Spectral asymptotics for a singularly perturbed fourth order locally periodic elliptic operator

Abstract

We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic operator with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable lead to localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order effective operators with constant coefficients.

Keywords

homogenization spectral problem higher order equations localization of eigenfunctions

1. Introduction

We study the spectral asymptotics of a self-adjoint fourth order elliptic operator with locally periodic coefficients. The problem is stated in a bounded domain, and we impose Dirichlet boundary conditions on the boundary of the domain. The problem can be viewed as a combination of homogenization and singular perturbation. Because of the rapidly varying coefficients, homogenization arguments can be applied after a proper rescaling of the equation. As a result, we obtain an effective problem stated in the full space, which is of fourth or second order, depending on the choice of the large parameters ( $ε^{- α}$ , $ε^{- β}$ in Eq. (1)). We focus on values of the parameters for which the eigenfunctions localize and prove that the eigenfunctions of the original problem converge to the properly rescaled eigenfunctions of the effective problem.

Similar problems for second order locally periodic elliptic operators, that are closely related to the present paper, were studied in [20,21]. Dependence of the problem on a slow variable, in the coefficients or in the geometry, together with the presence of a large parameter in the equation gives rise to the effect of localization of eigenfunctions. These results correspond to the so-called subcritical case, when eigenfunctions can be approximated by eigenfunctions of a harmonic oscillator operator.

A second order locally periodic elliptic operator with a large potential was studied in [4]. Homogenization of periodic elliptic systems with large potential was treated in [2]. In both cases, under a generic assumption on the ground state of an auxiliary cell problem, it was proved that the solution can be approximately factorized as the product of a fast oscillating cell eigenfunction and of a slowly varying solution of a scalar second order equation. These two cases correspond to the so-called critical case.

There is a vast literature devoted to the homogenization of elliptic systems and higher order elliptic equations in domains with microstructure. For the homogenization of linear elliptic systems we refer to [6,15,18,24]. Homogenization of boundary value problems for higher order equations in domains with fine-grained boundary were studied in [10,11,16,19]. Homogenization of linear higher order equations in perforated domains were studied in [7,12,23]; nonlinear higher order equations in perforated domains were considered in [9,13]. In [17] spectral asymptotics for a fourth order elliptic operator with rapidly oscillating coefficients were obtained. Spectral asymptotics for a biharmonic operator in a domain with a deeply indented boundary were constructed in [14].

2. Problem statement

Let Ω be a bounded domain in $R^{d}$ with Lipschitz boundary $\partial Ω$ . We consider the following Dirichlet spectral problem for a fourth order self-adjoint uniformly elliptic operator: $\{\begin{matrix} \partial_{i j} (a_{i j k l}^{ε} \partial_{k l} u^{ε}) - \frac{1}{ε^{α}} \partial_{i} (b_{i j}^{ε} \partial_{j} u^{ε}) + \frac{1}{ε^{β}} c^{ε} u^{ε} = λ^{ε} u^{ε}, & x \in Ω, \\ u^{ε} = \nabla u^{ε} \cdot n = 0, & x \in \partial Ω, \end{matrix}$ (1) where n denotes the exterior unit normal to $\partial Ω$ . We use summation convention over repeated indices and use “·” for the standard scalar product in $R^{d}$ ; $ε > 0$ is a small parameter; α, β are positive parameters.

Our main assumptions are:

The coefficients are real and of the form $a_{i j k l}^{ε} (x) = a_{i j k l} (x, \frac{x}{ε})$ , $b_{i j}^{ε} (x) = b_{i j} (x, \frac{x}{ε})$ and $c^{ε} (x) = c (x, \frac{x}{ε})$ , where the functions $a_{i j k l} (x, y), b_{i j} (x, y) \in C (\bar{Ω}; L^{\infty} (T^{d}))$ , $c (x, y) \in C^{2} (\bar{Ω}; L^{\infty} (T^{d}))$ are periodic in y; $T^{d}$ is the torus of measure one. As a consequence, there is $Λ > 0$ such that $| a_{i j k l} |, | b_{i j} |, | c | ⩽ Λ^{- 1}$ .

Symmetry condition: $a_{i j k l} = a_{k l i j}$ , $b_{i j} = b_{j i}$ .

The coefficients $a_{i j k l} (x, y)$ satisfy the uniform ellipticity condition in $Ω \times T^{d}$ : there is $Λ > 0$ such that, almost everywhere, $a_{i j k l} (x, y) ξ_{i j} ξ_{k l} ⩾ Λ | ξ |^{2} \forall ξ \in R^{d \times d} .$

The function $c (x, y)$ is assumed to be strictly positive almost everywhere in $Ω \times T^{d}$ , and its local average $\bar{c} (x) = \int_{T^{d}} c (x, y) d y$ has a unique global minimum at $x = 0 \in Ω$ , with a non-degenerate Hessian $H = \nabla \nabla \bar{c} (0)$ : $\bar{c} (x) = \bar{c} (0) + \frac{1}{2} H x \cdot x + o (| x |^{2}) .$

The coefficients $b_{i j} (x, y)$ satisfy the uniform ellipticity condition in $Ω \times T^{d}$ : there is $Λ^{'} > 0$ such that, almost everywhere, $b_{i j} (x, y) ξ_{i} ξ_{j} ⩾ Λ^{'} | ξ |^{2} \forall ξ \in R^{d} .$

For $Q \subset R^{d}$ we denote the $L^{2} (Q)$ -norm of u by $∥ u ∥_{2, Q}^{2} = \int_{Q} | u |^{2} d x$ , and $H_{0}^{2} (Q) = {v \in H^{2} (Q) : v = \nabla v \cdot n = 0 on \partial Q} .$

We consider the following bilinear form corresponding to (1) $A_{ε} (u, v) = \int_{Ω} a_{i j k l}^{ε} \partial_{i j} u \partial_{k l} v d x + \frac{1}{ε^{α}} \int_{Ω} b_{i j}^{ε} \partial_{i} u \partial_{j} v d x + \frac{1}{ε^{β}} \int_{Ω} c^{ε} u v d x .$ (2) The weak form of (1) reads: Find $λ^{ε} \in C$ and nonzero $u^{ε} \in H_{0}^{2} (Ω)$ such that $A_{ε} (u^{ε}, v) = λ^{ε} \int_{Ω} u^{ε} v d x$ (3) for all $v \in H_{0}^{2} (Ω)$ .

By the Riesz–Schauder, Hilbert–Schmidt theorems and the minmax principle ([8,22]), for each ε small enough, we have the following characterization of the spectrum.

Lemma 2.1.

Suppose that (H1)–(H4) are satisfied. Then for all sufficiently small $ε > 0$ , the eigenvalues $λ_{i}^{ε}$ of (1) are real and such that $0 < λ_{1}^{ε} ⩽ λ_{2}^{ε} ⩽ \dots, λ_{i}^{ε} \to \infty as i \to \infty,$ where each eigenvalue is counted as many times as its multiplicity. The eigenfunctions $u_{i}^{ε}$ form an orthonormal basis in $L^{2} (Ω)$ . All eigenvalues are of finite multiplicity and are characterized by the variational principle: $λ_{i}^{ε} = min \frac{A_{ε} (v, v)}{\int_{Ω} v^{2} d x},$ where the minimum is taken over all nonzero functions v in $H_{0}^{2} (Ω)$ that are orthogonal in $L^{2} (Ω)$ to the first $i - 1$ eigenfunctions $u_{1}^{ε}, \dots, u_{i - 1}^{ε}$ .

The goal of this paper is to describe the asymptotic behavior of the eigenpairs $(λ_{k}^{ε}, u_{k}^{ε})$ , as $ε \to 0$ . We restrict ourselves to the values of the parameters $α ⩾ 0$ , $β > 0$ (singular perturbation) such that $α < β$ (the concentration effect is observed) and $β < 4$ (subcritical case). The result is presented in the three Theorems 3.1, 4.1 and 5.1.

In the case $α = β$ we get no concentration, and the standard two-scale convergence can be applied to describe the asymptotics of eigenpairs. Depending on the value of α one gets either fourth order limit operator without second order terms ( $0 < α = β < 2$ ), or fourth order with second order terms ( $α = β = 0$ , $α = β = 2$ ), or just a second order limit operator ( $α = β > 2$ ).

The case $β = 4$ , $α < 4$ is the critical case, when the oscillations of the eigenfunctions are expected to be of order ε. As it is seen from [2], the technique to be used is different, and this case is to be considered elsewhere. In addition, the values of α, β such that $3 ⩽ β < 4$ and $α < β - 2$ are not covered by the present paper (the hatched region in Fig. 1), because the error coming from Lemma 3.3 while passing to the limit does not vanish, as $ε \to 0$ (see the proof of Lemma 3.7). In that region one has to consider the oscillations coming from the potential more carefully, which is also the case for the corresponding region for the second order operator, and it is to be considered elsewhere.

To describe the asymptotic behavior of eigenpairs $(λ_{k}^{ε}, u_{k}^{ε})$ as $ε \to 0$ , we divide the domain for the parameters $(α, β)$ into the following regions (see Fig. 1): $\begin{array}{rcl} R_{1} = {(α, β) : 0 ⩽ α < 1, 3 α < β < 3}, \\ R_{2} = {(α, β) : 0 < α < 1, β = 3 α}, \\ R_{3} = {(α, β) : 0 < α < 2, α < β < 3 α, β < α + 2}, \\ R_{4} = {(α, β) : α = 2, 2 < β < 4}, \\ R_{5} = {(α, β) : 2 < α < 4, α < β < 4} . \end{array}$ The reason for distinguishing these regions is that we get different asymptotics in each case. In short, in $R_{1}$ we get a fourth order equation in the limit without second order terms; in $R_{2}$ the limit equation contains both fourth and second order terms; in $R_{3}$ , $R_{4}$ , $R_{5}$ the limit equations are of the second order. We choose to consider in details one case $β = 3 α = 1$ , corresponding to region $R_{2}$ , since all the terms contribute in the limit (see Theorem 3.1). The spectral asymptotics in the other cases are described in Sections 4 and 5.

The results in this paper are presented as Theorems 3.1, 4.1 and 5.1. The rest of this paper is devoted to the proofs of these theorems.

Fig. 1.

The partition of the parameter region for $(α, β)$ .

3. A model problem: The case

(α, β) \in R_{2}

The result of this section is contained in the following theorem.

Theorem 3.1.
Let $(α, β) \in R_{2}$ and let $(λ_{k}^{ε}, u_{k}^{ε})$ be the kth eigenpair of (1) normalized by $∥ u_{k}^{ε} ∥_{2, Ω}^{2} = ε^{d α / 2}$ . Suppose that the conditions (H1)–(H4) are satisfied. Then we have the following representation: $λ_{k}^{ε} = \frac{\bar{c} (0)}{ε^{3 α}} + \frac{η_{k}^{ε}}{ε^{2 α}}, u_{k}^{ε} (x) = v_{k}^{ε} (\frac{x}{ε^{α / 2}}),$ where $(η_{k}^{ε}, v_{k}^{ε})$ are such that, as $ε \to 0$ ,
$η_{k}^{ε} \to η_{k}$ ,

up to a subsequence, $v_{k}^{ε}$ converges to $v_{k}$ weakly in $H^{2} (R^{d})$ and strongly in $L^{2} (R^{d})$ ,
where $η_{k}$ is the kth eigenvalue and $v_{k}$ is an eigenfunction corresponding to $η_{k}$ , normalized by $∥ v_{k} ∥_{2, R^{d}} = 1$ , of the uniformly elliptic effective problem $\partial_{i j} (a_{i j k l}^{eff} \partial_{k l} v) - \partial_{i} ({\bar{b}}_{i j} (0) \partial_{j} v) + \frac{1}{2} (H z \cdot z) v = η v, z \in R^{d},$ with $a_{i j k l}^{eff}$ defined by (18) , ${\bar{b}}_{i j} (0) = \int_{T^{d}} b_{i j} (0, y) d y$ , and $H = \nabla \nabla \bar{c} (0)$ .

The proof of Theorem 3.1 will occupy the rest of this section and is given for the case $β = 3 α = 1$ . The argument used is the same for the other values of $(α, β) \in R_{2}$ . Remark 1.
The localization of eigenfunctions appears in the current problem due to the simultaneous presence of a slow variable in the coefficients and a singular perturbation. Neither locally periodic coefficients, nor the singular perturbation alone will lead to the concentration effect. Note that it is not necessary that the dependence on the slow variable is in the coefficients of the operator: it can be in the geometry of the domain instead (see, for example, [20] for the case of locally periodic perforation).

3.1. Estimates for eigenvalues of the original problem

To motivate the change of variables we will make in the next subsection, we prove the following a priori estimates for the eigenvalues and the eigenfunctions of problem (1).

Lemma 3.2.
Suppose that (H1)–(H4) are satisfied. Let $(λ_{i}^{ε}, u_{i}^{ε})$ be the ith eigenpair of (1) with $β = 3 α = 1$ , normalized by $∥ u_{i}^{ε} ∥_{2, Ω} = 1$ . Then there exist positive constants $C_{1}$ , $C_{2} (i)$ such that for all sufficiently small $ε > 0$ , $- \frac{C_{1}}{ε^{2 / 3}} ⩽ λ_{i}^{ε} - \frac{\bar{c} (0)}{ε} ⩽ \frac{C_{2} (i)}{ε^{2 / 3}}, {∥ \nabla \nabla u_{i}^{ε} ∥}_{2, Ω} ⩽ \frac{C}{ε^{1 / 3}} .$

To prove Lemma 3.2 we will use the following estimate for integrals of oscillating functions. Lemma 3.3.
Let $g (x, y) \in C^{2} (\bar{Ω}; L^{\infty} (T^{d}))$ be such that $\int_{T^{d}} g (x, y) d y = 0$ for all $x \in \bar{Ω}$ . Then there exists a positive constant C such that $| \int_{Ω} g (x, \frac{x}{ε}) u v d x | ⩽ C ε^{2} (∥ Δ u ∥_{2, Ω} ∥ v ∥_{2, Ω} + ∥ u ∥_{2, Ω} ∥ Δ v ∥_{2, Ω})$ for all $u, v \in H_{0}^{2} (Ω)$ .
Proof.
Let $Ψ (x, y) \in C^{2} (\bar{Ω}; C^{1} (T^{d}))$ , periodic in y, be defined by $Δ_{y} Ψ (x, y) = g (x, y), y \in T^{d} .$ Since the local average of $g (x, y)$ is zero, Ψ is well-defined. By the Green formula, $\begin{array}{rcl} \int_{Ω} g (x, \frac{x}{ε}) u (x) v (x) d x \\ = \int_{Ω} (Δ_{y} Ψ) (x, \frac{x}{ε}) u (x) v (x) d x \\ = ε^{2} \int_{Ω} (Δ Ψ (x, \frac{x}{ε}) - 2 div ((\nabla_{x} Ψ) (x, \frac{x}{ε})) + (Δ_{x} Ψ) (x, \frac{x}{ε})) u (x) v (x) d x \\ = ε^{2} \int_{Ω} (Ψ (x, \frac{x}{ε}) Δ (u v) + 2 (\nabla_{x} Ψ) (x, \frac{x}{ε}) \cdot \nabla (u v) + (Δ_{x} Ψ) (x, \frac{x}{ε}) u v) d x . \end{array}$ After an application of the Green formula to the $\nabla u \cdot \nabla v$ term coming from $Δ (u v)$ , by the Hölder and triangle inequalities, we have $\begin{array}{rcl} | \int_{Ω} g (x, \frac{x}{ε}) u (x) v (x) d x | \\ ⩽ ε^{2} (∥ Ψ ∥_{L^{\infty} (Ω \times T^{d})} (∥ Δ u ∥_{2, Ω} ∥ v ∥_{2, Ω} + ∥ u ∥_{2, Ω} ∥ Δ v ∥_{2, Ω}) \\ + 2 ∥ \nabla_{x} Ψ ∥_{L^{\infty} (Ω \times T^{d})} (∥ \nabla u ∥_{2, Ω} ∥ v ∥_{2, Ω} + ∥ u ∥_{2, Ω} ∥ \nabla v ∥_{2, Ω}) \\ + ∥ Δ_{x} Ψ ∥_{L^{\infty} (Ω \times T^{d})} ∥ u ∥_{2, Ω} ∥ v ∥_{2, Ω}) . \end{array}$ The estimate follows from the regularity of Ψ and the Poincaré inequality. □
Proof of Lemma 3.2.
Let $v \in C_{0}^{\infty} (R^{d})$ be such that $∥ v ∥_{2, R^{d}} = 1$ , and set $v^{ε} (x) = v (ε^{- 1 / 6} x)$ . We assume that ε is small enough such that $supp v^{ε} \subset Ω$ . By the variational principle, $λ_{1}^{ε} ⩽ \frac{\int_{Ω} a_{i j k l}^{ε} \partial_{i j} v^{ε} \partial_{k l}^{2} v^{ε} d x + ε^{- 1 / 3} \int_{Ω} b_{i j}^{ε} \partial_{i} v^{ε} \partial_{j} v^{ε} d x}{\int_{Ω} {(v^{ε})}^{2} d x} + \frac{1}{ε} \frac{\int_{Ω} c^{ε} {(v^{ε})}^{2} d x}{\int_{Ω} {(v^{ε})}^{2} d x} .$ (4) By the boundedness of the coefficients, the first fraction in (4) is bounded by $C ε^{- 2 / 3}$ . The second fraction in (4) is estimated using (H4) and Lemma 3.3: $\begin{array}{rcl} \frac{1}{ε} \frac{\int_{Ω} c (x, x / ε) {(v^{ε})}^{2} d x}{\int_{Ω} {(v^{ε})}^{2} d x} & = & \frac{\bar{c} (0)}{ε} + \frac{1}{ε} \int_{R^{d}} (\bar{c} (ε^{1 / 6} x) - \bar{c} (0)) v^{2} d x \\ + \frac{1}{ε} \int_{R^{d}} (c (ε^{1 / 6} x, \frac{x}{ε^{5 / 6}}) - \bar{c} (ε^{1 / 6} x)) v^{2} d x \\ ⩽ & \frac{\bar{c} (0)}{ε} + C ε^{- 2 / 3} \end{array}$ for sufficiently small $ε > 0$ .

In order to obtain an estimate from below for the first eigenvalue $λ_{1}^{ε}$ , we need to estimate the second derivatives of the corresponding eigenfunctions. Let $u_{1}^{ε}$ denote any eigenfunction corresponding to $λ_{1}^{ε}$ , normalized by $∥ u_{1}^{ε} ∥_{2, Ω} = 1$ . Then, by (3), $λ_{1}^{ε} - \frac{1}{ε} \int_{Ω} c (x, \frac{x}{ε}) {(u_{1}^{ε})}^{2} d x = \int_{Ω} a_{i j k l}^{ε} \partial_{i j} u_{1}^{ε} \partial_{k l} u_{1}^{ε} d x + \frac{1}{ε^{1 / 3}} \int_{Ω} b_{i j}^{ε} \partial_{i} u_{1}^{ε} \partial_{j} u_{1}^{ε} d x .$ On the one hand, by the ellipticity of $a_{i j k l}$ and the boundedness of the coefficients, $\begin{array}{rclr} λ_{1}^{ε} - \frac{1}{ε} \int_{Ω} c (x, \frac{x}{ε}) {(u_{1}^{ε})}^{2} d x & ⩾ & Λ {∥ \nabla \nabla u_{1}^{ε} ∥}_{2, Ω}^{2} - C \frac{1}{ε^{1 / 3}} \int_{Ω} {| \nabla u_{1}^{ε} |}^{2} d x \\ ⩾ & Λ {∥ \nabla \nabla u_{1}^{ε} ∥}_{2, Ω}^{2} - C_{2} γ {∥ Δ u_{1}^{ε} ∥}_{2, Ω}^{2} - \frac{C_{2}}{4 γ} ε^{- 2 / 3} {∥ u_{1}^{ε} ∥}_{2, Ω}^{2} \\ ⩾ & C ({∥ \nabla \nabla u_{1}^{ε} ∥}_{2, Ω}^{2} - ε^{- 2 / 3}), & (5) \end{array}$ where $γ > 0$ in the Cauchy inequality is chosen small enough such that the resulting constant C is positive. One can choose γ that depends just on the ellipticity constant of $a_{i j k l}$ and the upper bound for $b_{i j}$ .

On the other hand, from the upper estimate for $λ_{1}^{ε}$ , $\begin{array}{rclr} λ_{1}^{ε} - \frac{1}{ε} \int_{Ω} c (x, \frac{x}{ε}) {(u_{1}^{ε})}^{2} d x \\ ⩽ \frac{\bar{c} (0)}{ε} - \frac{1}{ε} \int_{Ω} c (x, \frac{x}{ε}) {(u_{1}^{ε})}^{2} d x + C ε^{- 2 / 3} \\ = \frac{1}{ε} \int_{Ω} (\bar{c} (0) - \bar{c} (x)) {(u_{1}^{ε})}^{2} d x + \frac{1}{ε} \int_{Ω} (\bar{c} (x) - c (x, \frac{x}{ε})) {(u_{1}^{ε})}^{2} d x + C ε^{- 2 / 3} \\ ⩽ C (ε {∥ Δ u_{1}^{ε} ∥}_{2, Ω} + ε^{- 2 / 3}), & (6) \end{array}$ where we in the third step used that 0 is a minimum point for $\bar{c} (x)$ by (H4), and Lemma 3.3.

Combining (5) and (6), by the Cauchy inequality, we deduce that, ${∥ \nabla \nabla u_{1}^{ε} ∥}_{2, Ω}^{2} ⩽ C ε^{- 2 / 3} .$ (7)

We proceed with the estimate from below for $λ_{1}^{ε}$ . By (5) and (7) we have $\begin{array}{rcl} λ_{1}^{ε} & ⩾ & \frac{1}{ε} \int_{Ω} c (x, \frac{x}{ε}) {(u_{1}^{ε})}^{2} d x - \frac{C}{ε^{2 / 3}} \\ = & \frac{\bar{c} (0)}{ε} + \frac{1}{ε} \int_{Ω} (\bar{c} (x) - \bar{c} (0)) {(u_{1}^{ε})}^{2} d x + \frac{1}{ε} \int_{Ω} (c (x, \frac{x}{ε}) - \bar{c} (x)) {(u_{1}^{ε})}^{2} d x - \frac{C}{ε^{2 / 3}} . \end{array}$ By (H4) and Lemma 3.3, combined with (7), $λ_{1}^{ε} ⩾ \frac{\bar{c} (0)}{ε} - \frac{C}{ε^{2 / 3}}$ for all sufficiently small $ε > 0$ . In this way we have obtained the required estimates for the first eigenvalue $λ_{1}^{ε}$ . Since $λ_{1}^{ε}$ is the smallest eigenvalue, the estimate from below for $λ_{i}^{ε}$ , $i > 1$ , follows from the corresponding estimate for $λ_{1}^{ε}$ .

To estimate $λ_{i}^{ε} > λ_{1}^{ε}$ for $i > 1$ from above, one can use as a test function the projection of $v^{ε}$ onto the orthogonal complement of the span of the first $i - 1$ eigenvectors, with respect to the $L^{2} (Ω)$ inner product. Since the span is finite dimensional this projection is nonzero for all sufficiently small $ε > 0$ .

Let $m_{1}$ be the multiplicity of the first eigenvalue $λ_{1}^{ε} = λ_{2}^{ε} = \dots = λ_{m_{1}}^{ε}$ . We estimate from above $λ_{m_{1} + 1}^{ε}$ , similar arguments can be applied to estimate the other eigenvalues.

For $v \in C_{0}^{\infty} (R^{d})$ , we introduce $v^{ε} (x) = v (\frac{x}{ε^{1 / 6}})$ and denote $π_{ε, k} = \int_{Ω} v^{ε} u_{k}^{ε} d x$ , $k = 1, \dots, m_{1}$ . Then $V^{ε} (x) = v^{ε} (x) - \sum_{k} π_{ε, k} u_{k}^{ε} (x)$ is orthogonal in $L^{2} (Ω)$ to $span {u_{1}^{ε}, \dots, u_{m_{1}}^{ε}}$ . For convenience we assume the normalization condition ${∥ V^{ε} ∥}_{2, Ω}^{2} = {∥ v^{ε} ∥}_{2, Ω}^{2} - \sum_{k = 1}^{m_{1}} π_{ε, k}^{2} = ε^{d / 6} .$ (8) Using $V^{ε}$ as a test function in the variational principle, we deduce that $λ_{m_{1} + 1}^{ε} ⩽ ε^{- d / 6} (A_{ε} (v^{ε}, v^{ε}) - λ_{1}^{ε} \sum_{k} π_{ε, k}^{2}) .$ (9) By (H1)–(H3) and Lemma 3.3 we get $A_{ε} (v^{ε}, v^{ε}) - λ_{1}^{ε} \sum_{k} π_{ε, k}^{2} ⩽ \frac{\bar{c} (0)}{ε} {∥ v^{ε} ∥}_{2, Ω}^{2} - λ_{1}^{ε} \sum_{k} π_{ε, k}^{2} + C ε^{d / 6} ε^{- 2 / 3} .$ Due to (8) , (9) and the estimate from below for $λ_{1}^{ε}$ , $λ_{m_{1} + 1}^{ε} ⩽ \frac{\bar{c} (0)}{ε} + \frac{C_{1} \sum_{k} π_{ε, k}^{2}}{ε^{d / 6} ε^{2 / 3}} + \frac{C}{ε^{2 / 3}} .$ Due to the normalization condition for $u_{1}^{ε}$ , $\sum_{k} π_{ε, k}^{2} ⩽ m_{1} ε^{d / 6} ∥ v ∥_{2, R^{d}}^{2},$ thus $λ_{m_{1} + 1}^{ε} ⩽ \frac{\bar{c} (0)}{ε} + \frac{C_{2}}{ε^{2 / 3}},$ and the estimate is proved. □

3.2. Rescaling the problem and computing the asymptotics

Led by Lemma 3.2, we shift the spectrum of (1) by $\bar{c} (0) / ε$ and rescale such that the eigenvalues become bounded. Let $z = \frac{x}{ε^{1 / 6}}, v^{ε} (z) = u^{ε} (ε^{1 / 6} z), η^{ε} = ε^{2 / 3} (λ^{ε} - \frac{\bar{c} (0)}{ε}), Ω_{ε} = ε^{- 1 / 6} Ω .$ (10) Then (3) takes the form $\begin{array}{rclr} \int_{Ω_{ε}} {\hat{a}}_{i j k l}^{ε} \partial_{i j} v^{ε} \partial_{k l} φ d z + \int_{Ω_{ε}} {\hat{b}}_{i j}^{ε} \partial_{i} v^{ε} \partial_{j} φ d z + \frac{1}{ε^{1 / 3}} \int_{Ω_{ε}} ({\hat{c}}^{ε} - \bar{c} (0)) v^{ε} φ d z \\ = η^{ε} \int_{Ω_{ε}} v^{ε} φ d z, & (11) \end{array}$ for any $v \in H_{0}^{2} (Ω_{ε})$ , where ${\hat{a}}_{i j k l}^{ε} (z) = a (ε^{1 / 6} z, \frac{z}{ε^{5 / 6}}), {\hat{b}}_{i j}^{ε} (z) = b (ε^{1 / 6} z, \frac{z}{ε^{5 / 6}}), {\hat{c}}^{ε} (z) = c (ε^{1 / 6} z, \frac{z}{ε^{5 / 6}}) .$

For the rest of $R_{2}$ , one uses $z = ε^{- α / 2} x$ .

In order to describe the asymptotic behavior of eigenpairs $(η_{k}^{ε}, v_{k}^{ε})$ of (11), we consider the Green operator $\begin{array}{rcl} G_{ε} : L^{2} (Ω_{ε}) \to L^{2} (Ω_{ε}), \\ f_{ε} \mapsto V^{ε}, \end{array}$ where $V^{ε} \in H_{0}^{2} (Ω_{ε})$ is the unique solution to the boundary-value problem $\{\begin{matrix} \partial_{i j} ({\hat{a}}_{i j k l}^{ε} \partial_{k l} V^{ε}) + \partial_{i} ({\hat{b}}_{i j}^{ε} \partial_{j} V^{ε}) + \frac{{\hat{c}}^{ε} - \bar{c} (0)}{ε^{1 / 3}} V^{ε} + μ V^{ε} = f_{ε}, & z \in Ω_{ε}, \\ V^{ε} = \nabla V^{ε} \cdot n = 0, & z \in \partial Ω_{ε} . \end{matrix}$ (12) Here $μ > 0$ is a large enough constant, but depending just on the ellipticity constant Λ. The operator $G_{ε}$ can be considered as an operator from $L^{2} (R^{d})$ into itself by extending the corresponding solution $V^{ε}$ by zero outside $Ω_{ε}$ . The existence and uniqueness is ensured by the Riesz–Fréchet representation theorem since the corresponding symmetric quadratic form $\begin{array}{rclr} A_{ε} (V^{ε}, V^{ε}) & = & \int_{Ω_{ε}} {\hat{a}}_{i j k l}^{ε} \partial_{k l} V^{ε} \partial_{i j} V^{ε} d z + \int_{Ω_{ε}} {\hat{b}}_{i j}^{ε} \partial_{j} V^{ε} \partial_{i} V^{ε} d z \\ + \frac{1}{ε^{1 / 3}} \int_{Ω_{ε}} ({\hat{c}}^{ε} - \bar{c} (0)) {(V^{ε})}^{2} d z + μ \int_{Ω_{ε}} {(V^{ε})}^{2} d z & (13) \end{array}$ is coercive. Indeed, by (H1)–(H4) and Lemma 3.3 we have: $\begin{array}{rclr} A_{ε} (V^{ε}, V^{ε}) & ⩾ & Λ \int_{Ω_{ε}} {| \nabla \nabla V^{ε} |}^{2} d z - Λ^{- 1} \int_{Ω_{ε}} {| \nabla V^{ε} |}^{2} d z \\ - C_{0} ε^{4 / 3} {∥ V^{ε} ∥}_{2, Ω_{ε}} {∥ Δ V^{ε} ∥}_{2, Ω_{ε}} \\ + \frac{1}{ε^{1 / 3}} \int_{Ω_{ε}} (\bar{c} (ε^{1 / 6} z) - \bar{c} (0)) {(V^{ε})}^{2} d z + μ {∥ V^{ε} ∥}_{2, Ω_{ε}}^{2} . & (14) \end{array}$ Since the Hessian matrix H is positive definite, there exist a positive constant $K_{1}$ such that $\bar{c} (ε^{1 / 6} z) - \bar{c} (0) ⩾ K_{1} {| ε^{1 / 6} z |}^{2} .$ Due to the Dirichlet boundary conditions, ${∥ \nabla V^{ε} ∥}_{2, Ω_{ε}}^{2} ⩽ {∥ V^{ε} ∥}_{2, Ω_{ε}} {∥ Δ V^{ε} ∥}_{2, Ω_{ε}} .$ (15) Applying the Cauchy inequality $∥ V^{ε} ∥ ∥ Δ V^{ε} ∥ ⩽ δ ∥ V^{ε} ∥^{2} + ∥ Δ V^{ε} ∥^{2} / δ$ with $δ = Λ / 3$ in (14) we get $\begin{array}{rcl} A_{ε} (V^{ε}, V^{ε}) & ⩾ & \frac{Λ}{3} \int_{Ω_{ε}} {| \nabla \nabla V^{ε} |}^{2} d z + K_{1} \int_{Ω_{ε}} | z |^{2} {(V^{ε})}^{2} d z \\ + (μ - \frac{3}{Λ^{3}} - \frac{2 C_{0} ε^{4 / 3}}{Λ}) {∥ V^{ε} ∥}_{2, Ω_{ε}}^{2} . \end{array}$ For ε small enough, we can choose μ depending just on Λ and $| Ω |$ such that, for some positive constant $\tilde{C}$ , we have $A_{ε} (V^{ε}, V^{ε}) ⩾ \tilde{C} ({∥ \nabla \nabla V^{ε} ∥}_{2, Ω_{ε}}^{2} + \int_{Ω_{ε}} | z |^{2} {(V^{ε})}^{2} d z + {∥ V^{ε} ∥}_{2, Ω_{ε}}^{2}) .$ (16) Even though $A_{ε} (V^{ε}, V^{ε}) ⩾ C ∥ Δ V^{ε} ∥_{2, Ω_{ε}}$ is enough for coercivity of the quadratic form, we will make use of the last inequality. The addition of the constant μ has the effect of shifting the entire spectrum of (11) by μ.

We introduce also the limit Green operator $\begin{array}{rcl} G : L^{2} (R^{d}) \to L^{2} (R^{d}), \\ f \mapsto V, \end{array}$ where $V \in H^{2} (R^{d})$ is the unique solution to the equation $\partial_{i j} (a_{i j k l}^{eff} \partial_{k l} V) - \partial_{i} ({\bar{b}}_{i j} (0) \partial_{j} V) + \frac{1}{2} (H z \cdot z) V + μ V = f, z \in R^{d} .$ (17) Here H is the Hessian matrix of $\bar{c}$ at $x = 0$ (see (H4)), and the effective coefficients are defined by $\begin{matrix} a_{i j k l}^{eff} = \int_{T^{d}} (a_{i j m n} (0, y) \partial_{m n} N_{k l} (y) + a_{i j k l} (0, y)) d y, \\ {\bar{b}}_{i j} (x) = \int_{T^{d}} b_{i j} (x, y) d y, \end{matrix}$ (18) where the periodic functions $N_{k l} \in H^{2} (T^{d}) / R$ solve the following cell problems: $\partial_{i j} (a_{i j m n} (0, y) \partial_{m n} N_{k l} (y)) = - \partial_{i j} a_{i j k l} (0, y), y \in T^{d} .$ (19) Due to the periodicity of $a_{i j k l}$ in y, the above problem is well-posed, the solution $N_{k l}$ is unique up to an additive constant.

Lemma 3.4.

Under the assumptions (H1)–(H4) , $a^{eff}$ defined by (18) is coercive on $R^{d \times d}$ , i.e. there is a positive constant C such that $a_{i j k l}^{eff} ξ_{i j} ξ_{k l} ⩾ C | ξ |^{2}$ for all $ξ \in R^{d \times d}$ .

Proof.

Due to (H1)–(H4), $a^{eff}$ is well-defined. We rewrite (18) as $a_{i j k l}^{eff} = \int_{T^{d}} δ_{p i} δ_{q j} a_{p q r s} (0, y) (δ_{r k} δ_{s l} + \partial_{r s} N_{k l}) d y .$ Using $N_{i j}$ as a test function in Eq. (19) for $N_{k l}$ we obtain $\int_{T^{d}} a_{p q r s} (0, y) (δ_{r k} δ_{s l} + \partial_{r s} N_{k l}) \partial_{p q} N_{i j} d y = 0 .$ Thus $a_{i j k l}^{eff} = \int_{T^{d}} a_{p q r s} (0, y) (δ_{r k} δ_{s l} + δ_{r s} N_{k l}) (δ_{p i} δ_{q j} + \partial_{p q} N_{i j}) d y .$ The last equation shows that $a^{eff}$ is symmetric by (H2): $a_{i j k l}^{eff} = a_{k l i j}^{eff}$ . Moreover, with $ξ \in R^{d \times d}$ we obtain $\begin{array}{rcl} a_{i j k l}^{eff} ξ_{i j} ξ_{k l} & = & \int_{T^{d}} a_{p q r s} (0, y) (ξ_{k l} (δ_{r k} δ_{s l} + δ_{r s} N_{k l})) (ξ_{i j} (δ_{p i} δ_{q j} + \partial_{p q} N_{i j})) d y \\ ⩾ & Λ \sum_{p, q} \int_{T^{d}} {| ξ_{i j} (δ_{p i} δ_{q j} + \partial_{p q} N_{i j}) |}^{2} d y, \end{array}$ by the coerciveness of $a (0, y)$ guaranteed by (H3). The last inequality implies that $a^{eff}$ is positive definite. Indeed, if $ξ \in R^{d \times d}$ is such that $a_{i j k l}^{eff} ξ_{i j} ξ_{k l} = 0$ , then $| ξ_{i j} (δ_{p i} δ_{q j} + \partial_{p q} N_{i j}) | = 0$ for all p, q. It follows that $\nabla ξ_{i j} (y_{i} δ_{q j} + \partial_{q} N_{i j}) = 0$ for all q. Therefore, $ξ_{i j} (y_{i} δ_{q j} + \partial_{q} N_{i j})$ is constant and so necessarily $ξ_{i q} y_{i}$ is periodic by the periodicity of $N_{i j}$ . Hence $ξ_{i q} = 0$ for all i, q. We conclude that for all $ξ \in R^{d \times d} ∖ {0}$ , $a_{i j k l}^{eff} ξ_{i j} ξ_{k l} > 0 .$ By the compactness of the unit ball in $R^{d \times d}$ , there is a positive constant C such that $a_{i j k l}^{eff} ξ_{i j} ξ_{k l} ⩾ C | ξ |^{2}$ for all $ξ \in R^{d \times d}$ . □

The bilinear form corresponding to (17) takes the form $A_{eff} (u, v) = \int_{R^{d}} (a_{i j k l}^{eff} \partial_{k l} u \partial_{i j} v + {\bar{b}}_{i j} \partial_{j} u \partial_{i} v + \frac{1}{2} (H z \cdot z) u v + μ u v) d z,$ and it is coercive. Namely, there exists a positive constant $\hat{C}$ such that $A_{eff} (V, V) ⩾ \hat{C} (∥ \nabla \nabla V ∥_{2, R^{d}}^{2} + {∥ | z | V ∥}_{2, R^{d}}^{2} + ∥ V ∥_{2, R^{d}}^{2}) .$ Thus, by the Riesz–Fréchet representation theorem, the Green operator is well-defined. Using Lemma 3.4 we see that the operator G is self-adjoint. Moreover, due to the compact embedding of $H^{2} (R^{d}) \cap L^{2} (R^{d}, | z |^{2} d z)$ in $L^{2} (R^{d})$ , the operator G is compact as an operator in $L^{2} (R^{d})$ ; $L^{2} (R^{d}, | z |^{2} d z)$ is the weighted $L^{2}$ -space with the weight $| z |^{2}$ . As a direct consequence, we have the following result.

Lemma 3.5.

The spectrum of the limit problem $\partial_{i j} (a_{i j k l}^{eff} \partial_{k l} v) - \partial_{i} ({\bar{b}}_{i j} \partial_{j} v) + \frac{1}{2} (H z \cdot z) v = η v, z \in R^{d},$ (20) is real, discrete, and consists of a countably infinite number of eigenvalues, each of finite multiplicity: $η_{1} ⩽ η_{2} ⩽ \dots, η_{j} \to \infty as j \to \infty .$ The corresponding eigenfunctions $v_{j}^{ε}$ form an orthogonal basis in $L^{2} (R^{d})$ .

We proceed to the proof of the convergence of spectra. We will prove that the Green operator $G^{ε}$ converges uniformly to G in $L (L^{2} (R^{d}))$ . Then we apply the following result, the proof of the which can be found in [3, Lemma 2.6] (see also [5]), to conclude the desired convergence of eigenvalues and eigenfunctions.

Lemma 3.6.

Let $G_{ε}$ be a sequence of compact self-adjoint operators acting in $L^{2} (R^{d})$ . Assume that $G_{ε}$ converges uniformly to a compact self-adjoint operator G. Let $η_{k}^{ε}$ and $η_{k}$ be the kth eigenvalues of the operators $G_{ε}$ and G, respectively; $v_{k}^{ε}$ , $v_{k}$ are eigenfunctions corresponding to $η_{k}^{ε}$ , $η_{k}$ . Then as $ε \to 0$ ,

$η_{k}^{ε} \to η_{k}$ ,

up to a subsequence, $v_{k}^{ε}$ converges strongly in $L^{2} (R^{d})$ to $v_{k}$ .

The uniform convergence of the Green operators is a straightforward consequence of the convergence of the solutions to the corresponding boundary value problems with weakly converging data in $L^{2} (R^{d})$ , as has been pointed out in [3, Theorem 2.2].

Lemma 3.7.

Let $f_{ε}$ be a sequence converging weakly to f in $L^{2} (R^{d})$ , and let $V^{ε}$ be the unique solution of (12) . Then $V^{ε}$ converges weakly in $H^{2} (R^{d})$ and strongly in $L^{2} (R^{d})$ to the unique solution V of the effective problem (17). Moreover, $\begin{array}{rcl} \nabla V^{ε} \overset{2}{⇀} \nabla V (z) two-scale in L^{2} (R^{d}), \\ \nabla \nabla V^{ε} \overset{2}{⇀} \nabla \nabla V (z) + \nabla_{ζ} \nabla_{ζ} N_{k l} (ζ) \partial_{k l} V (z) two-scale in L^{2} (R^{d}), \end{array}$ where $N_{k l} \in H^{2} (T^{d}) / R$ , $k, l = 1, \dots, d$ , solve problem (19) .

Proof.

The proof consists of two parts. First, we derive a priori estimates for $V^{ε}$ . Second, we pass to the two-scale limit in order to obtain the effective problem.

The estimates for $V^{ε}$ follows from (16): ${∥ \nabla \nabla V^{ε} ∥}_{2, Ω_{ε}}^{2} + {∥ \nabla V^{ε} ∥}_{2, Ω_{ε}}^{2} + {∥ | z | V^{ε} ∥}_{2, Ω_{ε}}^{2} + {∥ V^{ε} ∥}_{2, Ω_{ε}}^{2} ⩽ C ∥ f_{ε} ∥_{2, Ω_{ε}}^{2} .$ (21) Having in hand the a priori estimate (21), we deduce (see, for example, Proposition 1.14 in [1]) that in $L^{2} (R^{d})$ we have the following two-scale convergences: $V^{ε} \overset{2}{⇀} V (z), \nabla V^{ε} \overset{2}{⇀} \nabla V (z), \nabla \nabla V^{ε} \overset{2}{⇀} \nabla \nabla V (z) + \nabla_{ζ} \nabla_{ζ} W (z, ζ),$ (22) where $W (z, ζ) \in L^{2} (R^{d}; H^{2} (T^{d}))$ . The strong convergence of $V^{ε}$ to V in $L^{2} (R^{d})$ follows also from (21), namely from the boundedness of weighted $L^{2}$ -norm, which gives compactness. We are going to pass to the limit in the weak formulation of (12): $\begin{array}{rclr} \int_{Ω_{ε}} {\hat{a}}_{i j k l}^{ε} \partial_{k l} V^{ε} \partial_{i j} Φ^{ε} d z + \int_{Ω_{ε}} {\hat{b}}_{i j}^{ε} \partial_{i} V^{ε} \partial_{j} Φ^{ε} d z + \frac{1}{ε^{1 / 3}} \int_{Ω_{ε}} ({\hat{c}}^{ε} - \bar{c} (ε^{1 / 6} z)) V^{ε} Φ^{ε} d z \\ + \frac{1}{ε^{1 / 3}} \int_{Ω_{ε}} (\bar{c} (ε^{1 / 6} z) - \bar{c} (0)) V^{ε} Φ^{ε} d z + μ \int_{Ω_{ε}} V^{ε} Φ^{ε} d z = \int_{Ω_{ε}} f_{ε} Φ^{ε} d z, & (23) \end{array}$ where $Φ^{ε} (z) = Φ (z, \frac{z}{ε^{5 / 6}})$ , $Φ (z, ζ) \in C ({\bar{Ω}}_{ε}; L^{\infty} (T^{d}))$ . The term containing $\bar{c} (ε^{1 / 6} z)$ is added and subtracted for convenience, since we are going to use Lemma 3.3 when passing to the limit. Due to the regularity assumptions (H1), the coefficients can be regarded as a part of a test function.

First we take a test function $Φ_{ε} = ε^{5 / 3} φ (z) ψ (\frac{z}{ε^{5 / 6}})$ , with $φ \in C_{0}^{\infty} (R^{d})$ , $ψ \in C^{\infty} (T^{d})$ . Then (23) transforms into $\begin{array}{rcl} \int_{Ω_{ε}} {\hat{a}}_{i j k l}^{ε} \partial_{k l} V^{ε} (φ (z) \partial_{i j} ψ (ζ) + ε^{5 / 6} \partial_{j} φ (z) \partial_{i} ψ (ζ) + ε^{5 / 6} \partial_{i} φ (z) \partial_{j} ψ (ζ) \\ + ε^{5 / 3} \partial_{i j} φ (z) ψ (ζ)) |_{ζ = \frac{z}{ε^{5 / 6}}} d z \\ + \int_{Ω_{ε}} {\hat{b}}_{i j}^{ε} \partial_{i} V^{ε} (ε^{5 / 6} φ (z) \partial_{i} ψ (ζ) + ε^{5 / 3} ψ (ζ) \partial_{i} φ (z)) |_{ζ = \frac{z}{ε^{5 / 6}}} d z \\ + ε^{4 / 3} \int_{Ω_{ε}} ({\hat{c}}^{ε} - \bar{c} (ε^{1 / 6} z)) V^{ε} (z) φ (z) ψ (\frac{z}{ε^{5 / 6}}) d z \\ + ε^{4 / 3} \int_{Ω_{ε}} (\bar{c} (ε^{1 / 6} z) - \bar{c} (0)) V^{ε} (z) φ (z) ψ (\frac{z}{ε^{5 / 6}}) d z \\ + μ ε^{5 / 3} \int_{Ω_{ε}} V^{ε} (z) φ (z) ψ (\frac{z}{ε^{5 / 6}}) d z = ε^{5 / 3} \int_{Ω_{ε}} f_{ε} (z) φ (z) ψ (\frac{z}{ε^{5 / 6}}) d z . \end{array}$ Using Lemma 3.3 and (22) we may pass to the limit, as $ε \to 0$ , using the two-scale convergence, and obtain $\begin{array}{rcl} \int_{R^{d}} \int_{T^{d}} a_{i j m n} (0, ζ) \partial_{ζ_{m} ζ_{n}} W (z, ζ) \partial_{i j} ψ (ζ) φ (z) d ζ d z \\ = - \int_{R^{d}} \int_{T^{d}} a_{i j k l} (0, ζ) \partial_{k l} V (z) \partial_{i j} ψ (ζ) φ (z) d ζ d z . \end{array}$ From the last identity we deduce that $W (z, ζ) = N_{k l} (ζ) \partial_{k l} V (z)$ , where the periodic functions $N_{k l} (ζ)$ solve (19).

Now we take $Φ_{ε} = φ (z) \in C_{0}^{\infty} (R^{d})$ as a test function in (23), and passing to the limit get the weak formulation of the effective problem (17): $\begin{array}{rcl} \int_{R^{d}} \int_{T^{d}} (a_{i j m n} (0, ζ) \partial_{m n} N_{k l} (ζ) + a_{i j k l} (0, ζ)) \partial_{i j} φ (z) d ζ d z \\ + \int_{R^{d}} \int_{T^{d}} b_{i j} (0, ζ) d ζ \partial_{j} V (z) \partial_{i} φ (z) d z + \frac{1}{2} \int_{R^{d}} (H z \cdot z) V (z) φ (z) d z \\ + μ \int_{R^{d}} V (z) φ (z) d z = \int_{R^{d}} f (z) φ (z) d z \end{array}$ for any $φ \in C_{0}^{\infty} (R^{d})$ . Lemma 3.7 is proved. □

Lemma 3.8.

The Green operator of (12) converges uniformly, in $L (L^{2} (R^{d}))$ , to the Green operator of (17), as $ε \to 0$ .

Proof.

Let $f_{ε} \in L^{2} (R^{d})$ be a maximizing sequence for ${sup}_{∥ f ∥_{2, R^{d}} = 1} ∥ (G^{ε} - G) f ∥_{2, R^{d}}$ . By compactness there is a subsequence $f_{ε}$ weakly converging to some f in $L^{2} (R^{d})$ . By Lemma 3.7, $G_{ε} f_{ε} \to G f$ strongly in $L^{2} (R^{d})$ , and by the compactness of G, $G f_{ε} \to G f$ strongly in $L^{2} (R^{d})$ . Hence $∥ G_{ε} - G ∥ ⩽ ∥ G_{ε} f_{ε} - G f ∥_{2, R^{d}} + ∥ G f_{ε} - G f ∥_{2, R^{d}} + o (1),$ as $ε \to 0$ , and the convergence along a subsequence follows. Since the limit $G_{ε} f_{ε}$ is unique by Lemma 3.7, the whole sequence converges. □

Due to the Lemma 3.8, the sequence of the Green operators of the rescaled problem (12) converges uniformly to the Green operator of the effective problem (17). Lemma 3.6 applied to the Green operators ensures the convergence of spectrum of the rescaled problem (11).

Lemma 3.9.

Let $(η_{k}^{ε}, v_{k}^{ε})$ be kth eigenpair of the rescaled spectral problem (11). Then under the assumptions (H1)–(H4), as $ε \to 0$ ,

$η_{k}^{ε} \to η_{k}$ , where $η_{k}$ is the kth eigenvalue of the effective problem (20) ,

along a subsequence $v_{k}^{ε}$ converges weakly in $H^{2} (R^{d})$ and strongly in $L^{2} (R^{d})$ to $v_{k}$ , where $v_{k}$ is the eigenfunction corresponding to $η_{k}$ under a proper orthonormalization.

The last lemma combined with (10) yields Theorem 3.1.

In the next sections we consider the cases when $(α, β)$ belong to $R_{1}$ , $R_{3}$ , $R_{4}$ , $R_{5}$ .

4. The case

(α, β) \in R_{1}

Recall that $R_{1} = {(α, β) : 0 ⩽ α < 1, 3 α < β < 3} .$

Theorem 4.1.
Let $(α, β) \in R_{1}$ and let $(λ_{k}^{ε}, u_{k}^{ε})$ be the kth eigenpair of (1) normalized by $∥ u_{k}^{ε} ∥_{2, Ω}^{2} = ε^{d β / 6}$ . Suppose that the conditions (H1)–(H4) are satisfied. Then we have the following representation: $u_{k}^{ε} = v_{k}^{ε} (\frac{x}{ε^{β / 6}}), λ_{k}^{ε} = \frac{\bar{c} (0)}{ε^{β}} + \frac{η_{k}^{ε}}{ε^{2 β / 3}},$ where $(η_{k}^{ε}, v_{k}^{ε})$ are such that as $ε \to 0$ ,
$η_{k}^{ε} \to η_{k}$ ,

up to a subsequence, $v_{k}^{ε}$ converges to $v_{k}$ weakly in $H^{2} (R^{d})$ and strongly in $L^{2} (R^{d})$ ,
where $η_{k}$ is the kth eigenvalue, and $v_{k}$ is an eigenfunction corresponding to $η_{k}$ normalized by $∥ v_{k} ∥_{2, R^{d}} = 1$ , of the uniformly elliptic effective spectral problem $\partial_{i j} (a_{i j k l}^{eff} \partial_{k l} v) + \frac{1}{2} (H z \cdot z) v = η v, z \in R^{d},$ with $a_{i j k l}^{eff}$ defined by (18), and $H = \nabla \nabla \bar{c} (0)$ .
Proof.
We shift the spectrum by $\bar{c} (0) / ε^{β}$ and make the following the change of variables: $γ = \frac{β}{6}, z = \frac{x}{ε^{γ}}, v (z) = u (ε^{γ} z), η^{ε} = ε^{2 β / 3} (λ^{ε} - \frac{\bar{c} (0)}{ε^{β}}), z \in Ω_{ε} = ε^{- γ} Ω .$ Then we obtain the rescaled problem $\{\begin{matrix} {\hat{A}}_{1}^{ε} v^{ε} = η^{ε} v^{ε}, & z \in Ω_{ε}, \\ v^{ε} = \nabla v^{ε} \cdot n = 0, & z \in \partial Ω_{ε}, \end{matrix}$ (24) where ${\hat{A}}_{1}^{ε} v^{ε} = \partial_{i j} ({\hat{a}}_{i j k l}^{ε} \partial_{k l} v^{ε}) - ε^{- α + β / 3} \partial_{i} ({\hat{b}}_{i j}^{ε} \partial_{j} v^{ε}) + ε^{- β / 3} ({\hat{c}}^{ε} - \bar{c} (0)) v^{ε}$ (25) and ${\hat{a}}_{i j k l}^{ε} = a_{i j k l} (ε^{β / 6} z, \frac{z}{ε^{1 - β / 6}}), {\hat{b}}_{i j}^{ε} = b_{i j} (ε^{β / 6} z, \frac{z}{ε^{1 - β / 6}}), {\hat{c}}^{ε} = c (ε^{β / 6} z, \frac{z}{ε^{1 - β / 6}}) .$ In order to describe the asymptotic behavior of the eigenpairs $(η_{k}^{ε}, v_{k}^{ε})$ , as $ε \to 0$ , we prove the uniform convergence of the corresponding Green operators and then use Lemma 3.6.

Let $μ > 0$ and $f_{ε}$ be a sequence converging weakly in $L^{2} (R^{d})$ to f. Consider the boundary value problem $\{\begin{matrix} {\hat{A}}_{1}^{ε} V^{ε} + μ V^{ε} = f_{ε}, & z \in Ω_{ε}, \\ V^{ε} = \nabla V^{ε} \cdot n = 0, & z \in \partial Ω_{ε} . \end{matrix}$ (26) By (H1)–(H4), for all sufficiently small $ε > 0$ , the Green operator of (26) is a compact, self-adjoint and positive operator on $L^{2} (R^{d})$ . Moreover, for the sequence of solutions $V^{ε}$ to (26) we have ${∥ \nabla \nabla V^{ε} ∥}_{2, R^{d}} + {∥ \nabla V^{ε} ∥}_{2, R^{d}} + {∥ V^{ε} ∥}_{2, R^{d}} + {∥ | z | V^{ε} ∥}_{2, R^{d}} ⩽ C .$ Thus, up to a subsequence, $V^{ε} \to V (z), \nabla V^{ε} \overset{2}{⇀} \nabla V (z), \nabla \nabla V^{ε} \overset{2}{⇀} \nabla \nabla V (z) + \nabla_{ζ} \nabla_{ζ} W (z, ζ),$ where $V \in L^{2} (R^{d})$ , $W (z, ζ) \in L^{2} (R^{d}; H^{2} (T^{d}))$ . Passing to the limit in the variational formulation of (26) we find that $V \in H^{2} (R^{d}) \cap L^{2} (R^{d}, | z |^{2} d z)$ is the unique solution to the equation $A_{1}^{eff} V + μ V = f, z \in R^{d},$ (27) where $A_{1}^{eff} V = \partial_{i j} (a_{i j k l}^{eff} \partial_{k l} V) + \frac{1}{2} (H z \cdot z) V,$ (28) and $a_{i j k l}^{eff}$ is coercive on $R^{d \times d}$ and given by (18) . The second order term vanishes because of the hypothesis $- α + β / 3 > 0$ and the boundedness of $\nabla V^{ε}$ . The Green operator of (27), as an operator on $L^{2} (R^{d})$ , is well-defined, is compact, self-adjoint, and positive. Due to the uniqueness of the solution to (27), the whole sequence $V^{ε}$ converges to V.

In this way the Green operator of (26) converges uniformly to the Green operator of (27), as $ε \to 0$ . By Lemma 3.6, the spectrum of (24) converges to the spectrum of the limit operator (28) in the sense of Kuratowsky convergence of subsets of R. Changing back the variables yields the desired result. □

5. The cases $(α, β) \in R_{3}, R_{4}, R_{5}$

Recall that $\begin{array}{rcl} R_{3} = {(α, β) : 0 < α < 2, α < β < 3 α, β < α + 2}, \\ R_{4} = {(α, β) : α = 2, 2 < β < 4}, \\ R_{5} = {(α, β) : 2 < α < 4, α < β < 4} . \end{array}$ For these regions, the limit problems are of second order and have the same form, but the effective coefficients and the corresponding cell problems are different. We assume the coerciveness of the matrix $b (x, y)$ so that the effective problems are well-posed. We gather the results for these cases in the following theorem.

Let the effective coefficients be defined by $b_{i j}^{eff} = \{\begin{matrix} \int_{T^{d}} b_{i j} (0, y) d y, & (α, β) \in R_{3}, \\ \int_{T^{d}} b_{i k} (0, y) (δ_{k j} + \partial_{k} M_{j}) d y, & (α, β) \in R_{4}, \\ \int_{T^{d}} b_{i k} (0, y) (δ_{k j} + \partial_{k} N_{j}) d y, & (α, β) \in R_{5}, \end{matrix}$ (29) where $M_{n} \in H^{2} (T^{d}) / R$ and $N_{n} \in H^{1} (T^{d}) / R$ are the unique solutions to the respective cell problems $\begin{array}{rclr} \partial_{y_{i} y_{j}} (a_{i j k l} (0, y) \partial_{y_{k} y_{l}} M_{n}) - \partial_{y_{i}} (b_{i j} (0, y) \partial_{y_{j}} M_{n}) = \partial_{y_{i}} b_{n i} (0, y), y \in T^{d}; & (30) \\ - \partial_{y_{i}} (b_{i j} (0, y) \partial_{y_{j}} N_{n}) = \partial_{y_{i}} b_{n i} (0, y), y \in T^{d}, & (31) \end{array}$ Theorem 5.1.

Let $(α, β) \in R_{3} \cup R_{4} \cup R_{5}$ and let $(λ_{k}^{ε}, u_{k}^{ε})$ be the kth eigenpair of (1) normalized by $∥ u_{k}^{ε} ∥_{2, Ω}^{2} = ε^{d (β - α) / 4}$ . Suppose that the conditions (H1)–(H5) are satisfied. Then the following representation holds: $u_{k}^{ε} (x) = v_{k}^{ε} (\frac{x}{ε^{(β - α) / 4}}), λ_{k}^{ε} = \frac{\bar{c} (0)}{ε^{β}} + \frac{η_{k}^{ε}}{ε^{(α + β) / 2}},$ where $(η_{k}^{ε}, v_{k}^{ε})$ are such that as $ε \to 0$ ,

$η_{k}^{ε} \to η_{k}$ ,

up to a subsequence, $v_{k}^{ε}$ converges to $v_{k}$ weakly in $H^{1} (R^{d})$ and strongly in $L^{2} (R^{d})$ ,

where

η_{k}

is the kth eigenvalue, and

v_{k}

is an eigenfunction corresponding to

η_{k}

normalized by

∥ v_{k} ∥_{2, R^{d}} = 1

, of the harmonic oscillator problem

- \partial_{i} (b_{i j}^{eff} \partial_{j} v) + \frac{1}{2} (H z \cdot z) v = η v, z \in R^{d},

(32)

with

b^{eff}

defined by (29), and

H = \nabla \nabla \bar{c} (0)

Proof.

We shift the spectrum by $\bar{c} (0) / ε^{β}$ and make the following the change of variables: $γ = \frac{β - α}{4}, z = \frac{x}{ε^{γ}}, v (z) = u (ε^{γ} z), η^{ε} = ε^{\frac{α + β}{2}} (λ^{ε} - \frac{\bar{c} (0)}{ε^{β}}), z \in Ω_{ε} = ε^{- γ} Ω .$ Then we obtain the rescaled problem $\{\begin{matrix} {\hat{A}}_{2}^{ε} v^{ε} = η^{ε} v^{ε}, & z \in Ω_{ε}, \\ v^{ε} = \nabla v^{ε} \cdot n = 0, & z \in \partial Ω_{ε}, \end{matrix}$ (33) where ${\hat{A}}_{2}^{ε} v^{ε} = ε^{α - 2 γ} \partial_{i j} ({\hat{a}}_{i j k l}^{ε} \partial_{k l} v^{ε}) - \partial_{i} ({\hat{b}}_{i j}^{ε} \partial_{j} v^{ε}) + ε^{α - β + 2 γ} ({\hat{c}}^{ε} - \bar{c} (0)) v^{ε}$ (34) and ${\hat{a}}_{i j k l}^{ε} = a_{i j k l} (ε^{γ} z, \frac{z}{ε^{1 - γ}}), {\hat{b}}_{i j}^{ε} = b_{i j} (ε^{γ} z, \frac{z}{ε^{1 - γ}}), {\hat{c}}^{ε} = c (ε^{γ} z, \frac{z}{ε^{1 - γ}}) .$ As above, to describe the asymptotic behavior of the eigenpairs $(η_{k}^{ε}, v_{k}^{ε})$ , as $ε \to 0$ , we prove the uniform convergence of the corresponding Green operators and then use Lemma 3.6.

Let $μ > 0$ and $f_{ε}$ be a sequence converging weakly in $L^{2} (R^{d})$ to f. Consider the boundary value problem $\{\begin{matrix} {\hat{A}}_{2}^{ε} V^{ε} + μ V^{ε} = f_{ε}, & z \in Ω_{ε}, \\ V^{ε} = \nabla V^{ε} \cdot n = 0, & z \in \partial Ω_{ε} . \end{matrix}$ (35) By (H1)–(H5), for all sufficiently small $ε > 0$ , the Green operator of (35) is a compact, self-adjoint and positive operator in $L^{2} (R^{d})$ . Moreover, for the sequence of solutions $V^{ε}$ to (35) we have $ε^{(3 α - β) / 4} {∥ \nabla \nabla V^{ε} ∥}_{2, R^{d}} + {∥ \nabla V^{ε} ∥}_{2, R^{d}} + {∥ V^{ε} ∥}_{2, R^{d}} + {∥ | z | V^{ε} ∥}_{2, R^{d}} ⩽ C .$ We proceed by dividing into the cases: $\frac{3 α - β}{4} \{\begin{matrix} < 1 - γ, & (α, β) \in R_{3}, \\ = 1 - γ, & (α, β) \in R_{4}, \\ > 1 - γ, & (α, β) \in R_{5}, \end{matrix}$ where $1 - γ$ will be the scale in the two-scale convergence.

5.1.

(α, β) \in R_{3}

Up to a subsequence, $V^{ε} \to V (z), \nabla V^{ε} \overset{2}{⇀} \nabla V (z), ε^{(3 α - β) / 4} \nabla \nabla V^{ε} \overset{2}{⇀} W (z, ζ),$ where $V \in H^{1} (R^{d})$ , $W (z, ζ) \in L^{2} (R^{d}; H^{2} (T^{d}))$ . Passing to the limit in the variational formulation of (35) we find that $V \in H^{1} (R^{d}) \cap L^{2} (R^{d}, | z |^{2} d z)$ is the unique solution to the equation $- \partial_{i} (b_{i j}^{eff} \partial_{j} V) + \frac{1}{2} (H z \cdot z) V + μ V = f, z \in R^{d},$ (36) where $b_{i j}^{eff} = {\bar{b}}_{i j} (0) = \int_{T^{d}} b_{i j} (0, y) d y$ is coercive on $R^{d}$ by (H5).

5.2.

(α, β) \in R_{4}

In this case $\frac{3 α - β}{4} = 1 - γ$ . Up to a subsequence, $V^{ε} \to V (z), \nabla V^{ε} \overset{2}{⇀} \nabla V (z) + \nabla_{ζ} W (z, ζ), ε^{(3 α - β) / 4} \nabla \nabla V^{ε} \overset{2}{⇀} \nabla_{ζ} \nabla_{ζ} W (z, ζ),$ in $L^{2} (R^{d})$ , where $V \in H^{1} (R^{d})$ , $W (z, ζ) \in L^{2} (R^{d}; H^{2} (T^{d}))$ . Passing to the limit in the variational formulation of (35) we find that $V \in H^{1} (R^{d}) \cap L^{2} (R^{d}, | z |^{2} d z)$ is the unique solution to the same equation (36) with $b_{i j}^{eff} = \int_{T^{d}} b_{i k} (0, y) (δ_{k j} + \partial_{k} M_{j}) d y,$ (37) where $M_{j} \in H^{2} (T^{d}) / R$ solves (31). By the periodicity of $b (x, y)$ , $M_{j}$ is well-defined.

Following the lines of Lemma 3.4, one can prove that under the assumptions (H1)–(H3) and (H5), $b^{eff}$ defined by (37) is coercive on $R^{d}$ .

5.3.

(α, β) \in R_{5}

Up to a subsequence, $V^{ε} \to V (z), \nabla V^{ε} \overset{2}{⇀} \nabla V (z) + \nabla_{ζ} N_{j} (z, ζ) \partial_{j} V (z), ε^{(3 α - β) / 4} \nabla \nabla V^{ε} \overset{2}{⇀} W_{1} (z, ζ),$ in $L^{2} (R^{d})$ , where $V \in H^{1} (R^{d})$ , $W_{1} (z, ζ) \in L^{2} (R^{d} \times T^{d})$ , and $N_{j} \in H^{1} (T^{d}) / R$ solves (31) . Passing to the limit in the variational formulation of (35) we find that $V \in H^{1} (R^{d}) \cap L^{2} (R^{d}, | z |^{2} d z)$ is the unique solution to (36) with $b_{i j}^{eff} = \int_{T^{d}} b_{i k} (0, y) (δ_{k j} + \partial_{k} N_{j}) d y .$ By the similar argument used in Lemma 3.4, $b^{eff}$ is coercive on $R^{d}$ .

In this way, in all the three cases, the Green operator of (35) converges uniformly to the Green operator of (36), as $ε \to 0$ . By Lemma 3.6, the spectrum of (33) converges to the spectrum of the limit operator (32). Changing back the variables yields the desired result. □

Footnotes

Acknowledgements

The first author was supported by the Norwegian Research Council under the Yggdrasil mobility program 2014, Project 227254/F11.

This work was done during the stay of A. Chechkina at Narvik University College in 2014, the hospitality of which is kindly acknowledged.

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