelepiped. (See [1].)In the last regime, hex ≥ C

ε2 , it was shown in [2] that if C is sufficiently large, all minimizers of theLawrence-Doniach energy are in the normal (nonsuperconducting) phase, that is, the order parameters

on the layers, unNn=0, are all identically equal to zero, and the induced magnetic field, ∇ × ~A, is

∗The authors were supported in part by NSF Grant DMS-1109459.

1

identically equal to the applied magnetic field. A similar result is known for the two-dimensionalGinzburg-Landau energy. (See [6].)

In this paper, we analyze the Lawrence-Doniach model in the second regime without imposinggauge periodicity assumptions. In two-dimensional superconductors, this regime for hex correspondsto a mixed state in which superconducting states and normal states (in the form of isolated vortices,i.e., zeros of un) coexist.

The Lawrence-Doniach model describes a layered superconductor occupying a cylinder D = Ω ×[0, L] with cross-section Ω and N + 1 equally spaced layers of material occupying Ωn ≡ Ω × ns,where Ω is a bounded simply connected Lipschitz domain in R2 and s = L

N is the interlayer distance.Assuming an applied magnetic field hex ~e3, the Lawrence-Doniach energy is given by

Gε,sLD(unNn=0,~A) = s

N∑n=0

∫Ω

[1

2|∇Anun|

2 +1

4ε2(1− |un|2)2

]dx

+ s

N−1∑n=0

∫Ω

1

2λ2s2|un+1 − uneı

∫ (n+1)sns

A3dx3 |2dx

+1

2

∫R3

|∇ × ~A− hex~e3|2dx

(1.1)

for (unNn=0,~A) such that

unNn=0 ∈ [H1(Ω;C)]N+1 and~A ∈ E ≡ ~C ∈ H1

loc(R3;R3) : (∇× ~C)− hex~e3 ∈ L2(R3;R3).(1.2)

Here ε > 0 is the reciprocal of the Ginzburg-Landau parameter and λ > 0 represents the Josephsonpenetration depth, which is assumed to be fixed throughout this study. The applied magnetic fieldhex~e3 is assumed to satisfy | ln ε| hex ε−2 as ε→ 0. The complex valued function un defined inΩ is the order parameter for the nth layer and |un(x1, x2)|2 is the density of superconducting electronpairs at each point (x1, x2, ns) on the nth layer. For a minimizer of the Lawrence-Doniach energy(1.1), |un(x1, x2)| ∼ 1 corresponds to a superconducting state at (x1, x2, ns), whereas |un(x1, x2)| = 0corresponds to a normal (nonsuperconducting) state at (x1, x2, ns), in which the density of super-

conducting electrons is zero. The vector field ~A = (A1, A2, A3) defined on R3 is called the magnetic

potential; its curl, ∇ × ~A, is the induced magnetic field. We let x = (x1, x2, x3), ∇ = (∂1, ∂2),x = (x1, x2), A = (A1, A2) and An(x) = (A1(x, ns), A2(x, ns)), the trace of A on the nth layer. Weset ∇Anun = ∇un − ıAnun on Ω. In the following, given two complex numbers u and v, we let

(u, v) = 12 (uv + uv) = <(uv), which is an inner product of u = u1 + ıu2 and v = v1 + ıv2 in C that

agrees with the inner product of (u1, u2) and (v1, v2) in R2.

Since ~A ∈ H1loc(R3;R3), it follows from the trace theorem and the Sobolev imbedding theo-

rem that its trace An ∈ H12

loc(R2;R2) ⊂ L4loc(R2;R2) and therefore the Lawrence-Doniach energy

Gε,sLD(unNn=0,~A) is well-defined and finite. The existence of minimizers in [H1(Ω;C)]N+1 × E was

shown by Chapman, Du and Gunzburger in [3]. Each minimizer of Gε,sLD corresponds to a physicallyrealistic state for the layered superconductor. The minimizer satisfies the Euler-Lagrange equationsassociated to the Lawrence-Doniach energy. This system of equations is called the Lawrence-Doniachsystem and it is given by

(∇ − ıAn)2un + 1ε2 (1− |un|2)un + Pn = 0 on Ω,

∇× (∇× ~A) = (j1, j2, j3) in R3,

(∇ − ıAn)un · ~n = 0 on ∂Ω,

∇× ~A− hex~e3 ∈ L2(R3;R3)

2

for all n = 0, 1, ..., N , where

Pn =

1

λ2s2 (u1Υ10 − u0) if n = 0,

1λ2s2 (un+1Υn+1

n + un−1Υnn−1 − 2un) if 0 < n < N ,

1λ2s2 (uN−1ΥN

N−1 − uN ) if n = N ,

Υn+1n = eı

∫ (n+1)sns

A3dx3 for n = 0, 1, ..., N − 1,

ji = −sN∑n=0

(∂iun − ıAinun,−ıun)χΩ(x1, x2)dx1dx2δns(x3) for i = 1, 2,

j3 = s

N−1∑n=0

1

λ2s2(un+1 − unΥn+1

n , ıunΥn+1n )χΩ(x1, x2)χ[ns,(n+1)s](x3).

It was proved in [2] that a minimizer (unNn=0,~A) of (1.1) satisfies |un| ≤ 1 a.e. in Ω for all n =

0, 1, ..., N .Two configurations (unNn=0,

~A) and (vnNn=0,~B) in [H1(Ω;C)]N+1 × E are called gauge equiv-

alent if there exists a function g ∈ H2loc(R3) such thatun(x) = vn(x)eıg(x,ns) in Ω,~A = ~B +∇g in R3.

(1.3)

Simple calculations show that Gε,sLD (and each term in Gε,sLD) is invariant under the above gauge trans-

formation, i.e., for two configurations (unNn=0,~A) and (vnNn=0,

~B) that are related by (1.3), we

have Gε,sLD(unNn=0,~A) = Gε,sLD(vnNn=0,

~B). Let ~a = ~a(x) be any fixed smooth vector field onR3 such that ∇ × ~a = ~e3 in R3. For example, we may choose ~a(x) = (0, x1, 0). It was also

proved in [2] that every pair (unNn=0,~A) ∈ [H1(Ω;C)]N+1 × E is gauge equivalent to another pair

(vnNn=0,~B) ∈ [H1(Ω;C)]N+1 ×K where

K = ~C ∈ E : ∇ · ~C = 0 and ~C − hex~a ∈ H1(R3) ∩ L6(R3;R3). (1.4)

Here the space H1(R3) represents the completion of C∞0 (R3;R3) with respect to the seminorm

‖~C‖H1(R3) = (

∫R3

|∇~C|2dx)12 .

In particular, any minimizer of Gε,sLD in the admissible space [H1(Ω;C)]N+1×E is gauge-equivalent toa minimizer in the space [H1(Ω;C)]N+1×K, called the “Coulomb gauge” for Gε,sLD. It was shown in [2]

that minimizers in the Coulomb gauge satisfy un ∈ C∞(Ω) and An ∈ H1loc(R2) for all n = 0, 1, ..., N .

Throughout this paper, we take ~a(x) = (0, x1, 0).Given the above definitions, our main results are the following:

Theorem 1. Assume | ln ε| hex ε−2 as ε → 0. Let (unNn=0,~A) ∈ [H1(Ω;C)]N+1 × K be a

minimizer of Gε,sLD. Then denoting the volume of D by |D|, we have∣∣∣∣Gε,sLD(unNn=0, ~A)− |D|2hex ln

1

ε√hex

∣∣∣∣ ≤ (Cs17 + oε(1))

|D|2hex ln

1

ε√hex

= oε,s(1)|D|2hex ln

1

ε√hex

3

as (ε, s)→ (0, 0) for some constant C independent of ε and s. In particular,

lim(ε,s)→(0,0)

Gε,sLD(unNn=0,~A)

hex ln 1ε√hex

=|D|2.

(See Theorem 3.1 and Theorem 4.2.)Here oε(1) denotes a quantity that converges to 0 as ε → 0 and oε,s(1) denotes a quantity that

converges to 0 as (ε, s)→ (0, 0). This theorem generalizes a result in the gauge periodic case studied byAlama, Bronsard and Sandier for the energy (1.1) in which the domain Ω is replaced by a parallelogram

P in R2, the integral of |∇× ~A−hex~e3|2 is taken over P×[0, L] instead of over R3, and the minimization

takes place among gauge periodic configurations (unNn=0,~A) in R3 with period P × [0, L]. (See [1].)

In that case, they further showed that for a minimizer of the gauge periodic problem, the orderparameters un are all equal and A3 is identically zero. In particular, the Josephson coupling term

s

N−1∑n=0

∫Ω

1


∫ (n+1)sns

A3dx3 |2dx (1.5)

vanishes. They also proved that A(x, ·) is periodic in x3 with period s and established certain sym-metries between the layers in A.

In the gauge periodic case, the results of Alama, Bronsard and Sandier indicate a close connectionbetween the Lawrence-Doniach energy and the two-dimensional Ginzburg-Landau energy GLε givenby

GLε(u, A) =1

2

∫Ω

[|∇Au|

2 +1

2ε2(1− |u|2)2

]dx+

1

2

∫R2

( ˆcurlA− hex)2dx (1.6)

for hex as assumed above where ˆcurl denotes the two-dimensional curl defined by ˆcurl(B1, B2) =∂1B

2 − ∂2B1. We remark that for a minimizer of the two-dimensional energy GLε, the magnetic

potential A satisfies ˆcurlA = hex in R2 \ Ω. (See Lemma 2.1 in [6].) Therefore the minimum of GLεis equal to the minimum of Fε given by

Fε(u, A) =1

2

∫Ω

[|∇Au|

2 +1

2ε2(1− |u|2)2

]dx+

1

2

∫Ω

( ˆcurlA− hex)2dx.

(See Prop. 3.4 in [9] for bounded simply connected smooth domains and Prop. 2.1 in this paper forbounded simply connected Lipschitz domains.) The main idea in our proof of the upper bound inTheorem 1 is to construct a test function that is an extension of N + 1 copies in each layer, Ω×ns,of a two-dimensional configuration (u, (A1, A2)(x, ns)) used by Sandier and Serfaty in Chapter 8 of[9] to provide an upper bound on the minimal two-dimensional energy Fε.

A matching lower bound is much more difficult to establish. To obtain it, we prove in Lemma 4.1that, for a minimizer (unNn=0,

~A) ∈ [H1(Ω;C)]N+1 ×K of Gε,sLD, we have

1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

| ˆcurlA(x, x3)− ˆcurlAn(x)|2dxdx3 ≤ ε,s(1)Mε, (1.7)

where Mε = |D|2 hex ln 1

ε√hex

. The proof of (1.7) uses single layer potential representation formulas for

~A proved by Bauman and Ko in [2] as well as a priori estimates for single layer potentials (see [4] and[10]) and harmonic functions. This estimate plays a crucial role in the proof of the lower bound, as it

implies that the three-dimensional integral 12

∫D| ˆcurlA(x, x3) − hex|2dx can be approximated within

oε,s(1)Mε by the sum of two-dimensional integrals, ΣN−1n=0

s2

∫Ω| ˆcurlAn(x)−hex|2dx. As a result of (1.7),

4

the energy for a minimizer from the first term in (1.1) plus the magnetic term 12

∫D| ˆcurlA− hex|2dx

can be approximated to leading order by the sum of sFε(un, An). Combining this observation withthe lower bound estimate for Fε proved in [9], we obtain the lower bound estimate in Theorem 1.

As a consequence of our proof of Theorem 1, we conclude that the Josephson coupling term (1.5)contributes a lower order energy to the total Lawrence-Doniach energy. In fact, we obtain

Theorem 2. Assume | ln ε| hex 1ε2 as ε → 0. Let (unNn=0,

~A) ∈ [H1(Ω;C)]N+1 × K be aminimizer of Gε,sLD. Then

s

N−1∑n=0

∫Ω

1


∫ (n+1)sns

A3dx3 |2dx+1

2

∫R3\D

|∇ × ~A− hex~e3|2dx

+1

2

∫D

[(∂A3

∂x2− ∂A2

∂x3)2 + (

∂A1

∂x3− ∂A3

∂x1)2

]dx ≤ oε,s(1)Mε

as (ε, s)→ (0, 0).

(See Section 4 for the proof.)Theorems 1 and 2 and the estimate (1.7) imply a strong influence up to leading order of the two-

dimensional energy Fε on the minimal Lawrence-Doniach energy. In fact, we prove in Corollary 4.3that for a minimizer (unNn=0,

~A) ∈ [H1(Ω;C)]N+1 ×K of the Lawrence-Doniach energy, we have

Gε,sLD(unNn=0, ~A) =

[N−1∑n=0

sFε(un, An)

][1 + oε,s(1)].

Another consequence of Theorems 1 and 2 is the following:

Corollary. Under the assumptions of Theorem 1, we have

‖∇ ×~A

hex− ~e3‖L2(R3;R3) → 0,

and

1

N + 1

N∑n=0

µnhex→ dx in H−1(Ω)

as (ε, s)→ (0, 0), where µn is the vorticity on the nth layer defined as

µn = ˆcurl(ıun, ∇Anun) + ˆcurlAn.

(See Corollary 4.4.)The convergence of the average scaled vorticity in the layers to the Lebesgue measure generalizes

a result for minimizers of the two-dimensional Ginzburg-Landau energy Fε studied by Sandier andSerfaty (see Cor. 8.1 in [9]). They showed that for minimizers of Fε, the scaled vorticity measureµhex

converges to dx in H−1(Ω) as ε→ 0. The vorticity measure µn in each layer is a gauge-invariantversion of the Jacobian determinant of un, and is analagous to the vorticity in fluids. If un is given inpolar coordinates by ρne

iθn , then µn = ˆcurlρ2n(∇θn − An)+ ˆcurlAn. The above corollary indicates

that on average there are numerous vortices and they have an approximately uniform distribution.More detailed results on the nature and number of vortices for minimizers of the Lawrence-Doniachenergy is an interesting open problem to which the results of this paper should be relevant.

Recall that another model for certain high-temperature anisotropic superconductors is the three-dimensional anisotropic Ginzburg-Landau model. In this model, a mass tensor with unequal principal

5

values is introduced to account for the anisotropic structure in the superconductor. (See [3] and

[7] for more background information.) For a given admissible function (ψ, ~A) in H1(D;C) × E, theanisotropic Ginzburg-Landau energy GεAGL is given by

GεAGL(ψ, ~A) =1

2

∫D

[|∇Aψ|

2 +1

λ2|( ∂

∂x3− ıA3)ψ|2

]dx

+

∫D

(1− |ψ|2)2

4ε2dx+

1

2

∫R3

|∇ × ~A− hex~e3|2dx.(1.8)

Here λ is the same constant as in the Lawrence-Doniach energy. The connection between the Lawrence-Doniach energy Gε,sLD and the anisotropic Ginzburg-Landau energy GεAGL when ε is fixed and s tendsto zero was studied in [2], [3], and [11]. In particular, it was shown in [3] that under this assumption,a subsequence of minimizers of the Lawrence-Doniach energy form a minimizing sequence of theanisotropic Ginzburg-Landau energy. Gamma convergence of the Lawrence-Doniach energy in thiscase to the anisotropic Ginzburg-Landau energy was proved in [11].

Our last result concerns the asymptotic behavior of the two energies as both ε and s tend to zero.We prove that, under an additional assumption on s versus ε, the difference between the two minimumenergies is negligible compared to the leading term in the minimal Lawrence-Doniach energy. Moreprecisely, we have the following theorem:

Theorem 3. Assume | ln ε| hex ε−2 as ε → 0. Let (unNn=0,~A) ∈ [H1(Ω;C)]N+1 × K be a

minimizer of Gε,sLD and let (ζ, ~B) ∈ H1(D;C)×K be a minimizer of GεAGL. If in addition we assumethat s ≤ Cε for all ε sufficiently small where C is a constant independent of ε, then

|GεAGL(ζ, ~B)− Gε,sLD(unNn=0,~A)| ≤ ε(1)

|D|2hex ln

1

ε√hex

as ε→ 0. Hence

|GεAGL(ζ, ~B)− |D|2hex ln

1

ε√hex| ≤ ε(1)

|D|2hex ln

1

ε√hex

as ε→ 0.

(See Theorem 5.6 and Theorem 5.7.)In the case when ε is fixed, the discrete nature of the layering in the Lawrence-Doniach model is

eliminated as s→ 0 and therefore it is very natural that the discrete model reduces to the continuousone. The situation considered here is more delicate, since the interlayer distance s is allowed to be atthe same order as the characteristic vortex size ε, in which case the discrete nature of the Lawrence-Doniach model plays a more important role. Note that the assumption s ≤ Cε is only needed inTheorem 3. The previous two theorems concerning the minimum Lawrence-Doniach energy hold evenfor the very discrete case (e.g., 1 s ε).

Our paper is organized as follows: In Section 2 we state some preliminary results concerning thesingle layer potential representation formulas for A and An. In Section 3 we prove the upper boundon the minimal Lawrence-Doniach energy. In Section 4 we prove (1.7) and use it to prove the lowerbound on the minimal Lawrence-Doniach energy, as well as the corollaries stated above. Finally inSection 5 we prove the comparison result between the minimal Lawrence-Doniach energy and theminimal three-dimensional anisotropic Ginzburg-Landau energy and its consequence as summarizedin Theorem 3.

2 Preliminaries

As noted in the introduction, the Lawrence-Doniach energy is invariant under the gauge transformation(1.3) and minimizers of Gε,sLD are gauge-equivalent to a minimizer in the “Coulomb gauge”. It was

6

proved by Bauman and Ko in [2] that, for a minimizer (unNn=0,~A) of Gε,sLD in the “Coulomb gauge”,

the magnetic potential ~A has an explicit representation formula using single layer potentials. Recallthe definition of the space H1(R3) in the introduction. From [2], each ~C ∈ H1(R3) has a representativein L6(R3;R3) such that

‖~C‖L6(R3;R3) ≤ 2‖~C‖H1(R3) (2.1)

and

‖~C‖2H1(R3)

=

∫R3

(|∇ · ~C|2 + |∇ × ~C|2)dx. (2.2)

We remark that the Lawrence-Doniach energy Gε,sLD considered here is different from that studiedin [3], [2] and [11], via a simple rescaling in the energy and in the magnetic potentials. (The scalingused here is the same as that used by Sandier and Serfaty in [9] which has been very successful inanalyzing minimizers for the two-dimensional Ginzburg-Landau energy as ε tends to zero.)

More precisely, setting κ = 1ε and letting GκLD be the Lawrence-Doniach energy studied in [3] with

ψn as the order parameter for the nth layer and ~ALD as the magnetic potential for GκLD, respectively,we have

Gε,sLD(unNn=0,~A) = κ2

2 GκLD(ψnNn=0,

~ALD),

un = ψn and ~A = κ ~ALD.(2.3)

Similar rescaling holds for the anisotropic Ginzburg-Landau energy, i.e.,GεAGL(ψ, ~A) = κ2

2 GκEM (ψEM , ~AEM ),

ψ = ψEM and ~A = κ ~AEM ,(2.4)

where GκEM is the anisotropic Ginzburg-Landau (or effective mass) energy introduced in [3], and ψEMand ~AEM are the order parameter and the magnetic potential for GκEM , respectively. The aboveformulas will be used in Section 5.

The analysis in [2] (after appropriate rescaling) applies here without any difficulty. In particular,we have representation formulas for A1, A2, A1

n and A2n for a minimizer of the Lawrence-Doniach

energy in the Coulomb gauge as in Lemma 3.1, Theorem 3.2 and Corollary 3.3 in [2]. To state theseformulas, we first define the single layer potential for our setting. For each k ∈ 0, 1, ..., N, and fora given function g ∈ Lp(Ω× ks) with 1 < p <∞, we define the operator Sk by

[Sk(g)](x) =

∫Ω×ks

c

|x−Q|g(Q)dσ(Q)

for x in R3\[Ω× ks] where dσ denotes the surface measure on the plane and c = − 14π . (See [4] and

[10] for results on layer potentials in smooth and Lipschitz domains, respectively.) Let (unNn=0,~A) ∈

[H1(Ω;C)]N+1 ×K be a minimizer of Gε,sLD. Define hik in L2(Ω) by

hik(x) = s(∂iuk − ıAikuk,−ıuk)χΩ(x)

and define gik in L2(Ω× ks) bygik(x) = χΩ×ks(x)hik(x)

for i = 1, 2. Then the single layer potential of gik is

[Sk(gik)](x) =

∫Ω×ks

c

|x−Q|gik(Q)dσ(Q)

=

∫Ω

c

|x− (y, ks)|hik(y)dy.

(2.5)

7

With the above definitions, from the formulas in [2], we have

Ai(x)− hexai(x) =

N∑k=0

Sk(gik)(x) in L2loc(R3) (2.6)

and

Ain(x)− hexain(x) = tin(x, ns) +

N∑k=0k 6=n

Sk(gik)(x, ns) a.e. in R2 (2.7)

for i = 1, 2, where tin(x, ns) is the trace of [Sn(gin)](x) on R2 × ns, which is given by

tin(x, ns) =

∫Ω×ns

c

|(x, ns)−Q|gin(Q)dσ(Q)

=

∫Ω

c

|x− y|hin(y)dy,

and ain(x) = ai(x, ns) corresponds to the trace of ai in R2 × ns.In order to state further properties that will be used later, we need some definitions and results

from [2] (based on the theory of single layer potentials in [4] and [10]) concerning nontangential limitsand nontangential maximal functions. For fixed R > 0 and 0 < θ < π/2, let

Γ ≡ ΓR,θ = x ∈ R3 : |x| < R and |x · ~e3| > |x|cosθ

be a cone nontangential to the plane x3 = 0 with vertex at the origin. Denote by

Γ+ = x ∈ Γ : x3 > 0 and Γ− = x ∈ Γ : x3 < 0.

For (x, ns) ∈ R2 × ns, let

Γ(x, ns) = y ∈ R3 : y − (x, ns) ∈ Γ.

Similarly, denote byΓ+(x, ns) = y ∈ R3 : y − (x, ns) ∈ Γ+

andΓ−(x, ns) = y ∈ R3 : y − (x, ns) ∈ Γ−.

For a function u defined in Γ(x, ns), define the nontangential limit (n.t.limit) of u(y) as y → (x, ns)by

n.t.limity→(x,ns)

u(y) = limy→(x,ns)

u(y) : y ∈ Γ(x, ns),

provided the limit exists. Also we have the following definition of the nontangential maximal functionof u at (x, ns), denoted by u∗(x, ns) = u∗R,θ(x, ns) for each n in 0, 1, · · · , N.

u∗(x, ns) = sup|u(y)| : y ∈ ΓR,θ(x, ns).

By Theorem 3.2 in [2], Sn(gin) ∈W 1,2loc (R3) ∩ C∞(R3 \ Ωn) and tin(x, ns) ∈W 1,2

loc (R2 × ns).Throughout this paper, we let θ = π

4 and R = 1. Also, let R0 be a fixed constant satisfying

R0 ≥ 2(diam Ω),

8

where diam Ω is the diameter of Ω. It follows that the nontangential maximal functions of Sn(gin)and ∇Sn(gin) are in L2

loc(R2 × ns) and

‖(Sn(gin))∗‖L2(Ω×ns) + ‖(∇Sn(gin))∗‖L2(Ω×ns) ≤ C‖gin‖L2(Ω×ns) (2.8)

for some constant C depending only on R0. (This property of the constant C uses the fact that Ω isa subset of a disk of radius R0 in R2. See [2].) Also tin(x, ns) and ∇tin(x, ns) are the nontangentiallimits of Sn(gin) and ∇Sn(gin), respectively, pointwise a.e. in R2 × ns and in L2

loc(R2 × ns), andwe have

∇Sn(gin)(x) =

∫R2

−c(x− (y, ns))

|x− (y, ns)|3hin(y)dy a.e. in R3

and

(∇tin)(x, ns) = P.V.

∫R2

−c(x− y)

|x− y|3hin(y)dy a.e. in R2 × ns,

where P.V. denotes the principal-valued integral. In addition, we have

‖tin‖L2(Ω) + ‖∇tin‖L2(Ω) ≤ C‖hin‖L2(Ω) (2.9)

for some constant C depending only on R0.The above representation formulas and properties of the single layer potential will be used in

Section 4 in our proof of the lower bound for the minimum Lawrence-Diniach energy.We conclude this section with the following proposition concerning the minimum of the energies

GLε and Fε over bounded simply connected Lipschitz domains, which is a modification of Proposition3.4 in [9].

Proposition 2.1. Assume that Ω ⊂ R2 is a bounded simply connected Lipschitz domain. Let

X = (v, b) ∈ H1(Ω;C)×H1loc(R2;R2) : (curlb− hex) ∈ L2(R2)

andXΩ = (v, b) ∈ H1(Ω;C)×H1(Ω;R2).

Then we havemin

(v,b)∈XGLε(v, b) = min

(v,b)∈XΩ

Fε(v, b).

Proof. Note that for any function (v, b) in X, we have (v, b|Ω) ∈ XΩ. From this and the definitions ofGLε and Fε, we obtain

min(v,b)∈X

GLε(v, b) ≥ min(v,b)∈XΩ

Fε(v, b).

Given a minimizer (v, b) ∈ XΩ of Fε, let φ solve ∆φ = H, where H = (curlb− hex) ·χΩ ∈ L2(R2). Wemay take φ to be the Newtonian potential of H. By standard estimates for Newtonian potentials, wehave φ ∈ H2

loc(R2). Define b = (−∂2φ, ∂1φ) + (0, hexx1) ∈ H1loc(R2;R2). Direct calculations show that

curl(b|Ω) = curlb in Ω, where b|Ω is the restriction of b on Ω. Therefore there exists f ∈ H2(Ω) suchthat b|Ω = b +∇f in Ω. Define v = veif . Simple calculations using that H = curlb− hex and gaugeinvariance imply that

GLε(v, b) = Fε(v, b|Ω) = Fε(v, b) = min(v,b)∈XΩ

Fε(v, b).

Hence min(v,b)∈X GLε(v, b) ≤ min(v,b)∈XΩFε(v, b) and equality must hold.

9

3 Upper bound

From here on in the paper, we let C0 denote any constant that is independent of ε, s, Ω, L, D and R0

for all ε and s sufficiently small. Recall that in our notation, V denotes a two-dimensional vector V =(V1, V2) and ~W denotes a three-dimensional vector W = (W1,W2,W3). Also ˆcurl V = ∂1V2 − ∂2V1.We will denote by ∇ and ∆ the operators (∂1, ∂2) and (∂1)2 + (∂2)2, respectively.

In this section we prove the following upper bound on the minimum Lawrence-Doniach energy:

Theorem 3.1. Assume | ln ε| hex 1ε2 as ε→ 0. Let (unNn=0,

~A) be a minimizer of Gε,sLD. Thenwe have

Gε,sLD(unNn=0, ~A) ≤ |D|2hex(ln

1

ε√hex

)(1 + c(ε, s)

)for all ε and s sufficiently small, where

c(ε, s) =s

L+

C

ln 1ε√hex

= oε,s(1).

Here C is a constant depending only on R0 and L.

The main idea in the proof is to construct a test configuration with vanishing Josephson couplingterm, such that its Lawrence-Doniach energy Gε,sLD from inside the domain D is a sum of N+1 identicalcopies of the two-dimensional Ginzburg-Landau energy Fε in Ω. In this way we may apply the upperbound estimate for the minimal two-dimensional Ginzburg-Landau energy from Proposition 8.1 in[9] to obtain the leading energy of Gε,sLD. The technical difficulty comes from extending the magnetic

potential ~A outside the domain D appropriately so that the energy contribution from R3 \D is of alower order compared to that in D.

Proof of Theorem 3.1. We first construct a test configuration of the two-dimensional Ginzburg-Landau

energy Fε, as in the proof of Proposition 8.1 in [9]. Let θ =√

hex2π and Lε = 1

θZ×1θZ in R2. Let hε(x)

be a solution in R2 (periodic with respect to Lε) of

−∆hε + hε = 2π∑b∈Lε

δb.

Define ρε(x) by

ρε(x) =

0 if |x− b| ≤ ε for some b ∈ Lε,|x−b|ε − 1 if ε ≤ |x− b| ≤ 2ε for some b ∈ Lε,

1 otherwise.

Let Aε(x) = (A1ε(x), A2

ε(x)) be the solution in R2 of ˆcurlAε = hε, and ϕε(x) be the solution (well-defined modulo 2π) of −∇⊥hε = ∇ϕε − Aε in R2 \ Lε, where ∇⊥ = (−∂2, ∂1). (See the introductionof [8] for a construction of such a ϕε.) Let uε(x) = ρε(x)eıϕε(x) in R2. By the proof of Proposition 8.1in [9], there exists x0 ∈ Kε = (− 1

2θ ,12θ )× (− 1

2θ ,12θ ) such that

Fε(ux0ε , A

x0ε ) ≤ |Ω|

2hex(ln

1

ε√hex

+ C0) (3.1)

for all ε sufficiently small, where ux0ε (x) = uε(x− x0) and Ax0

ε (x) = Aε(x− x0).Using the above test configuration (ux0

ε , Ax0ε ) for the two-dimensional Ginzburg-Landau energy Fε,

we next construct a test configuration (vnNn=0,~B) in [H1(Ω;C)]N+1 × E for the Lawrence-Doniach

energy Gε,sLD. Let φ = φ(x) solve

∆φ(x) = H(x) in R2, (3.2)

10

where H(x) = (hx0ε (x) − hex) · χΩ(x) and hx0

ε (x) = hε(x − x0). Since hε ∈ Lploc(R2) for every p > 0,we have H ∈ Lp(R2) and is supported in Ω. By Theorem 9.9 in [5], we may choose φ to be theNewtonian potential of H, i.e., φ = Γ2 ∗H, where Γ2(x) = 1

2π ln |x| is the fundamental solution of ∆

in R2. By standard estimates on the Newtonian potential, we have φ ∈W 2,ploc (R2) for 1 < p <∞ and

∇φ = ∇Γ2 ∗H. In particular, φ ∈ H2loc(R2). Let R = max2(diam Ω), 1 and denote by BR

2a disk in

R2, with radius R2 such that Ω ⊂ BR

2. Let BR and BR+1 be disks concentric with BR

2with radii R and

R + 1 respectively. To construct ~B, we need two cut-off functions. Choose ξ = ξ(x) ∈ C∞0 (R2) suchthat ξ ≥ 0, ξ(x) ≡ 1 for x ∈ BR and ξ(x) ≡ 0 for x ∈ R2\BR+1 and |∇ξ| ≤ 2. Let η = η(x3) ∈ C∞0 (R)be such that η ≥ 0, η is symmetric with respect to x3 = L

2 , η(x3) ≡ 1 if − s2 ≤ x3 ≤ L+ s2 and η(x3) ≡ 0

if x3 ≤ − s2 − d or x3 ≥ L+ s2 + d satisfying |η′(x3)| ≤ 2

d for some fixed positive number d. Now define

~B(x) = hex~a(x) + η(x3)ξ(x)(−∂2φ(x), ∂1φ(x), 0) ∈ H1loc(R3;R3),

where recall that ~a(x) = (0, x1, 0) satisfies ∇× ~a = ~e3 in R3. By simple calculations we have

∇× ~B = hex~e3 + (−η′ξ∂1φ,−η′ξ∂2φ, η(ξ∆φ+ ∇ξ · ∇φ)). (3.3)

Note that for x = (x, x3) ∈ Ω× [0, L], ~B(x) = [hex~a+ (−∂2φ, ∂1φ, 0)](x) is independent of x3. Denoteby BΩ the restriction of B = (B1, B2) on Ω× x3 for any x3 ∈ [0, L], i.e.,

BΩ = (−∂2φ · χΩ(x), (hexx1 + ∂1φ) · χΩ(x)).

Clearly BΩ satisfiesˆcurlBΩ = hex + (hx0

ε − hex) = hx0ε = ˆcurlAx0

ε

on Ω. Since Ω is simply connected, there exists a function f ∈ H2(Ω) such that BΩ = Ax0ε + ∇f .

Define vn(x) = v(x) ≡ ux0ε (x)eıf for all n = 0, 1, ..., N . Since (v, BΩ) is gauge equivalent to (ux0

ε , Ax0ε )

for the two-dimensional Ginzburg-Landau energy Fε, by (3.1), we have

Fε(v, BΩ) = Fε(ux0ε , A

x0ε ) ≤ |Ω|

2hex(ln

1

ε√hex

+ C0). (3.4)

We show that the test configuration (vnNn=0,~B) defined above gives us the desired upper bound for

the Lawrence-Doniach energy Gε,sLD.It is easy to see that, since the vn’s are all equal and B3 ≡ 0, the Josephson coupling term (1.5)

vanishes. Therefore

Gε,sLD(vnNn=0,~B) =s

N∑n=0

∫Ω

[1

2|∇Bnvn|

2 +1

4ε2(1− |vn|2)2

]dx

+1

2

∫R3

∣∣∣∇× ~B − hex~e3

∣∣∣2 dx.It follows from (3.3) and the definitions of ξ and η that

1

2

∫R3

|∇ × ~B − hex~e3|2dx =1

2

∫R3

[|η′|2|ξ|2|∇φ|2 + η2(ξ∆φ+ ∇ξ · ∇φ)2

]dx

=1

2

∫ − s2− s2−d

∫R2

[|η′|2|ξ|2|∇φ|2 + η2(ξ∆φ+ ∇ξ · ∇φ)2

]dxdx3

+1

2

∫ L+ s2 +d

L+ s2

∫R2

[|η′|2|ξ|2|∇φ|2 + η2(ξ∆φ+ ∇ξ · ∇φ)2

]dxdx3

+1

2

∫ L+ s2

− s2

∫R2

(ξ∆φ+ ∇ξ · ∇φ)2dxdx3.

11

Since (ξ∆φ+ ∇ξ · ∇φ)2 does not depend on x3 and Ns = L, we have

1

2

∫ L+ s2

− s2

∫R2

(ξ∆φ+ ∇ξ · ∇φ)2dxdx3 = s(N + 1) · 1

2

∫R2

(ξ∆φ+ ∇ξ · ∇φ)2dx.

Also since Bn = BΩ and vn = v for all n = 0, 1, ..., N , we have

s

N∑n=0

∫Ω

[1

2|∇Bnvn|

2 +1

4ε2(1− |vn|2)2

]dx

= s(N + 1)

∫Ω

[1

2|∇BΩ

v|2 +1

4ε2(1− |v|2)2

]dx.

Therefore we may write Gε,sLD(vnNn=0,~B) as a sum:

Gε,sLD(vnNn=0, ~B) = I1 + I2 + I3,

where

I1 = s(N + 1)

∫Ω

[1

2|∇BΩ

v|2 +1

4ε2(1− |v|2)2

]dx+

1

2

∫R2

(ξ∆φ+ ∇ξ · ∇φ)2dx

,

I2 =1

2

∫ L+ s2 +d

L+ s2

∫R2

[|η′|2|ξ|2|∇φ|2 + η2(ξ∆φ+ ∇ξ · ∇φ)2

]dxdx3,

and

I3 =1

2

∫ − s2− s2−d

∫R2

[|η′|2|ξ|2|∇φ|2 + η2(ξ∆φ+ ∇ξ · ∇φ)2

]dxdx3.

By the symmetry property of η, it is obvious that I2 = I3. In the following lemmas, we prove severalestimates from which we obtain (in Lemma 3.5) that

I1 + 2I2 ≤|D|2hex(ln

1

ε√hex

)(1 +s

L+

C

ln 1ε√hex

)

for all ε and s sufficiently small where C is a constant depending only on R0 and L. This concludesthe proof.

Our first lemma concerns the L2 norm of the function H(x) = (hx0ε (x) − hex) · χΩ(x). Note that

H is independent of s.

Lemma 3.2. For the function H defined above, we have

‖H‖2L2(R2) = ‖hx0ε − hex‖2L2(Ω) ≤ C0|Ω|hex

for all ε sufficiently small and C0 as described above.

Proof. Let Kε = (− 12θ ,

12θ ) × (− 1

2θ ,12θ ) and K x0

ε be the translation of Kε by x0. Then we show thatfor all ε sufficiently small,

‖H‖2L2(R2) ≤ 2|Ω||K x0

ε |‖hx0

ε − hex‖2L2(Kx0ε )

= 2|Ω||Kε|‖hε − hex‖2L2(Kε)

. (3.5)

12

Indeed, let Ki be the collection of cubes formed by the lattice Lε in R2 such that Ki ∩ Ω 6= ∅ andΩ ⊂

⋃iKi. Since hx0

ε − hex is periodic with respect to Lε, we have

‖H‖2L2(R2) = ‖hx0ε − hex‖2L2(Ω) ≤

∑i

‖hx0ε − hex‖2L2(Ki)

=

∑i

|Ki|

|K x0ε |‖hx0

ε − hex‖2L2(Kx0ε ).

When ε is sufficiently small,∑i

|Ki| ≤ 2|Ω|, from which (3.5) follows immediately. By Proposition 3.2

in [8], we have‖hε − hex‖2L2(Kε)

≤ C0.

Combining the above inequality with claim (3.5) and using the fact that |Kε| = ( 1θ )2 = 2π

hex, we have

‖H‖2L2(R2) ≤ 2C0|Ω||Kε|

= 2C0hex2π|Ω| = C0|Ω|hex

for all ε sufficiently small.

Lemma 3.3. Let φ(x) be the Newtonian potential of H in R2 and ξ(x) be defined as above. Then wehave ∫

R2

(∇ξ · ∇φ)2dx ≤ C|Ω|hex

for all ε sufficiently small and some constant C depending only on R0.

Proof. Recall that ∇φ(x) =∫

Ω∇xΓ2(x− y)H(y)dy and ξ and φ are independent of s. Also recall that,

by the definition of ξ, ∇ξ is supported on BR+1 \BR and |∇ξ|2 ≤ 4. Therefore for all ε sufficientlysmall, ∫

R2

(∇ξ · ∇φ)2dx =

∫BR+1\BR

(∇ξ · ∇φ)2dx ≤ 4

∫BR+1\BR

∣∣∣∣∫Ω

∇xΓ2(x− y)H(y)dy

∣∣∣∣2 dx.It follows from the inequality |∇xΓ2(x− y)| ≤ C0

|x−y| that∫R2

(∇ξ · ∇φ)2dx ≤ C0

∫BR+1\BR

[∫Ω

1

|x− y|· |H(y)|dy

]2

dx.

Since Ω ⊂ BR2

, we have |x− y| ≥ R2 , which along with Holder’s inequality implies[∫

Ω

1

|x− y|· |H(y)|dy

]2

≤ 4

R2|Ω| · ‖H‖2L2(Ω)

and thus∫R2

(∇ξ · ∇φ)2dx ≤ C0 · |BR+1 \BR| ·4

R2|Ω| · ‖H‖2L2(Ω) = C0 ·

(2R+ 1)

R2· |Ω| · ‖H‖2L2(Ω).

Since R ≥ 1, it is clear that 2R+1R2 = 2

R + 1R2 ≤ 3. Hence, it follows from Lemma 3.2 that∫

R2

(∇ξ · ∇φ)2dx ≤ C0|Ω| · ‖H‖2L2(Ω) ≤ C|Ω|hex

for all ε sufficiently small and some constant C depending only on R0.

13

Lemma 3.4. For I1 as defined above, we have

I1 ≤|D|2hex ln

1

ε√hex

+s

L

|D|2hex ln

1

ε√hex

+ C|D|2hex (3.6)

for all ε and s sufficiently small and some constant C depending only on R0.

Proof. Using the equation (3.2) and the definition of ξ(x), we have

1

2

∫R2

(ξ∆φ+∇ξ · ∇φ)2dx ≤∫R2

(ξ∆φ)2dx+

∫R2

(∇ξ · ∇φ)2dx

=

∫Ω

(hx0ε − hex)2dx+

∫R2

(∇ξ · ∇φ)2dx.

From this and lemmas 3.2 and 3.3 we obtain

1

2

∫R2

(ξ∆φ+ ∇ξ · ∇φ)2dx ≤ C|Ω|hex. (3.7)

Thus from the definition of I1,

I1 ≤s(N + 1)

∫Ω

[1

2|∇BΩ

v|2 +1

4ε2(1− |v|2)2

]dx+ C|Ω|hex

≤s(N + 1)Fε(v, BΩ) + s(N + 1) · C|Ω|hex.

By (3.4) and the identity sN |Ω| = |D| we have

I1 ≤s(N + 1)|Ω|2hex(ln

1

ε√hex

+ C0) + s(N + 1) · C|Ω|hex

≤|D|2hex ln

1

ε√hex

+s

L

|D|2hex ln

1

ε√hex

+ C|D|2hex

for all ε and s sufficiently small and some constant C depending only on R0.

Lemma 3.5. I2 defined as above satisfies I2 ≤ C |D|2 hex and

I1 + 2I2 ≤|D|2hex(ln

1

ε√hex

)(1 +s

L+

C

ln 1ε√hex

)

for all ε and s sufficiently small and some constant C depending only on R0 and L.

Proof. We write I2 asI2 = I2,1 + I2,2,

where

I2,1 =1

2

∫ L+ s2 +d

L+ s2

∫R2

|η′|2|ξ|2|∇φ|2dxdx3

and

I2,2 =1

2

∫ L+ s2 +d

L+ s2

∫R2

η2(ξ∆φ+ ∇ξ · ∇φ)2dxdx3.

Recall that, by our choice of η and ξ, we have |η′| ≤ 2d and |ξ| ≤ 1 is supported on BR+1. Therefore

I2,1 ≤1

2

4

d2d

∫R2

|ξ|2|∇φ|2dx ≤ 2

d

∫BR+1

|∇φ|2dx.

14

Using ∇φ(x) =∫

Ω∇xΓ2(x− y)H(y)dy and Holder’s inequality, we obtain∫

BR+1

|∇φ|2dx ≤∫BR+1

(∫Ω

|∇xΓ2(x− y)|dy)(∫

Ω

|∇xΓ2(x− y)| · |H(y)|2dy)dx.

For any fixed x ∈ BR+1 and for every y ∈ Ω ⊂ BR2

, we have 0 ≤ |x− y| ≤ (R+ 1) + R2 = 3R

2 + 1 ≤ 5R2

since R ≥ 1. Letting z = x− y, we get∫Ω

|∇xΓ2(x− y)|dy ≤ C0

∫Ω

1

|x− y|dy ≤ C0

∫B(0, 5R2 )

1

|z|dz = C0 ·R, (3.8)

where B(0, 5R2 ) is the disk centered at the origin with radius 5R

2 in R2. Similarly, by the symmetry ofΓ2 in x and y and almost exactly the same arguments, we get∫

BR+1

|∇xΓ2(x− y)|dx ≤ C0 ·R (3.9)

for any y ∈ Ω. It follows from (3.8), Fubini’s theorem, (3.9) and Lemma 3.2 that

I2,1 ≤2

d(C0R)2 · ‖H‖2L2(Ω) ≤

C0R2

d· |Ω|

2hex.

For I2,2 we use (3.7) and the fact that |η| ≤ 1 to get

I2,2 ≤d

2

∫R2

(ξ∆φ+ ∇ξ · ∇φ)2dx ≤ d · C |Ω|2hex.

Therefore we obtain

I2 = I2,1 + I2,2 ≤ C(R2

d+ d)

|Ω|2hex ≤ C

|D|2hex (3.10)

for all ε and s sufficiently small and some constant C depending only on R0 and L.Now combining inequalities (3.6) and (3.10) we have

I1 + 2I2 ≤|D|2hex ln

1

ε√hex

+s

L

|D|2hex ln

1

ε√hex

+ C|D|2hex

=|D|2hex ln

1

ε√hex

(1 + c(ε, s)),

where

c(ε, s) =s

L+

C

ln 1ε√hex


4 Lower bound

In this section we prove the lower bound in Theorem 1. This relies on approximating the energy ofthe magnetic term | ˆcurlA− hex|2 by the sum of its traces | ˆcurlAn− hex|2 on the layers Ωn in the thindomains Ω× [ns, (n+ 1)s). We first show that the error from this approximation is indeed of a lowerorder compared to the leading order term of the total energy.

15

Lemma 4.1. Assume | ln ε| hex 1ε2 as ε → 0. Let (unNn=0,

~A) ∈ [H1(Ω;C)]N+1 × K be aminimizer of Gε,sLD. Then we have

1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

| ˆcurlA(x, x3)− ˆcurlAn(x)|2dxdx3 ≤ Cs27Mε

for all ε and s sufficiently small, where Mε = |D|2 hex ln 1

ε√hex

and C is a constant depending only on

R0 and L.

Proof. We shall use the single layer potential representation formulas for ~A(x)−hex~a(x) and An(x)−hexan(x) proved by Bauman and Ko in [2] for (unNn=0,

~A) as above. First note that since ~a(x) =(0, x1, 0) is independent of x3, we have

A(x, x3)− An(x) =(A(x, x3)− hexa(x, x3)

)−(An(x)− hexan(x)

)for (x, x3) in Ω× [ns, (n+ 1)s). Therefore

ˆcurlA(x, x3)− ˆcurlAn(x) = ˆcurl(A− hexa)(x, x3)− ˆcurl(An − hexan)(x)

=∂

∂x1

((A2 − hexa2)(x, x3)− (A2

n − hexa2n)(x)

)− ∂

∂x2

((A1 − hexa1)(x, x3)− (A1

n − hexa1n)(x)

)and it follows from this, (2.6) and (2.7) and the regularity results described in Section 2 that

ˆcurlA(x, x3)− ˆcurlAn(x)

=

N∑k=0k 6=n

∂

∂x1(Sk(g2

k)(x, x3)− Sk(g2

k)(x, ns))

+∂

∂x1

(Sn(g2

n)(x, x3)− t2n(x, ns))

−

N∑k=0k 6=n

∂

∂x2

(Sk(g1

k)(x, x3)− Sk(g1k)(x, ns)

)+

∂

∂x2

(Sn(g1

n)(x, x3)− t1n(x, ns))

in L2loc(R3), where hik(x) = s(∂iuk − ıAikuk,−ıuk)χΩ(x) and gik(x) = χΩ×ks(x)hik(x) for i = 1, 2.

Since |uk| ≤ 1 a.e. in Ω (see [2]), we have

|hik| ≤ s|∂iuk − ıAikuk|

and‖gik‖2L2(Ω×ks) = ‖hik‖2L2(Ω) ≤ s

2‖∂iuk − ıAikuk‖2L2(Ω).

Applying the elementary inequality (a−b)2 ≤ 2a2+2b2 to the representation formula for ˆcurlA(x, x3)−ˆcurlAn(x) and taking the sum of the integrals, we obtain

1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

| ˆcurlA(x, x3)− ˆcurlAn(x)|2dxdx3 ≤ E1 + E2,

16

where

E1 =

N−1∑n=0

∫ (n+1)s

ns

∫Ω

∣∣ N∑k=0k 6=n

∂

∂x1

(Sk(g2


)+

∂

∂x1

(Sn(g2

n)(x, x3)− t2n(x, ns))∣∣2dxdx3

and

E2 =

N−1∑n=0

∫ (n+1)s

ns

∫Ω

∣∣ N∑k=0k 6=n

∂

∂x2

(Sk(g1


)+

∂

∂x2

(Sn(g1

n)(x, x3)− t1n(x, ns))∣∣2dxdx3.

In the following we analyze E1 and the analysis for E2 will be similar. First define

∆n,k(x, x3) =∂

∂x1[Sk(g2

k)(x, x3)− Sk(g2k)(x, ns)]

for n 6= k and

∆n,n(x, x3) =∂

∂x1[Sn(g2

n)(x, x3)− t2n(x, ns)].

Note that for n 6= k, ∆n,k is C∞ in R3 \ Ωk since it is harmonic there. By the Cauchy-Schwartzinequality,

E1 ≤ (N + 1)

N−1∑n=0

∫ (n+1)s

ns

∫Ω

N∑k=0

|∆n,k(x, x3)|2dxdx3

= (N + 1)

N∑k=0

N−1∑n=0

∫ (n+1)s

ns

∫Ω

|∆n,k(x, x3)|2dxdx3.

(4.1)

Let 12 < α < 1 be a constant to be chosen later. For every k fixed in 0, 1, ..., N, we write

N−1∑n=0

∫ (n+1)s

ns

∫Ω

|∆n,k(x, x3)|2dxdx3 = E1,1 + E1,2, (4.2)

where

E1,1 = E1,1,k =∑

|n−k|≤s−α

∫ (n+1)s

ns

∫Ω

|∆n,k(x, x3)|2dxdx3

and

E1,2 = E1,2,k =∑

|n−k|>s−α

∫ (n+1)s

ns

∫Ω

|∆n,k(x, x3)|2dxdx3.

The sums above are taken over n in the indicated subsets of 0, 1, ..., N −1. If |n−k| ≤ s−α, we have|ns−ks| ≤ s1−α → 0 as s→ 0. Therefore, for s sufficiently small and for each n 6= k in 0, 1, ..., N−1satisfying |n− k| ≤ s−α, the following holds for any (x, x3) ∈ Ω× [ns, (n+ 1)s):

|∆n,k(x, x3)| ≤ | ∂∂x1

[Sk(g2k)(x, x3)]|+ | ∂

∂x1[Sk(g2

k)(x, ns)]|

≤ 2[∂

∂x1Sk(g2

k)]∗(x, ks)

17

and thus ∫ (n+1)s

ns

∫Ω

|∆n,k(x, x3)|2dxdx3 ≤ 4s‖[ ∂

∂x1Sk(g2

k)]∗(x, ks)‖2L2(Ω×ks),

where from Section 2, [ ∂∂x1

Sk(g2k)]∗(x, ks) is the nontangential maximal function of the tangential

derivative ∂∂x1

Sk(g2k) at the point (x, ks) on the kth layer Ωk. By (2.8) we have

‖[ ∂

∂x1Sk(g2

k)]∗‖2L2(Ω×ks) ≤ C‖g2k‖2L2(Ω×ks) ≤ Cs

2‖∂2uk − ıA2kuk‖2L2(Ω),

where C is a constant depending only on R0. For n = k, we know from (2.9) that

‖∇t2k‖L2(Ω) ≤ C‖g2k‖L2(Ω×ks).

Hence

E1,1 ≤∑

|n−k|≤s−α4sC‖g2

k‖2L2(Ω×ks) ≤ C · s−α · s‖g2

k‖2L2(Ω×ks)

=C · s1−α‖g2k‖2L2(Ω×ks)

(4.3)

for all s sufficiently small and some constant C depending only on R0.In order to estimate E1,2, consider n in 0, 1, ..., N − 1 such that |n − k| > s−α. Recall that

Sk(gik) is harmonic in R3 \ Ωk. Without loss of generality, we may assume k + s−α < n ≤ N − 1.(The analysis for 0 ≤ n < k − s−α is similar.) Let DU = (x, x3) ∈ D : x3 ≥ ks + s1−α. Then it isclear that Ω× [ns, (n+ 1)s] ⊂ DU for every n satisfying the above assumptions. Take some bounded

smooth domain Dk ⊂ (x, x3) ∈ R3 : x3 > ks+ s1−α

2 in R3 such that Ω×ks+ s1−α

2 is a flat portion

of the boundary of Dk, DU ⊂ Dk and dist(DU , ∂Dk) ≥ s1−α

2 . Then Sk(gik) is harmonic in Dk and foreach x ∈ Dk, it follows from (2.5) and Holder’s inequality that

|Sk(gik)(x)| =|∫

Ω

c

|x− (y, ks)|hik(y)dy|

≤ 1

4π

∫Ω

1

|x3 − ks|· |hik(y)|dy

≤ 1

4π

∫Ω

2

s1−α · |hik(y)|dy ≤ C0

s1−α |Ω|12 ‖hik‖L2(Ω),

and therefore supDk

|Sk(gik)| ≤ C0

s1−α |Ω|12 ‖hik‖L2(Ω). For (x, x3) ∈ Ω × [ns, (n + 1)s], we have (since

(x, x3) ∈ Ω× [ns, (n+ 1)s] ⊂⊂ R3 \ Ωk and ∆n,k is harmonic in R3 \ Ωk)

|∆n,k(x, x3)| ≤ supΩ×[ns,(n+1)s]

| ∂2

∂x1∂x3Sk(g2

k)| · |x3 − ns|

≤s · supDU

| ∂2

∂x1∂x3Sk(g2

k)|.

By Theorem 2.10 in [5] and the fact that dist(DU , ∂Dk) ≥ s1−α

2 , we have

supDU

| ∂2

∂x1∂x3Sk(g2

k)| ≤ (12

s1−α )2 supDk

|Sk(g2k)| ≤ C0

s3−3α|Ω| 12 ‖h2

k‖L2(Ω).

Hence we obtain|∆n,k(x, x3)| ≤ C0s

3α−2|Ω| 12 ‖h2k‖L2(Ω)

18

and therefore∫Ω

|∆n,k(x, x3)|2dx ≤ C0s6α−4|Ω| · ‖h2

k‖2L2(Ω) · |Ω| = C0|Ω|2s6α−4‖h2k‖2L2(Ω).

To get the best rate of convergence in s as s → 0, we may take α = 57 so that 1 − α = 6α − 4 = 2

7 .The above estimate then becomes∫

Ω

|∆n,k(x, x3)|2dx ≤ C0|Ω|2s27 ‖h2

k‖2L2(Ω).

Integrating over [ns, (n+ 1)s] and observing that the cardinality of the indices for the summationon n in E1,2 is less than N and Ns = L is fixed, we obtain

E1,2 ≤ L · C0|Ω|2s27 ‖h2

k‖2L2(Ω) ≤ Cs27 ‖h2

k‖2L2(Ω) (4.4)

for some constant C depending only on R0 and L. Combining (4.1), (4.2), (4.3) and (4.4) yields

E1 ≤ (N + 1)

N∑k=0

Cs27 ‖h2

k‖2L2(Ω). (4.5)

Similarly we have

E2 ≤ (N + 1)

N∑k=0

Cs27 ‖h1

k‖2L2(Ω). (4.6)

To finish the proof of the lemma, note that

N∑k=0

(‖h1k‖2L2(Ω) + ‖h2

k‖2L2(Ω))

≤ s2N∑k=0

(‖∂1uk − ıA1kuk‖2L2(Ω) + ‖∂2uk − ıA2

kuk‖2L2(Ω))

= s2N∑k=0

‖∇Akuk‖2L2(Ω).

Since s2

N∑k=0

‖∇Akuk‖2L2(Ω) is part of the Lawrence-Doniach energy, it follows from Theorem 3.1 that

N∑k=0

(‖h1k‖2L2(Ω) + ‖h2

k‖2L2(Ω)) ≤ 2sGε,sLD(unNn=0, ~A) ≤ 2sMε(1 + oε,s(1)).

Hence, it follows from this, (4.5) and (4.6) that

1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

| ˆcurlA(x, x3)− ˆcurlAn(x)|2dxdx3 ≤ E1 + E2

≤(N + 1)Cs27

N∑k=0

(‖h1k‖2L2(Ω) + ‖h2

k‖2L2(Ω))

≤(N + 1)Cs27 · sMε(1 + oε,s(1)) ≤ Cs 2

7Mε


19

The above lemma provides the main step in our proof of the lower bound on the minimal Lawrence-Doniach energy.

Theorem 4.2. Assume | ln ε| hex 1ε2 as ε → 0. Let (unNn=0,

~A) ∈ [H1(Ω;C)]N+1 × K be aminimizer of Gε,sLD. Then we have

Gε,sLD(unNn=0,~A) ≥ |D|

2hex(ln

1

ε√hex

)(1− oε(1)− Cs 1

7

)for all ε and s sufficiently small, where C is a constant depending only on R0 and L.

Proof. By dropping the nonnegative Josephson coupling term and the square of the L2 norm of thefirst two components of ∇× ~A− hex~e3, it is clear that

Gε,sLD(unNn=0, ~A) ≥ sN∑n=0

∫Ω

[1

2|∇Anun|

2 +1

4ε2(1− |un|2)2

]dx

+1

2

∫R3

( ˆcurlA− hex)2dx.

Then

1

2

∫R3

(ˆcurlA(x)− hex

)2dx ≥ 1

2

∫D

(ˆcurlA(x)− hex

)2dx

=1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

(ˆcurlA(x, x3)− hex

)2dxdx3

=1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

[ˆcurl(A(x, x3)− An(x)

)+(

ˆcurlAn(x)− hex)]2

dxdx3.

Applying the elementary inequality (a+ b)2 ≥ a2 − 2|a| · |b| yields

1

2

∫D

( ˆcurlA− hex)2dx ≥ 1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

( ˆcurlAn − hex)2dxdx3

−N−1∑n=0

∫ (n+1)s

ns

∫Ω

| ˆcurl(A− An)| · | ˆcurlAn − hex|dxdx3.

(4.7)

Therefore

Gε,sLD(unNn=0, ~A)

≥sN−1∑n=0

∫Ω

[1

2|∇Anun|

2 +1

4ε2(1− |un|2)2

]dx+

1

2

∫D

( ˆcurlA− hex)2dx

≥sN−1∑n=0

∫Ω

[1

2|∇Anun|

2 +1

4ε2(1− |un|2)2

]dx+

1

2

∫Ω

( ˆcurlAn − hex)2dx

−N−1∑n=0

∫ (n+1)s

ns

∫Ω

| ˆcurl(A− An)| · | ˆcurlAn − hex|dxdx3.

(4.8)

It was proved in [2] that un ∈ H1(Ω;C) and An ∈ H1loc(R2;R2) for all n = 0, 1, ..., N−1. Therefore each

pair (un, An) is in the admissible set for the minimization of the two-dimensional Ginzburg-Landau

20

By combining Theorems 3.1 and 4.2 we obtain Theorem 1.

Proof of Theorem 2. By (4.7) and (4.12), we see that the leading term in our lower bound of theminimum Lawrence-Doniach energy comes from two terms, since

Gε,sLD(unNn=0, ~A) ≥sN∑n=0

∫Ω

[1

2|∇Anun|

2 +1

4ε2(1− |un|2)2

]dx

+1

2

∫D

( ˆcurlA− hex)2dx

≥|D|2hex ln

1

ε√hex

(1− oε(1)− Cs 17 )

(4.13)

for some constant C depending only on R0 and L and for all ε and s sufficiently small. As a result of(4.13) and Theorem 3.1, we conclude that

s

N−1∑n=0

∫Ω

1


∫ (n+1)sns

A3dx3 |2dx+1

2

∫R3\D

|∇ × ~A− hex~e3|2dx

+1

2

∫D

[(∂A3

∂x2− ∂A2

∂x3)2 + (

∂A1

∂x3− ∂A3

∂x1)2]dx ≤ oε,s(1)Mε.

This proves Theorem 2.

Corollary 4.3. Under the assumptions of Theorem 2, we have

∣∣Gε,sLD(unNn=0, ~A)−N−1∑n=0

sFε(un, An)∣∣ ≤ oε,s(1) ·Mε

as (ε, s)→ (0, 0).

Proof. By (4.8) and (4.12), we have

Gε,sLD(unNn=0,~A) ≥

N−1∑n=0

sFε(un, An)− oε,s(1) ·Mε

for all ε and s sufficiently small. By the Cauchy-Schwartz inequality, Lemma 4.1, Theorem 3.1 and(4.11), we have ∣∣∣∣∣12

∫D

| ˆcurlA− hex|2dx−N−1∑n=0

s

2

∫Ω

| ˆcurlAn − hex|2dx

∣∣∣∣∣ ≤ Cs 17Mε

for some constant C depending only on R0 and L. Combining this with Theorem 2 and the definitionof Gε,sLD, we obtain ∣∣Gε,sLD(unNn=0,

~A)−N−1∑n=0

sFε(un, An)∣∣ ≤ oε,s(1) ·Mε.

22

Corollary 4.4. Under the assumptions of Theorem 1, we have

‖∇ ×~A

hex− ~e3‖L2(R3;R3) → 0,

and

1

N + 1

N∑n=0

µnhex→ dx in H−1(Ω)

as (ε, s)→ (0, 0), where µn is the vorticity on the nth layer defined as

µn = ˆcurl(ıun, ∇Anun) + ˆcurlAn.

Proof. The convergence of ∇×~A

hexto ~e3 in L2(R3;R3) follows immediately from Theorem 3.1 and the

assumption that | ln ε| hex 1ε2 .

By the regularity results proved by Bauman and Ko in [2], we know that (ıun, ∇Anun) ∈ L2(Ω;R2)

and An ∈ H1(Ω;R2). Thus

µn = ˆcurl(ıun, ∇Anun) + ˆcurlAn ∈ H−1(Ω),

and since |un| ≤ 1, we have

‖µn − hex‖2H−1(Ω) ≤ ‖∇Anun‖2L2(Ω) + ‖ ˆcurlAn − hex‖2L2(Ω).

Therefore, by Theorem 3.1 and (4.11), we get

s

N∑n=0

‖µn − hex‖2H−1(Ω) ≤ sN∑n=0

‖∇Anun‖2L2(Ω) + s

N∑n=0

‖ ˆcurlAn − hex‖2L2(Ω)

= s

N∑n=0

‖∇Anun‖2L2(Ω) +

N∑n=0

∫ (n+1)s

ns

∫Ω


≤ 2Mε(1 + oε,s(1)) + C0Mε ≤ C0Mε.

This implies that

s

N∑n=0

‖ µnhex− dx‖2H−1(Ω) ≤

C0Mε

h2ex

→ 0 (4.14)

as (ε, s)→ (0, 0). Note that

‖ 1

N + 1

N∑n=0

µnhex− dx‖2H−1(Ω) =‖ 1

N + 1

N∑n=0

(µnhex− dx)‖2H−1(Ω)

=1

(N + 1)2‖N∑n=0

(µnhex− dx)‖2H−1(Ω).

By the Cauchy-Schwartz inequality and (4.14), we obtain

‖ 1

N + 1

N∑n=0

µnhex− dx‖2H−1(Ω) ≤

N + 1

(N + 1)2

N∑n=0

‖ µnhex− dx‖2H−1(Ω)

=1

s(N + 1)· s

N∑n=0

‖ µnhex− dx‖2H−1(Ω) → 0

as (ε, s)→ (0, 0), since sN = L is the height of the domain D which is fixed.

23

5 Comparison results

In this section we prove a comparison result between the minimum Lawrence-Doniach energy and theminimum three-dimensional anisotropic Ginzburg-Landau energy under the assumption that s ≤ C0εfor some constant C0 independent of ε, s, Ω, L, D and R0 for all ε and s sufficiently small. Recall thedefinition of the anisotropic Ginzburg-Landau energy GεAGL given in (1.8) in the introduction. Directcalculations show that GεAGL is invariant under the gauge transformation

ξ(x) = ψ(x)eıg(x) in Ω,~B = ~A+∇g in R3

for some g ∈ H2loc(R3). Recall the rescaling formulas (2.4) for the anisotropic Ginzburg-Landau

energies, from which we may translate estimates from [3], [2] and [11] to our scaling. As pointed

out in [2], every minimizer (ψ, ~A) ∈ H1(D;C) × E of GεAGL is gauge equivalent to another pair inH1(D;C) × K, where the spaces E and K are defined in (1.2) and (1.4) respectively. The space

H1(D;C) ×K fixes a “Coulomb gauge” for (ψ, ~A) as in the study of the Lawrence-Doniach energy.Our goal of this section is to prove Theorem 3. First we prove several lemmas that will be used foran upper bound on the minimal three-dimensional anisotropic Ginzburg-Landau energy.

Lemma 5.1. Let (unNn=0,~A) ∈ [H1(Ω;C)]N+1 ×K be a minimizer of Gε,sLD. Then

‖A3‖2L6(D) ≤ C0 ·Mε

for all ε and s sufficiently small and some constant C0 independent of ε, s, Ω, L, D and R0.

Proof. Since a3 = 0, by (2.1), (2.2) and Theorem 3.1 we have

‖A3‖2L6(D) = ‖A3 − hexa3‖2L6(D) ≤ ‖ ~A− hex~a‖2L6(D)

≤ 4‖ ~A− hex~a‖2H1(R3)= 4

∫R3

|∇ × ( ~A− hex~a)|2dx

≤ 8Gε,sLD(unNn=0, ~A) ≤ C0 ·Mε

for C0 as described above and all ε and s sufficiently small.

Lemma 5.2. Under the assumptions above and in addition assuming s ≤ C0ε, we have

N−1∑n=0

∫Ω

1

s|un+1 − un|4dx ≤ oε(1)Mε

as ε→ 0.

Proof. We write

|un+1 − un| = |(un+1 − uneı∫ (n+1)sns

A3dx3) + (uneı∫ (n+1)sns

A3dx3 − un)|.

Using the triangle inequality and (a+ b)4 ≤ 24(a4 + b4) for a, b > 0, we obtain

N−1∑n=0

∫Ω

1

s|un+1 − un|4dx ≤ J1 + J2,

where

J1 =

N−1∑n=0

∫Ω

24

s|un+1 − uneı

∫ (n+1)sns

A3dx3 |4dx

24

and

J2 =

N−1∑n=0

∫Ω

24

s|uneı

∫ (n+1)sns

A3dx3 − un|4dx.

Note that, since |un+1| ≤ 1 and |uneı∫ (n+1)sns

A3dx3 | = |un| ≤ 1,

J1 ≤N−1∑n=0

∫Ω

24

s· 22|un+1 − uneı

∫ (n+1)sns

A3dx3 |2dx.

By Theorem 2 and our assumption that λ is fixed, it follows that J1 ≤ oε,s(1)Mε.For J2, following an idea in [11] and Holder’s inequality, we have

J2 ≤24

s

N−1∑n=0

∫Ω

∣∣∫ (n+1)s

ns

A3dx3

∣∣4dx ≤ 24

s

N−1∑n=0

∫Ω

s3

∫ (n+1)s

ns

(A3)4dx3dx

= 24s2

∫D

(A3)4dx ≤ 24s2(∫D

(A3)6dx) 2

3 |D| 13 = 24s2|D| 13 ‖A3‖4L6(D).

Since s ≤ C0ε and ε2Mε → 0 as ε → 0, these imply that s2Mε → 0 as ε → 0. Therefore, Lemma 5.1implies that J2 ≤ oε,s(1)Mε and hence the lemma is proved.

In the following we will need several additional lemmas which follow from Lemma 5.1, Lemma 5.2and calculations from the proof of Lemma 5.5 in [3] (after appropriate rescaling). We remark thatLemma 5.2 is a stronger estimate than the analogous estimate in [3] and [11] (in which ε was fixed),and it is a key ingredient in our proof of Theorem 3.

Lemma 5.3. Assume | ln ε| hex ε−2 and s ≤ C0ε as ε→ 0. Let (unNn=0,~A) ∈ [H1(Ω;C)]N+1×

K be a minimizer of Gε,sLD. Define ψ ∈ H1(D;C) to be

ψ(x, x3) ≡N−1∑n=0

[(1− tn)un(x) + tnun+1(x)

]χ[ns,(n+1)s](x3),

where tn = x3−nss . Then we have

1

2

∫D

1

λ2|( ∂

∂x3− ıA3)ψ|2dx

≤ sN−1∑n=0

∫Ω

1


∫ (n+1)sns

A3dx3 |2dx+ oε(1)Mε

(5.1)

as ε→ 0.

Proof. By the definition of ψ, we have

1

2

∫D

1

λ2|( ∂

∂x3− ıA3)ψ|2dx− s

N−1∑n=0

∫Ω

1


∫ (n+1)sns

A3dx3 |2dx

=1

2λ2s2

N−1∑n=0

∫Ω

∫ (n+1)s

ns

∣∣(un+1 − un)(1− ıstnA3)− ısA3un∣∣2dx3

− s∣∣(un+1 − un) + un(1− eı

∫ (n+1)sns

A3dx3)∣∣2dx

=(R11 − R21) + (R12 − R22) + (R13 − R23),

(5.2)

25

where

R11 − R21 =1

2λ2s2

N−1∑n=0

∫Ω

[∫ (n+1)s

ns

∣∣(un+1 − un)(1− ıstnA3)∣∣2dx3 − s|un+1 − un|2

]dx,

R12 − R22 =1

2λ2s2

N−1∑n=0

∫Ω

2<[∫ (n+1)s

ns

(un+1 − un)(1− ıstnA3)ısA3undx3

− s(un+1 − un)un(1− e−ı

∫ (n+1)sns

A3dx3)]dx,

and

R13 − R23 =1

2λ2s2

N−1∑n=0

∫Ω

[∫ (n+1)s

ns

s2(A3)2|un|2dx3 − s|un|2 · |1− eı∫ (n+1)sns

A3dx3 |2]dx.

Here we use the splitting in the proof of Lemma 5.5 in [3]. Each R1j−R2j corresponds to the quantityR1j −R2j in the proof of Lemma 5.5 in [3] for j = 1, 2, 3.

Note that |1− ıstnA3|2 = 1 + (stnA3)2 and, by the definition of tn, stn = x3 − ns. Therefore we

have

R11 − R21

=1

2λ2s2

N−1∑n=0

∫Ω

[∫ (n+1)s

ns

|un+1 − un|2(1 + (x3 − ns)2(A3)2

)dx3 − s|un+1 − un|2

]dx.

From the estimates for |R11 −R21| in the proof of Lemma 5.5 in [3] and using the rescaling relations(2.3) and (2.4) (in particular, note that ψn = un and κAz = A3), we have∣∣∣∣∣ 1

s2

N−1∑n=0

∫Ω

[∫ (n+1)s

ns

|un+1 − un|2(1 + (x3 − ns)2(A3)2

)dx3 − s|un+1 − un|2

]dx

∣∣∣∣∣≤2s

12

[s

N−1∑n=0

∫Ω

(|un+1 − un|

s)2dx

] 12 · s 1

2 ·[∫ L

0

∫Ω

|A3|4dxdx3

] 12 .

(5.3)

It follows from (5.3) that

R11 − R21 ≤s

λ2

(N−1∑n=0

1

s

∫Ω

|un+1 − un|2dx) 1

2

·(∫

D

(A3)4dx

) 12

. (5.4)

Using similar calculations as in the proof of Lemma 5.2, it is not hard to show that

N−1∑n=0

∫Ω

1

s|un+1 − un|2dx ≤ C ·Mε

for some constant C depending only on |D| for all ε sufficiently small. Then applying Holder’sinequality to

∫D

(A3)4dx as in the proof of Lemma 5.2 and using Lemma 5.1, we deduce from (5.4)that

R11 − R21 ≤ Cs ·√Mε ·Mε

for some constant C depending only on |D|. From the proof of Lemma 5.2, s√Mε → 0 as ε → 0. It

follows thatR11 − R21 ≤ oε(1)Mε. (5.5)

26

for some constant C depending only on |D| for all ε sufficiently small, and therefore

s

λ2‖∂A

3

∂x3‖L2(D)‖A3‖L2(D) ≤ CsMε = oε(1)Mε.

It is easy to see that1

2λ2

s2

12

∫D

(A3)4dx ≤ oε(1)Mε

and thereforeR13 − R23 ≤ oε(1)Mε. (5.10)

Combining the estimates (5.5), (5.8) and (5.10) and using (5.2), we obtain (5.1).

Lemma 5.4. Assuming the hypotheses of Lemma 5.3, we have

1

2

∫D

|∇Aψ|2dx ≤ s

2

N∑n=0

∫Ω

|∇Anun|2dx+ oε(1)Mε (5.11)

as ε→ 0.

Proof. We calculate 12

∫D|∇Aψ|2dx:

1

2

∫D

|∇Aψ|2dx =

1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

∣∣∣∣∇A[(1− tn)un(x) + tnun+1(x)]∣∣∣∣2dxdx3

=1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

∣∣∣∣(1− tn)∇Aun(x) + tn∇Aun+1(x)

∣∣∣∣2dxdx3.

Using the triangle inequality and the convexity of the function x2, we obtain

1

2

∫D

|∇Aψ|2dx ≤ 1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

[(1− tn)|∇Aun(x)|2 + tn|∇Aun+1(x)|2

]dxdx3

=1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

[(1− tn)

∣∣∣∣∇Anun(x) + ı(An − A)un

∣∣∣∣2]dxdx3

+1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

[tn

∣∣∣∣∇An+1un+1(x) + ı(An+1 − A)un+1

∣∣∣∣2]dxdx3.

Expanding the above quadratic terms and applying Young’s inequality ab ≤ sa2

2 + 12sb

2 to the crossproduct terms, we obtain

1

2

∫D

|∇Aψ|2dx ≤1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

[(1− tn)(1 + s)|∇Anun|

2

+ (1− tn)(1 + s−1)|An − A|2]dxdx3

+1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

[tn(1 + s)|∇An+1

un+1|2

+ tn(1 + s−1)|An+1 − A|2]dxdx3.

(5.12)

28

Note that∫ (n+1)s

nstndx3 =

∫ (n+1)s

ns(1 − tn)dx3 = s

2 . UsingN∑n=0

∫Ω

(1−|un|2)2

4ε2 dx as an upper bound for

bothN−1∑n=0

∫Ω

(1−|un|2)2

4ε2 dx andN−1∑n=0

∫Ω

(1−|un+1|2)2

4ε2 dx, we deduce from (5.12) that

1

2

∫D

|∇Aψ|2dx ≤ s

2

N∑n=0

∫Ω

|∇Anun|2dx+R4 (5.13)

where

R4 =1 + s−1

2

N−1∑n=0

∫ (n+1)s

ns

∫Ω

[(1− tn)|An − A|2 + tn|An+1 − A|2

]dxdx3

+s2

2

N∑n=0

∫Ω

|∇Anun|2dx.

Using the idea in [11], we have

N−1∑n=0

∫ (n+1)s

ns

∫Ω

(1− tn)|An − A|2dxdx3

≤N−1∑n=0

∫ (n+1)s

ns

∫Ω

(1− tn)

(∫ (n+1)s

ns

| ∂A∂x3|dx3

)2

dxdx3

≤s2· s

N−1∑n=0

∫Ω

∫ (n+1)s

ns

∣∣ ∂A∂x3

∣∣2dx3dx =s2

2

∫D

∣∣ ∂∂x3

(A− hexa)∣∣2dx.

It follows from (2.2) that∫D

∣∣ ∂∂x3

(A− hexa)∣∣2dx ≤ ∫

R3

∣∣∇( ~A− hex~a)∣∣2dx =

∫R3

∣∣∇× ( ~A− hex~a)∣∣2dx

and by Theorem 3.1,

N−1∑n=0

∫ (n+1)s

ns

∫Ω

(1− tn)|An − A|2dxdx3 ≤ s2Mε(1 + oε(1)).

From this it is clear that R4 ≤ oε(1)Mε. Thus (5.11) follows from (5.13).

Lemma 5.5. Assuming the hypotheses of Lemma 5.3, we have∫D

(1− |ψ|2)2

4ε2dx ≤ s

N∑n=0

∫Ω

(1− |un|2)2

4ε2dx+ oε(1)Mε (5.14)

as ε→ 0.

Proof. By the definition of ψ and direct calculations we obtain, for (x, x3) ∈ Ω× [ns, (n+ 1)s],

1− |ψ|2 = (1− tn)(1− |un|2) + tn(1− |un+1|2) + tn(1− tn)|un − un+1|2.

29

Explicit calculations using the definition of tn gives∫D

(1− |ψ|2)2

4ε2dx =

1

4ε2s

15

N−1∑n=0

∫Ω

[5(1− |un|2)2 + 5(1− |un+1|2)2

+ 5(1− |un|2)(1− |un+1|2) +1

2|un − un+1|4

+5

2(1− |un|2)|un − un+1|2 +

5

2(1− |un+1|2)|un − un+1|2

]dx.

Applying Young’s inequality ab ≤ δ(ε)a2+ 1δ(ε)b

2 to (1−|un|2)|un−un+1|2 and (1−|un+1|2)|un−un+1|2

for some δ(ε) > 0 depending only on ε and to be chosen later, and applying the elementary inequality

ab ≤ a2

2 + b2

2 to (1− |un|2)(1− |un+1|2), we obtain

∫D

(1− |ψ|2)2

4ε2dx ≤ 1

4ε2s

15

N−1∑n=0

∫Ω

[15

2

((1− |un|2)2 + (1− |un+1|2)2

)+

5δ(ε)

2

((1− |un|2)2 + (1− |un+1|2)2

)]dx

+1

4ε2s

15

N−1∑n=0

∫Ω

(1

2+

5

δ(ε)

)|un − un+1|4dx

=R51 +R52.

(5.15)

Note that

R51 ≤ sN∑n=0

∫Ω

(1− |un|2)2

4ε2dx+

δ(ε)

3· s

N∑n=0

∫Ω

(1− |un|2)2

4ε2dx

where we have usedN∑n=0

∫Ω

(1−|un|2)2

4ε2 dx to bound both

N−1∑n=0

∫Ω

(1− |un|2)2

4ε2dx

andN−1∑n=0

∫Ω

(1− |un+1|2)2

4ε2dx

from above again. Also note that, by the assumption s ≤ C0ε and Lemma 5.2, we have

R52 ≤ C0(1

2+

5

δ(ε))

N−1∑n=0

∫Ω

1

s|un − un+1|4dx ≤ C0(

1

2+

5

δ(ε)) · c(ε)Mε,

where c(ε) depends only on ε for all ε sufficiently small and satisfies c(ε)→ 0 as ε→ 0. Now we choose

δ(ε) to satisfy δ(ε)→ 0 and c(ε)δ(ε) → 0 as ε→ 0. (For example, we may choose δ(ε) =

√c(ε).) Then we

obtain

R51 ≤ sN∑n=0

∫Ω

(1− |un|2)2

4ε2dx+ oε(1) · s

N∑n=0

∫Ω

(1− |un|2)2

4ε2dx (5.16)

30

andR52 ≤ oε(1)Mε (5.17)

as ε→ 0. Combining (5.16) and (5.17) with (5.15) and using Theorem 3.1, we obtain (5.14).

Theorem 5.6. Assume | ln ε| hex ε−2 and s ≤ C0ε as ε→ 0. Let (unNn=0,~A) ∈ [H1(Ω;C)]N+1×

K be a minimizer of Gε,sLD and (ζ, ~B) ∈ H1(D;C)×K be a minimizer of GεAGL. We have

GεAGL(ζ, ~B) ≤ Gε,sLD(unNn=0, ~A)(1 + oε(1)) (5.18)

as ε→ 0.

Proof. Let ψ be defined as in Lemma 5.3 and let ~A ≡ ~A. We show that GεAGL(ψ, ~A) satisfies (5.18).

Since we are taking the same magnetic potential ~A for Gε,sLD and GεAGL, the magnetic terms in the twoenergies are equal. Then the comparison result (5.18) follows from (5.1), (5.11) and (5.14).

Next we prove a lower bound for the minimal anisotropic Ginzburg-Landau energy GεAGL. Theproof is based on a slicing method in which we use the lower bound for the two-dimensional Ginzburg-Landau energy proved by Sandier and Serfaty in [9] on each cross section Ω×x3, and integrate overx3 to get the desired lower bound for GεAGL.

Theorem 5.7. Assume | ln ε| hex ε−2 as ε → 0. Let (ζ, ~B) ∈ H1(D;C)× E be a minimizer ofGεAGL. Then we have

GεAGL(ζ, ~B) ≥Mε(1− oε(1))

as ε→ 0.

Proof. Dropping appropriate terms in GεAGL, we see that

GεAGL(ζ, ~B) ≥∫D

[1

2|∇Bζ|

2 +(1− |ζ|2)2

4ε2

]dx+

1

2

∫D

(ˆcurlB − hex

)2dx

=

∫ L

0

∫Ω

[1

2|∇Bζ|

2 +(1− |ζ|2)2

4ε2

]dx+

1

2

∫Ω

(ˆcurlB − hex

)2dx

dx3.

Since ζ ∈ H1(D;C) and B ∈ H1loc(R3;R2), we have that for almost every x3 ∈ [0, L], ζ(·, x3) ∈

H1(Ω× x3;C) and B(·, x3) ∈ H1(Ω× x3;R2). By Theorem 8.1 in [9], we have

min(v,b)∈H1×H1

Fε(v, b) ≥|Ω|2hex ln

1

ε√hex

(1− c(ε))

for some constant c(ε) depending only on ε such that c(ε) → 0 as ε → 0. Thus, for almost everyx3 ∈ [0, L], we have∫

Ω

[1

2|∇Bζ|

2 +(1− |ζ|2)2

4ε2]dx+

1

2

∫Ω

( ˆcurlB − hex)2dx ≥ |Ω|2hex ln

1

ε√hex

(1− c(ε)).

Integrating from 0 to L, we obtain

GεAGL(ζ, ~B) ≥ L |Ω|2hex ln

1

ε√hex

(1− c(ε)) = Mε(1− c(ε))

for all ε sufficiently small. The theorem follows from this and the property of c(ε) mentioned above.

Theorem 3 follows by combining Theorems 5.6, 5.7, 3.1 and 4.2.

31

References

[1] S. Alama, L. Bronsard and E. Sandier, On the Lawrence-Doniach model of superconduc-tivity: magnetic fields parallel to the axes, J. Eur. Math. Soc., 14 (2012), pp. 1825-1857.

[2] P. Bauman and Y. Ko, Analysis of solutions to the Lawrence-Doniach system for layeredsuperconductors, SIAM J. Math. Anal., 37 (2005), pp. 914-940.

[3] S. J. Chapman, Q. Du and M. D. Gunzburger, On the Lawrence-Doniach and anisotropicGinzburg-Landau models for layered superconductors, SIAM J. Appl. Math., 55 (1995), pp. 156-174.

[4] E. B. Fabes, M. Jodeit JR and N. M. Riviere, Potential techniques for boundary valueproblems on C1-domains, Acta Math., 141 (1978), pp. 165-186.

[5] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Clas-sics in Mathematics, reprint of the 1998 ed., Springer-Verlag, Berlin, 2001.

[6] T. Giorgi and D. Phillips, The breakdown of superconductivity due to strong fields for theGinzburg-Landau model, SIAM Review, 44 (2002), pp. 237-256.

[7] Y. Iye, How anisotropic are the cuprate high Tc superconductors?, Comments Cond. Mat. Phys.,16 (1992), pp. 89-111.

[8] E. Sandier and S. Serfaty, On the energy of type-II superconductors in the mixed phase, Rev.Math. Phys., 12 (2000), pp. 1219-1257.

[9] E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Progress inNonlinear Differential Equations and Their Applications 70, Birkhauser, Boston, 2007.

[10] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equationin Lipschitz domains, J. Funct. Anal., 59 (1984), pp. 572-611.

[11] C. Yuan, Gamma-convergence of the Lawrence-Doniach energy to a three-dimensionalanisotropic Ginzburg-Landau energy, Ph.D. thesis, Purdue University, West Lafayette, IN, 2008.

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