Analysis of Minimizers of the Lawrence-Doniach Energy for Superconductors in Applied Fields

We analyze minimizers of the Lawrence-Doniach energy for layered superconductors occupying a bounded generalized cylinder, $ \Omega\times[0,L]$, in $\mathbb{R}^3$, where $\Omega$ is a bounded simply connected Lipschitz domain in $\mathbb{R}^2$. For an applied magnetic field $\vec{H}_{ex}=h_{ex}\vec{e}_{3}$ that is perpendicular to the layers with $\left|\ln\epsilon\right|\ll h_{ex}\ll\epsilon^{-2}$ as $\epsilon\rightarrow 0$, where $\epsilon$ is the reciprocal of the Ginzburg-Landau parameter, we prove an asymptotic formula for the minimum Lawrence-Doniach energy as $\epsilon$ and the interlayer distance $s$ tend to zero. Under appropriate assumptions on $s$ versus $\epsilon$, we establish comparison results between the minimum Lawrence-Doniach energy and the minimum three-dimensional anisotropic Ginzburg-Landau energy. As a consequence, our asymptotic formula also describes the minimum three-dimensional anisotropic energy as $\epsilon$ tends to zero.


Introduction
The Lawrence-Doniach model was formulated by Lawrence and Doniach in 1971 as a macroscopic model for layered superconductors. While the standard Ginzburg-Landau model has been well accepted as a macroscopic model for isotropic superconductors, it does not account for the anisotropy in three-dimensional high temperature superconducting materials. For these materials, depending on 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 nonlinear 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)) has been intensively investigated. In this case, an analysis of the behavior of energy minimizers and their vortex structure in a perpendicular applied magnetic field with modulus h ex in different regimes (e.g., h ex ∼ | ln ǫ|, | ln ǫ| ≪ h ex ≪ ǫ −2 , or h ex ≥ C ǫ 2 ) is now well understood. (See [9].) However, for the Lawrence-Doniach energy (see (1.1)), an analysis for h ex in the first two regimes has been done only for the gauge-periodic problem, in which the superconductor is assumed to occupy all of R 3 and the gauge invariant quantities are assumed to be periodic with respect to a given parallelepiped. (See [1].) In the last regime, h ex ≥ C ǫ 2 , it was shown in [2] that if C is sufficiently large, all minimizers of the Lawrence-Doniach energy are in the normal (nonsuperconducting) phase, that is, the order parameters on the layers, {u n } N n=0 , are all identically equal to zero, and the induced magnetic field, ∇ × A, is identically equal to the applied magnetic field. A similar result is known for the two-dimensional Ginzburg-Landau energy. (See [6].) In this paper, we analyze the Lawrence-Doniach model in the second regime without imposing gauge periodicity assumptions. In two-dimensional superconductors, this regime for h ex corresponds to a mixed state in which superconducting states and normal states (in the form of isolated vortices, i.e., zeros of u n ) 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 Ω n ≡ Ω × {ns}, where Ω is a bounded simply connected Lipschitz domain in R 2 and s = L N is the interlayer distance. Assuming an applied magnetic field h ex e 3 , the Lawrence-Doniach energy is given by n=0 Ω 1 2λ 2 s 2 |u n+1 − u n e ı (n+1)s ns A 3 dx3 | 2 dx (1.2) Here ǫ > 0 is the reciprocal of the Ginzburg-Landau parameter and λ > 0 represents the Josephson penetration depth, which is assumed to be fixed throughout this study. The applied magnetic field h ex e 3 is assumed to satisfy | ln ǫ| ≪ h ex ≪ ǫ −2 as ǫ → 0. The complex valued function u n defined in Ω is the order parameter for the nth layer and |u n (x 1 , x 2 )| 2 is the density of superconducting electron pairs at each point (x 1 , x 2 , ns) on the nth layer. For a minimizer of the Lawrence-Doniach energy (1.1), |u n (x 1 , x 2 )| ∼ 1 corresponds to a superconducting state at (x 1 , x 2 , ns), whereas |u n (x 1 , x 2 )| = 0 corresponds to a normal (nonsuperconducting) state at (x 1 , x 2 , ns), in which the density of superconducting electrons is zero. The vector field A = (A 1 , A 2 , A 3 ) defined on R 3 is called the magnetic potential; its curl, ∇ × A, is the induced magnetic field. We let x = (x 1 , x 2 , x 3 ),∇ = (∂ 1 , ∂ 2 ), x = (x 1 , x 2 ),Â = (A 1 , A 2 ) andÂ n (x) = (A 1 (x, ns), A 2 (x, ns)), the trace ofÂ on the nth layer. We set∇Â n u n =∇u n − ıÂ n u n on Ω. In the following, given two complex numbers u and v, we let (u, v) = 1 2 (ūv + uv) = ℜ(uv), which is an inner product of u = u 1 + ıu 2 and v = v 1 + ıv 2 in C that agrees with the inner product of (u 1 , u 2 ) and (v 1 , v 2 ) in R 2 .
Since A ∈ H 1 loc (R 3 ; R 3 ), it follows from the trace theorem and the Sobolev imbedding theorem that its traceÂ n ∈ H 1 2 loc (R 2 ; R 2 ) ⊂ L 4 loc (R 2 ; R 2 ) and therefore the Lawrence-Doniach energy G ǫ,s LD ({u n } N n=0 , A) is well-defined and finite. The existence of minimizers in [H 1 (Ω; C)] N +1 × E was shown by Chapman, Du and Gunzburger in [3]. Each minimizer of G ǫ,s LD corresponds to a physically realistic state for the layered superconductor. The minimizer satisfies the Euler-Lagrange equations associated to the Lawrence-Doniach energy. This system of equations is called the Lawrence-Doniach system and it is given by (∇ − ıÂ n ) 2 u n + 1 ǫ 2 (1 − |u n | 2 )u n + P n = 0 on Ω, ∇ × (∇ × A) = (j 1 , j 2 , j 3 ) in R 3 , (∇ − ıÂ n )u n · n = 0 on ∂Ω, ∇ × A − h ex e 3 ∈ L 2 (R 3 ; R 3 ) for all n = 0, 1, ... , N , where It was proved in [2] that a minimizer ({u n } N n=0 , A) of (1.1) satisfies |u n | ≤ 1 a.e. in Ω for all n = 0, 1, ... , N . Two Simple calculations show that G ǫ,s LD (and each term in G ǫ,s LD ) is invariant under the above gauge transformation, i.e., for two configurations ({u . Let a = a(x) be any fixed smooth vector field on R 3 such that ∇ × a = e 3 in R 3 . For example, we may choose a(x) = (0, x 1 , 0). It was also proved in [2] that every pair ({u Here the spaceȞ 1 (R 3 ) represents the completion of C ∞ 0 (R 3 ; R 3 ) with respect to the seminorm In particular, any minimizer of G ǫ,s LD in the admissible space [H 1 (Ω; C)] N +1 × E is gauge-equivalent to a minimizer in the space [H 1 (Ω; C)] N +1 × K, called the "Coulomb gauge" for G ǫ,s LD . It was shown in [2] that minimizers in the Coulomb gauge satisfy u n ∈ C ∞ (Ω) andÂ n ∈ H 1 loc (R 2 ) for all n = 0, 1, ... , N . Throughout this paper, we take a(x) = (0, x 1 , 0).
Given the above definitions, our main results are the following: Then denoting the volume of D by |D|, we have as (ǫ, s) → (0, 0) for some constant C independent of ǫ and s. In particular, (See Theorem 3.1 and Theorem 4.2.) Here o ǫ (1) denotes a quantity that converges to 0 as ǫ → 0 and o ǫ,s (1) denotes a quantity that converges to 0 as (ǫ, s) → (0, 0). This theorem generalizes a result in the gauge periodic case studied by Alama, Bronsard and Sandier for the energy (1.1) in which the domain Ω is replaced by a parallelogram P in R 2 , the integral of |∇× A−h ex e 3 | 2 is taken over P ×[0, L] instead of over R 3 , and the minimization takes place among gauge periodic configurations ({u n } N n=0 , A) in R 3 with period P × [0, L]. (See [1].) In that case, they further showed that for a minimizer of the gauge periodic problem, the order parameters u n are all equal and A 3 is identically zero. In particular, the Josephson coupling term vanishes. They also proved thatÂ(x, ·) is periodic in x 3 with period s and established certain symmetries between the layers inÂ.
In the gauge periodic case, the results of Alama, Bronsard and Sandier indicate a close connection between the Lawrence-Doniach energy and the two-dimensional Ginzburg-Landau energy GL ǫ given by for h ex as assumed above whereĉurl denotes the two-dimensional curl defined byĉurl(B 1 , B 2 ) = ∂ 1 B 2 − ∂ 2 B 1 . We remark that for a minimizer of the two-dimensional energy GL ǫ , the magnetic potentialÂ satisfiesĉurlÂ = h ex in R 2 \ Ω. (See Lemma 2.1 in [6].) Therefore the minimum of GL ǫ is equal to the minimum of F ǫ given by (See Prop. 3.4 in [9] for bounded simply connected smooth domains and Prop. 2.1 in this paper for bounded simply connected Lipschitz domains.) The main idea in our proof of the upper bound in Theorem 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, (A 1 , A 2 )(x, ns)) used by Sandier and Serfaty in Chapter 8 of [9] to provide an upper bound on the minimal two-dimensional energy F ǫ . A matching lower bound is much more difficult to establish. To obtain it, we prove in Lemma 4.1 that, for a minimizer ( where M ǫ = |D| 2 h ex ln 1 ǫ √ 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 1 2 D |ĉurlÂ(x, x 3 ) − h ex | 2 dx can be approximated within o ǫ,s (1)M ǫ by the sum of two-dimensional integrals, Σ N −1 n=0 s 2 Ω |ĉurlÂ n (x)−h ex | 2 dx. As a result of (1.7), the energy for a minimizer from the first term in (1.1) plus the magnetic term 1 2 D |ĉurlÂ − h ex | 2 dx can be approximated to leading order by the sum of sF ǫ (u n ,Â n ). Combining this observation with the lower bound estimate for F ǫ 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 (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 twodimensional energy F ǫ on the minimal Lawrence-Doniach energy. In fact, we prove in Corollary 4.3 that for a minimizer ({u Another consequence of Theorems 1 and 2 is the following: Corollary. Under the assumptions of Theorem 1, we have as (ǫ, s) → (0, 0), where µ n is the vorticity on the nth layer defined as µ n =ĉurl(ıu n ,∇Â n u n ) +ĉurlÂ n . (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 F ǫ studied by Sandier and Serfaty (see Cor. 8.1 in [9]). They showed that for minimizers of F ǫ , the scaled vorticity measure µ hex converges to dx in H −1 (Ω) as ǫ → 0. The vorticity measure µ n in each layer is a gauge-invariant version of the Jacobian determinant of u n , and is analagous to the vorticity in fluids. If u n is given in polar coordinates by ρ n e iθn , then µ n =ĉurl{ρ 2 n (∇θ n −Â n )} +ĉurlÂ n . 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-Doniach energy 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 threedimensional anisotropic Ginzburg-Landau model. In this model, a mass tensor with unequal principal 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 H 1 (D; C) × E, the anisotropic Ginzburg-Landau energy G ǫ AGL is given by (1.8) Here λ is the same constant as in the Lawrence-Doniach energy. The connection between the Lawrence-Doniach energy G ǫ,s LD and the anisotropic Ginzburg-Landau energy G ǫ AGL when ǫ is fixed and s tends to 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 the anisotropic Ginzburg-Landau energy. Gamma convergence of the Lawrence-Doniach energy in this case to the anisotropic Ginzburg-Landau energy was proved in [11].
Our last result concerns the asymptotic behavior of the two energies as both ǫ and s tend to zero. We prove that, under an additional assumption on s versus ǫ, the difference between the two minimum energies is negligible compared to the leading term in the minimal Lawrence-Doniach energy. More precisely, we have the following theorem: If in addition we assume that s ≤ Cǫ for all ǫ sufficiently small where C is a constant independent of ǫ, then (See Theorem 5.6 and Theorem 5.7.) In the case when ǫ 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 continuous one. The situation considered here is more delicate, since the interlayer distance s is allowed to be at the same order as the characteristic vortex size ǫ, in which case the discrete nature of the Lawrence-Doniach model plays a more important role. Note that the assumption s ≤ Cǫ is only needed in Theorem 3. The previous two theorems concerning the minimum Lawrence-Doniach energy hold even for the very discrete case (e.g., 1 ≫ s ≫ ǫ).
Our paper is organized as follows: In Section 2 we state some preliminary results concerning the single layer potential representation formulas forÂ andÂ n . In Section 3 we prove the upper bound on the minimal Lawrence-Doniach energy. In Section 4 we prove (1.7) and use it to prove the lower bound on the minimal Lawrence-Doniach energy, as well as the corollaries stated above. Finally in Section 5 we prove the comparison result between the minimal Lawrence-Doniach energy and the minimal three-dimensional anisotropic Ginzburg-Landau energy and its consequence as summarized in Theorem 3.

Preliminaries
As noted in the introduction, the Lawrence-Doniach energy is invariant under the gauge transformation (1.3) and minimizers of G ǫ,s LD are gauge-equivalent to a minimizer in the "Coulomb gauge". It was proved by Bauman and Ko in [2] that, for a minimizer ({u n } N n=0 , A) of G ǫ,s LD in the "Coulomb gauge", the magnetic potential A has an explicit representation formula using single layer potentials. Recall the definition of the spaceȞ 1 (R 3 ) in the introduction. From [2], each C ∈Ȟ 1 (R 3 ) has a representative in L 6 (R 3 ; R 3 ) such that We remark that the Lawrence-Doniach energy G ǫ,s LD considered here is different from that studied in [3], [2] and [11], via a simple rescaling in the energy and in the magnetic potentials. (The scaling used here is the same as that used by Sandier and Serfaty in [9] which has been very successful in analyzing minimizers for the two-dimensional Ginzburg-Landau energy as ǫ tends to zero.) More precisely, setting κ = 1 ǫ and letting G κ LD be the Lawrence-Doniach energy studied in [3] with ψ n as the order parameter for the nth layer and A LD as the magnetic potential for G κ LD , respectively, Similar rescaling holds for the anisotropic Ginzburg-Landau energy, i.e., where G κ EM is the anisotropic Ginzburg-Landau (or effective mass) energy introduced in [3], and ψ EM and A EM are the order parameter and the magnetic potential for G κ EM , respectively. The above formulas 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 A 1 , A 2 , A 1 n and A 2 n 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 these formulas, we first define the single layer potential for our setting. For each k ∈ {0, 1, ... , N }, and for a given function g ∈ L p (Ω × {ks}) with 1 < p < ∞, we define the operator S k by where dσ denotes the surface measure on the plane and c = − 1 4π . (See [4] and [10] for results on layer potentials in smooth and Lipschitz domains, respectively.) With the above definitions, from the formulas in [2], we have and a i n (x) = a i (x, ns) corresponds to the trace of a i in R 2 × {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 limits and nontangential maximal functions. For fixed R > 0 and 0 < θ < π/2, let Γ ≡ Γ R,θ = {x ∈ R 3 : |x| < R and |x · e 3 | > |x|cosθ} be a cone nontangential to the plane {x 3 = 0} with vertex at the origin. Denote by Similarly, denote by 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.limit provided the limit exists. Also we have the following definition of the nontangential maximal function . Throughout this paper, we let θ = π 4 and R = 1. Also, let R 0 be a fixed constant satisfying where diam Ω is the diameter of Ω. It follows that the nontangential maximal functions of S n (g i n ) and ∇S n (g i n ) are in L 2 loc (R 2 × {ns}) and for some constant C depending only on R 0 . (This property of the constant C uses the fact that Ω is a subset of a disk of radius R 0 in R 2 . See [2].) Also t i n (x, ns) and∇t i n (x, ns) are the nontangential limits of S n (g i n ) and∇S n (g i n ), respectively, pointwise a.e. in R 2 × {ns} and in L 2 loc (R 2 × {ns}), and we have where P.V. denotes the principal-valued integral. In addition, we have for some constant C depending only on R 0 . 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 GL ǫ and F ǫ over bounded simply connected Lipschitz domains, which is a modification of Proposition 3.4 in [9].
Then we have min Proof. Note that for any function (v, b) in X, we have (v, b| Ω ) ∈ X Ω . From this and the definitions of GL ǫ and F ǫ , we obtain min .
. We may take φ to be the Newtonian potential of H. By standard estimates for Newtonian potentials, we . Direct calculations show that curl(b| Ω ) = curlb in Ω, whereb| Ω is the restriction ofb on Ω. Therefore there exists f ∈ H 2 (Ω) such thatb| Ω = b + ∇f in Ω. Defineṽ = ve if . Simple calculations using that H = curlb − h ex and gauge invariance imply that and equality must hold.

Upper bound
From here on in the paper, we let C 0 denote any constant that is independent of ǫ, s, Ω, L, D and R 0 for all ǫ and s sufficiently small. Recall that in our notation,V denotes a two-dimensional vector V = (V 1 , V 2 ) and W denotes a three-dimensional vector 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: for all ǫ and s sufficiently small, where Here C is a constant depending only on R 0 and L.
The main idea in the proof is to construct a test configuration with vanishing Josephson coupling term, such that its Lawrence-Doniach energy G ǫ,s LD from inside the domain D is a sum of N +1 identical copies of the two-dimensional Ginzburg-Landau energy F ǫ in Ω. In this way we may apply the upper bound estimate for the minimal two-dimensional Ginzburg-Landau energy from Proposition 8.1 in [9] to obtain the leading energy of G ǫ,s LD . The technical difficulty comes from extending the magnetic potential A outside the domain D appropriately so that the energy contribution from R 3 \ D is of a lower order compared to that in D.
We show that the test configuration ({v n } N n=0 , B) defined above gives us the desired upper bound for the Lawrence-Doniach energy G ǫ,s LD . It is easy to see that, since the v n 's are all equal and B 3 ≡ 0, the Josephson coupling term (1.5) vanishes. Therefore It follows from (3.3) and the definitions of ξ and η that Since (ξ∆φ +∇ξ ·∇φ) 2 does not depend on x 3 and N s = L, we have Also sinceB n =B Ω and v n = v for all n = 0, 1, ... , N , we have Therefore we may write G ǫ,s LD ({v n } N n=0 , B) as a sum: where and By the symmetry property of η, it is obvious that I 2 = I 3 . In the following lemmas, we prove several estimates from which we obtain (in Lemma 3.5) that for all ǫ and s sufficiently small where C is a constant depending only on R 0 and L. This concludes the proof.
Our first lemma concerns the L 2 norm of the function H(x) = (hx 0 ǫ (x) − h ex ) · χ Ω (x). Note that H is independent of s.
for all ǫ sufficiently small and C 0 as described above.
Proof. Let K ǫ = (− 1 2θ , 1 2θ ) × (− 1 2θ , 1 2θ ) and Kx 0 ǫ be the translation of K ǫ byx 0 . Then we show that for all ǫ sufficiently small, Indeed, let {K i } be the collection of cubes formed by the lattice L ǫ in R 2 such that K i ∩ Ω = ∅ and Ω ⊂ i K i . Since hx 0 ǫ − h ex is periodic with respect to L ǫ , we have When ǫ is sufficiently small, i |K i | ≤ 2|Ω|, from which (3.5) follows immediately. By Proposition 3.2 Combining the above inequality with claim (3.5) and using the fact that |K ǫ | = ( 1 θ ) 2 = 2π hex , we have for all ǫ sufficiently small. Proof. Recall that∇φ(x) = Ω∇x Γ 2 (x−ŷ)H(ŷ)dŷ and ξ and φ are independent of s. Also recall that, by the definition of ξ,∇ξ is supported on B R+1 \ B R and |∇ξ| 2 ≤ 4. Therefore for all ǫ sufficiently small, It follows from the inequality |∇xΓ 2 (x −ŷ)| ≤ C0 |x−ŷ| that Since Ω ⊂ B R 2 , we have |x −ŷ| ≥ R 2 , which along with Hölder's inequality implies and thus for all ǫ sufficiently small and some constant C depending only on R 0 . Lemma 3.4. For I 1 as defined above, we have for all ǫ and s sufficiently small and some constant C depending only on R 0 .
Proof. Using the equation (3.2) and the definition of ξ(x), we have From this and lemmas 3.2 and 3.3 we obtain Thus from the definition of I 1 , By (3.4) and the identity sN |Ω| = |D| we have for all ǫ and s sufficiently small and some constant C depending only on R 0 .
Lemma 3.5. I 2 defined as above satisfies I 2 ≤ C |D| 2 h ex and for all ǫ and s sufficiently small and some constant C depending only on R 0 and L.
Proof. We write I 2 as where Recall that, by our choice of η and ξ, we have |η ′ | ≤ 2 d and |ξ| ≤ 1 is supported on B R+1 . Therefore Using∇φ(x) = Ω∇x Γ 2 (x −ŷ)H(ŷ)dŷ and Hölder's inequality, we obtain For any fixedx ∈ B R+1 and for everyŷ ∈ Ω ⊂ B R 2 , we have 0 ≤ |x −ŷ| ≤ (R + 1) where B(0, 5R 2 ) is the disk centered at the origin with radius 5R 2 in R 2 . Similarly, by the symmetry of Γ 2 inx andŷ and almost exactly the same arguments, we get for anyŷ ∈ Ω. It follows from (3.8), Fubini's theorem, (3.9) and Lemma 3.2 that For I 2,2 we use (3.7) and the fact that |η| ≤ 1 to get Therefore we obtain for all ǫ and s sufficiently small and some constant C depending only on R 0 and L. Now combining inequalities (3.6) and (3.10) we have for all ǫ and s sufficiently small and some constant C depending only on R 0 and L.

Lower bound
In this section we prove the lower bound in Theorem 1. This relies on approximating the energy of the magnetic term |ĉurlÂ − h ex | 2 by the sum of its traces |ĉurlÂ n − h ex | 2 on the layers Ω n in the thin domains Ω × [ns, (n + 1)s). We first show that the error from this approximation is indeed of a lower order compared to the leading order term of the total energy. Proof. We shall use the single layer potential representation formulas for A(x) − h ex a(x) andÂ n (x) − h exân (x) proved by Bauman and Ko in [2] for ({u n } N n=0 , A) as above. First note that since a(x) = (0, x 1 , 0) is independent of x 3 , we havê and it follows from this, (2.6) and (2.7) and the regularity results described in Section 2 that in Ω (see [2]), we have Applying the elementary inequality (a−b) 2 ≤ 2a 2 +2b 2 to the representation formula forĉurlÂ(x, x 3 )− curlÂ n (x) and taking the sum of the integrals, we obtain In the following we analyze E 1 and the analysis for E 2 will be similar. First define for n = k and ∆ n,n (x, Note that for n = k, ∆ n,k is C ∞ in R 3 \ Ω k since it is harmonic there. By the Cauchy-Schwartz inequality, ns Ω where 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| ≤ s 1−α → 0 as s → 0. Therefore, for s sufficiently small and for each n = k in {0, 1, ... , N − 1} satisfying |n − k| ≤ s −α , the following holds for any (x, x 3 ) ∈ Ω × [ns, (n + 1)s): and thus where from Section 2, [ ∂ ∂x1 S k (g 2 k )] * (x, ks) is the nontangential maximal function of the tangential derivative ∂ ∂x1 S k (g 2 k ) at the point (x, ks) on the kth layer Ω k . By (2.8) we have where C is a constant depending only on R 0 . For n = k, we know from (2.9) that ∇ t 2 k L 2 (Ω) ≤ C g 2 k L 2 (Ω×{ks}) . Hence for all s sufficiently small and some constant C depending only on R 0 . In order to estimate E 1,2 , consider n in {0, 1, ... , N − 1} such that |n − k| > s −α . Recall that Then it is clear that Ω × [ns, (n + 1)s] ⊂ D U for every n satisfying the above assumptions. Take some bounded is harmonic in D k and for each x ∈ D k , it follows from (2.5) and Hölder's inequality that and therefore sup ∈ Ω × [ns, (n + 1)s], we have (since (x, x 3 ) ∈ Ω × [ns, (n + 1)s] ⊂⊂ R 3 \ Ω k and ∆ n,k is harmonic in By Theorem 2.10 in [5] and the fact that dist(D U , ∂D k ) ≥ s 1−α 2 , we have Hence we obtain |∆ n,k (x, and therefore To get the best rate of convergence in s as s → 0, we may take α = 5 7 so that 1 − α = 6α − 4 = 2 7 . The above estimate then becomes Integrating over [ns, (n + 1)s] and observing that the cardinality of the indices for the summation on n in E 1,2 is less than N and N s = L is fixed, we obtain for some constant C depending only on R 0 and L. Combining (4.1), (4.2), (4.3) and (4.4) yields Similarly we have To finish the proof of the lemma, note that is part of the Lawrence-Doniach energy, it follows from Theorem 3.1 that Hence, it follows from this, (4.5) and (4.6) that for all ǫ and s sufficiently small and some constant C depending only on R 0 and L.
The above lemma provides the main step in our proof of the lower bound on the minimal Lawrence-Doniach energy.
for all ǫ and s sufficiently small, where C is a constant depending only on R 0 and L.
Proof. By dropping the nonnegative Josephson coupling term and the square of the L 2 norm of the first two components of ∇ × A − h ex e 3 , it is clear that Applying the elementary inequality (a + b) 2 ≥ a 2 − 2|a| · |b| yields ns Ω |ĉurl(Â −Â n )| · |ĉurlÂ n − h ex |dxdx 3 .
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 the minimum Lawrence-Doniach energy comes from two terms, since for some constant C depending only on R 0 and L and for all ǫ and s sufficiently small. As a result of (4.13) and Theorem 3.1, we conclude that This proves Theorem 2.

Comparison results
In this section we prove a comparison result between the minimum Lawrence-Doniach energy and the minimum three-dimensional anisotropic Ginzburg-Landau energy under the assumption that s ≤ C 0 ǫ for some constant C 0 independent of ǫ, s, Ω, L, D and R 0 for all ǫ and s sufficiently small. Recall the definition of the anisotropic Ginzburg-Landau energy G ǫ AGL given in (1.8) in the introduction. Direct calculations show that G ǫ AGL is invariant under the gauge transformation for some g ∈ H 2 loc (R 3 ). 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) ∈ H 1 (D; C) × E of G ǫ AGL is gauge equivalent to another pair in H 1 (D; C) × K, where the spaces E and K are defined in (1.2) and (1.4) respectively. The space H 1 (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 for an upper bound on the minimal three-dimensional anisotropic Ginzburg-Landau energy.
L 6 (D) ≤ C 0 · M ǫ for all ǫ and s sufficiently small and some constant C 0 independent of ǫ, s, Ω, L, D and R 0 .
In the following we will need several additional lemmas which follow from Lemma 5.1, Lemma 5.2 and calculations from the proof of Lemma 5.5 in [3] (after appropriate rescaling). We remark that Lemma 5.2 is a stronger estimate than the analogous estimate in [3] and [11] (in which ǫ was fixed), and it is a key ingredient in our proof of Theorem 3.
where t n = x3−ns s . Then we have Proof. By the definition of ψ, we have Here we use the splitting in the proof of Lemma 5.5 in [3]. EachR 1j −R 2j corresponds to the quantity R 1j − R 2j in the proof of Lemma 5.5 in [3] for j = 1, 2, 3.
Note that |1 − ıst n A 3 | 2 = 1 + (st n A 3 ) 2 and, by the definition of t n , st n = x 3 − ns. Therefore we haveR From the estimates for |R 11 − R 21 | in the proof of Lemma 5.5 in [3] and using the rescaling relations (2.3) and (2.4) (in particular, note that ψ n = u n and κA z = A 3 ), we have It follows from (5.3) that Using similar calculations as in the proof of Lemma 5.2, it is not hard to show that for some constant C depending only on |D| for all ǫ sufficiently small. Then applying Hölder's inequality to D (A 3 ) 4 dx as in the proof of Lemma 5.2 and using Lemma 5.1, we deduce from (5.4) thatR 11 −R 21 ≤ Cs · M ǫ · M ǫ for some constant C depending only on |D|. From the proof of Lemma 5.2, s √ M ǫ → 0 as ǫ → 0. It follows thatR dx.