Spectral properties of an elliptic operator with double-contrast coefficients near a hyperplane

1. Introduction

The problem we are going to study lies on the intersection of spectral theory and homogenization theory for partial differential operators. We recall that one of the central problems of homogenization theory is to study the asymptotic behaviour as ε → 0 of the solution u ε to the problem A ε u ε = f in Ω , u ε = 0 on ∂ Ω , where ε > 0 is a small parameter, A ε : = − div ( a ε ( x ) ∇ ) , { a ε ( x ) } ε > 0 is a family of real measurable functions satisfying a − ε ⩽ a ε ( x ) ⩽ a + ε with positive constants a ± ε and becoming highly oscillating as ε → 0 . The typical example is a ε ( x ) = a ( x ε − 1 ) , where a ( x ) is a fixed Z n -periodic function. It is well-known (see, e.g., [1,14]) that if (1.1) inf ε a − ε > 0 , sup ε a + ε < ∞ , then the family { u ε } ε is compact in L 2 ( Ω ) , and if u ε → u as ε = ε k → 0 then u ( x ) is a solution of the problem A u = f in Ω , u = 0 on ∂ Ω , where A : = − div ( A ( x ) ∇ ) , A ( x ) is some bounded matrix-function, bounded away from zero, which depends, in general, on the subsequence ε k . If a ε ( x ) = a ( x ε − 1 ) then the whole sequence u ε converges, and in this case A ( x ) is a constant matrix. As we see the limit differential operator has qualitatively the same form as the initial one provided (1.1) holds.

If, on the contrary, conditions (1.1) are violated, for example there exist subsets D ε ⊂ Ω with non-zero measure such that lim ε → 0 sup x ∈ D ε a ε ( x ) = ∞ or (1.2) lim ε → 0 inf x ∈ D ε a ε ( x ) = 0 , then the limit operator may have a more complicated form, which depends essentially on the structure of the domains D ε . We refer to the monograph [28], where various problems of this type are studied. In particular, the authors consider the case, when condition (1.2) holds on the union D ε of thin shells, distributed periodically, with period ε, in the domain Ω. They study the behaviour of linear evolution equations involving such operators A ε . Semi-linear evolution equations were investigated in [13,36,37]. Spectral properties of such operators were studied in [22].

In all papers mentioned above the case of bulk distribution of shells was considered. In the present work we are interested in the case of surface distribution of shells, i.e., the shells are located in a neighbourhood of some hyperplane.

We note that our research is inspired, in particular, by spectral problems for periodic differential operators posed in a waveguide-like domain. These applications are discussed later in the introduction.

Below we briefly present our main results. Let Ω be a bounded domain in R n ( n ⩾ 2 ), ε > 0 be a small parameter. In Ω we consider the operator (1.3) A ε = − 1 b ε ( x ) div ( a ε ( x ) ∇ ) subject to Dirichlet boundary conditions on ∂ Ω . Here the functions a ε and b ε are bounded above and bounded away from zero uniformly in ε everywhere except a small neighbourhood of some hyperplane Γ having non-empty intersection with the domain Ω. More precisely, we denote by D ε = { D i ε } a family of thin spherical shells distributed periodically, with period ε, along Γ (counted by the parameter i). Each shell has an external radius R ε = R ε (here R ∈ ( 0 , 1 / 2 ) is a constant), the thickness of the shells is d ε = o ( ε ) as ε → 0 . By B ε = { B i ε } we denote the union of balls surrounded by these shells (see Fig. 1). When ε → 0 the number N ( ε ) of shells goes to infinity as ε → 0 , namely (1.4) N ( ε ) ∼ ε 1 − n | Γ | (hereinafter we will use the same notation | · | for the volume of a domain in R n and as well for the area of an ( n − 1 ) -dimensional hypersurface in R n ). We define the functions a ε ( x ) and b ε ( x ) as follows: a ε ( x ) is equal to 1 for x ∈ Ω ∖ D ε and equal to the constant α ε > 0 for x ∈ D ε , b ε ( x ) is equal to 1 for x ∈ Ω ∖ B ε and equal to the constant β ε > 0 for x ∈ B ε .

OPEN IN VIEWER

Fig. 1.

The domain Ω and the family of shells D i ε .

The operator A ε describes vibrations of a body occupying the domain Ω, the functions a ε ( x ) and b ε ( x ) are its stiffness and mass density, respectively. We note, that the most interesting effects occur if α ε = O ( d ε ) , β ε = O ( ε − 1 ) as ε → 0 . In this case A ε describes vibrations of a body with a lot of tiny heavy inclusions B i ε surrounded by thin soft layers D i ε .

Operators of the form (1.3) occur in many other areas of physics. For example, in the case n = 3 the operator A ε governs the propagation of acoustic waves in an inhomogeneous medium with mass density ( a ε ( x ) ) − 1 and compressibility b ε ( x ) .1

Asymptotics of eigenvibrations of a body with a mass density perturbed near a hypersurface was studied in a lot of papers – see, e.g., [10–12,26,27,29] and references therein; we refer also to the monographs [35,39]. The case of simultaneously perturbed density and stiffness (“double-contrast”) was investigated in [20] (see also [2], where the case of bulk distribution of double-contrast inclusions was studied). In all these articles the geometry of a set supporting the perturbation differs essentially from that one considered in the current paper.

The spectrum σ ( A ε ) of A ε is purely discrete. Our goal is to describe its behaviour as ε → 0 supposing that the following conditions hold: (1.5) lim ε → 0 ( d ε ) 2 α ε = 0 , (1.6) lim ε → 0 q ε = q ∈ [ 0 , ∞ ] , where q ε = α ε n R d ε ε β ε , (1.7) lim ε → 0 r ε = r ∈ [ 0 , ∞ ) , where r ε = R n ϰ n ε β ε . Here by ϰ n we denote the volume of the n-dimensional unit ball. We note, that q is allowed to be infinite. In terms of powers of ε conditions (1.5)–(1.7) can be rewritten as follows. We set d ε = ε γ , β ε = ε b , α ε = ε a . Since d ε = o ( ε ) then γ > 1 . We fix γ ∈ ( 1 , ∞ ) as a parameter and look how q and r depend on a and b. Condition (1.5) implies 2 γ − a > 0 ; condition (1.7) requires 1 + b ⩾ 0 . Thus, on the ( a , b ) -plane we are restricted to the set { a < 2 γ , b ⩾ − 1 } . The interior of this set is marked by the grey color on Fig. 2. We also introduce the following points on this plane: A = ( − ∞ , − 1 ) , B = ( γ , − 1 ) , C = ( 2 γ , − 1 ) , D = ( 2 γ , γ − 1 ) , E = ( 2 γ , ∞ ) . Due to (1.6), we obtain:

OPEN IN VIEWER

Fig. 2.

( a , b ) -plane.

Condition (1.5) means that the eigenfunctions cannot concentrate on the shells (cf. (3.45)). The case α ε = O ( ( d ε ) 2 ) differs essentially and will not be studied in this paper.

The parameter q characterizes the “strength” of the coupling between the domain Ω ε : = Ω ∖ ⋃ i ( D i ε ∪ B i ε ‾ ) and the union of the balls B i ε . We postpone the precise statement to Section 3 (see Remark 3.1) because first we need to introduce some more notations.

The finiteness of r implies the uniform (with respect to ε) boundedness of the “mass” M B ε of B ε , namely, using (1.4) and the fact that d ε = o ( ε ) , we obtain: M B ε : = ∫ ⋃ i B i ε b ε ( x ) d x = β ε ∑ i | B i ε | = β ε ( R ε − d ε ) n ϰ n N ( ε ) = r ε ( R ε − d ε ) n N ( ε ) R n ε ∼ r | Γ | as ε → 0 .

In spite of the fact that A ε contains many parameters the behaviour of its spectrum as ε → 0 depends essentially only on q being finite or infinite and r being positive or zero.

At this point we present the results in a formal way; more precise statements are formulated in the next section using the language of operator theory. Below the convergence of spectra is understood in the Hausdorff sense, see Definition 2.1. One has:

Note, that in the case r = 0 the problems (1.8) and (1.10) reduce to the eigenvalue problem for the Dirichlet Laplacian in Ω.

Also we note, that instead of Dirichlet conditions on ∂ Ω one can consider Neumann ones. In this case the limit spectral problem will have either the form (1.8) or (1.10), but with Neumann conditions on ∂ Ω instead of Dirichlet ones. The proofs are completely analogous. The same is valid for mixed conditions too. The change of boundary conditions on ∂ Ω will influence the spectral threshold – in the Dirichlet case it is strictly positive (provided q > 0 ), while in the Neumann case it is equal to zero; nevertheless the structure of the spectrum of the limit problem remains the same (cf. Remark 3.2).

In the last part of the paper we consider the same problem for a waveguide-like domain Ω: Ω = R × ( d − , d + ) , Γ = { x = ( x 1 , 0 ) : x 1 ∈ R } . where d − < 0 , d + > 0 . Here we are interested in the case q > 0 , r > 0 only (i.e., we deal with point B on Fig. 2).

Due to the periodicity of A ε its spectrum is a locally finite union of compact intervals (bands). In general the bands may overlap and the natural question arising here is whether gaps open up in the spectrum (i.e., whether there is an open interval ( α , β ) ⊂ ( 0 , ∞ ) such that ( α , β ) ∩ σ ( A ε ) = ∅ , while α , β ∈ σ ( A ε ) ). This problem is interesting for applications since the presence of gaps is important for the description of wave processes which are governed by the differential operators under consideration. Namely, if the wave frequency belongs to a gap, then the corresponding wave cannot propagate in the medium without attenuation. This feature is a dominant requirement for so-called photonic crystals which are materials with periodic dielectric structure attracting much attention in recent years (see, e.g., [16,25]).

It was proved in [22] that in the case Ω = R n and a bulk distribution of shells, the spectrum of A ε has a gap when ε is small enough.

Our goal is to study whether gaps will open up in case of our waveguide-like domain. We will prove that the spectrum of A ε converges in the Hausdorff sense to the spectrum of some operator A which is defined by the same differential expression as in the case of a bounded domain, and its spectrum is as follows: if q < min { π 2 d − 2 , π 2 d + 2 } then σ ( A ) = [ α 1 , q ] ∪ [ α 2 , ∞ ) , otherwise σ ( A ) = [ α 1 , ∞ ) . Here α 1 , α 2 are some positive numbers satisfying 0 < α 1 < q < α 2 . Thus if the waveguide is thin enough (i.e., if d − < π q − 1 / 2 and d + < π q − 1 / 2 ) we have a gap in the spectrum of A (and, consequently, σ ( A ε ) has a gap when ε is small enough).

Periodic perturbations of the Laplacian in wavegide-like domains leading to opening of spectral gaps were also studied in [3–5,8,9,19,33,34,41]. In all these papers (except [4,19]) spectral gaps appear because of a perturbation of the boundary of the waveguide (for example by making small holes periodically distributed along the waveguide [9] or by dividing the waveguide into two parts coupled by a periodic system of small windows [5]). In the paper [4] the authors considered small perturbations of the Laplace operator in a cylindrical domain by second-order differential operators with periodic coefficients; they gave sufficient conditions on this perturbation for gap opening. These conditions are not valid for the operators considered in the present work. In the paper [19] the authors perturbed the Laplace operator by a singular potential supported by a family of periodically distributed surfaces.

Let us note that we restrict ourselves to the two-dimensional case since our study is originally motivated by applications to photonic crystals. Namely, it is well-known (see, e.g., [16, Chapter 1]), that in a 3 D crystal, which is homogeneous in one direction, the propagation of suitably polarized electro-magnetic waves is governed by scalar 2 D operators of the form (1.3) posed on crystal cross-section.

Nevertheless, it will be visible from the proof that there are no mathematical peculiarities in the two-dimensional case and the results remain qualitatively the same for higher dimensions.

The paper is organized as follows. In Section 2 we set the problem and formulate the main result (Theorem 2.1), also we prove the estimate (1.11) (Theorem 2.2). Theorem 2.1 will be proven in Section 3: the case q < ∞ in Section 3.2 and the case q = ∞ in Section 3.4. Finally, in Section 4 we consider the case of a waveguide.