Homogenization of spectral problems in bounded domains with doubly high contrasts

Homogenization of a spectral problem in a bounded domain with a high contrast in both stiffness and density is considered. For a special critical scaling, two-scale asymptotic expansions for eigenvalues and eigenfunctions are constructed. Two-scale limit equations are derived and relate to certain non-standard self-adjoint operators. In particular they explicitly display the first two terms in the asymptotic expansion for the eigenvalues, with a surprising bound for the error of order \epsilon^{5/4} proved.


Introduction
Homogenization for problems with physical properties which are not only highly oscillatory but also highly heterogeneous has long been documented to display unusual effects, for example the memory effects observed by E. Ya. Khruslov [9,13,14]. Of particular interest in this context are the double-porosity models where the parameter of high-contrast δ is critically scaled again the periodicity size ε, δ ∼ ε 2 , e.g. [2,4]. Those have been treated both by a high-contrast version of the classical method of asymptotic expansions, e.g. [16,17,7,12] and using the techniques of two-scale convergence, e.g. [19,20,5]. In particular, for spectral problems in bounded [19] and unbounded [20] periodic domains V.V. Zhikov studied the spectral convergence, introduced two-scale limit operator, developed the techniques of two-scale resolvent convergence and two-scale compactness. In [12] the spectral convergence of eigenvalues in the gaps of Floquet-Bloch spectrum due to defects in double-porosity type media were studied, and [5] supplemented this by the analysis of eigenfunction convergence based on an analysis of a uniform exponential decay.
In this work we study spectral problems of double-porosity type in a bounded domain Ω where the high contrast might occur not only in the "stiffness" coefficient but also in the "density", and argue that this leads to some interesting new effects. Namely, referring to the next section for precise technical formulations, for the spectral problem (1) − div (a ε (x) ∇u ε ) = λ ε ρ ε (x) u ε , with Dirichlet boundary conditions on the exterior boundary, most generally, both a ε and ρ ε are ε-periodic, a ε = ρ ε = 1 in the connected matrix and a ε ∼ ε α , ρ ε ∼ ε β in the disconnected inclusions. (Outside homogenization, the above resembles problems of vibrations with high contrasts in both density and stiffness, e.g. [3].) The double-porosity corresponds to α = 2 and β = 0. For β = 0, it is not hard to see that it is α = β + 2 when the spectral problems at the macro and micro-scales are coupled in a non-trivial way. To explore this, we choose β = −1 and α = 1 Figure 1. The geometry and the periodicity cell and show that this leads to some unusually coupled two-scale limit behaviors of the eigenfunctions and the eigenvalues. Namely, although the limit behavior of the eigenfunctions is still somewhat similar to that of double porosity, i.e. the two-scale limit is a function of only slow variable x in the matrix and a function of both x and the fast variable y in the inclusions, the limit equations themselves are quite different. We show that there exist asymptotic series of eigenvalues λ ε ∼ λ 0 + ελ 1 with λ 0 being any eigenvalue of a non-standard self-adjoint "microscopic" inclusion problem, Theorem 3.1, whose eigenfunctions are directly related to the two-scale limit w 0 (x, y) in the matrix. In fact, λ 0 is either a solution of β(λ 0 ) = |Q 1 |λ 0 , where β(λ) is a function introduced by Zhikov [19], or is an eigenvalue of the Dirichlet Laplacian in the inclusion Q 0 with a zero mean eigenfunction. In the matrix, u ε ∼ v 0 (x), where v 0 is an eigenfunction of the homogenized operator in Ω, whose eigenvalue ν determines the second term λ 1 in the asymptotics of λ ε , see (57). This is first derived via formal asymptotic expansions, but then we prove a non-standard error bound: |λ ε − λ 0 − ελ 1 | ≤ Cε 5/4 , see Theorems 4.6 & 4.7. The proof employs a combination of a high contrast boundary layer analysis with maximum principle and estimates in Hilbert spaces with ε-dependent weights. We finally briefly discuss further refinement of the results via the technique of two-scale convergence. Namely, some version of the compactness result holds, cf. [19], indicating at the presence of gaps in the spectrum for small enough ε, see Theorem 5.1.
The paper is organized as follows. The next section formulates the problem and introduces necessary notation, Section 3 executes formal asymptotic expansion and derives associated homogenized equations. Section 4 proves the error bounds and Section 5 discusses the two-scale convergence approach. Some technical details are assembled in the appendices.

Problem statement and notations
We consider a model of eigenvibrations for a body occupying a bounded domain Ω in R n (n = 2, 3, . . . ) containing a periodic array of small inclusions, see Figure 1. The size of inclusions is controlled by a small positive parameter ε, ε → 0. First we introduce necessary notation.
Let Q = [0, 1] n be a reference periodicity cell in R n . Let Q 0 be a periodic set of "inclusions", i.e. Q 0 + m = Q 0 , ∀m ∈ Z n , and Q 0 = Q 0 ∩ Q is a reference inclusion lying inside Q with C 2 -smooth boundary Γ, see Figure 1. Let Q 1 = Q\Q 0 , Introducing y = x/ε we refer to y as to a fast variable, as opposes to the slow variable x. In the x-variable the periodicity cell is εQ = [0, ε) n . If y ∈ Q j then x = εy ∈ εQ j , j = 0, 1. We denote Ω ε 0 := Ω ∩ εQ 0 , Ω ε 1 := Ω ∩ εQ 1 = Ω\Ω ε 0 , Γ ε := εΓ ∩ Ω, see Figure 1. The trace on Γ ε of function f : Ω ε j → R n is denoted by f | j . Let n y be the outer unit normal to Q 0 on its boundary Γ and let n x denote the similar normal on Γ ε .
Let stiffness a ε and density ρ ε be as follows with a small positive ε.
We study the asymptotic behaviour of self-adjoint spectral problem as ε → 0. If Γ and ∂Ω are smooth enough then variational problem (2) can be equivalently represented in a classical formulation implying that at the interfaces the transmission conditions are satisfied

Formal asymptotic expansions
We seek formal asymptotic expansions for the eigenvalues λ ε and eigenfunctions u ε in the form Here all the functions v j (x, y), w j (x, y), j ≥ 0, are required to be periodic in the "fast" variable y; v 0 and w 0 are not simultaneously identically zero In a standard way, the ansatz (6), (7) is then formally substituted into (3)-(5). In particular, from (3), for (x, y) ∈ Ω × Q 1 , we obtain (with ∆ y and ∆ x denoting the Laplace operators in y and x, respectively, and summation henceforth implied with respect to repeated indices), and for (x, y) ∈ Further, the first of conditions (5) transforms to (15) v j (x, y) Similarly, the other transmission condition (5) yield ∂v 0 ∂n y y∈Γ = 0, The above has employed the identity (19) ∂u where ∂ ∂ny := n y · ∇ y , ∂ ∂nx := n y · ∇ x , with ∇ y and ∇ x standing for gradients in y and x, respectively.
Finally, (4) suggests (The boundary layer problem does not generally permit satisfying (4) by v j and w j for j ≥ 1, as also clarified later.) Combining (9) and (16), together with the periodicity conditions in y, implies that v 0 is a constant with respect to y, i.e.
Then, (10) and (17) form the following boundary value problem for v 1 The latter is solvable if and only if Considering next (12) and (15) gives We notice that (23)-(24) together with (8) constitutes restrictions on possible values of λ 0 . Those are described by Theorem 3.1 below. Before, let us consider an auxiliary Dirichlet problem Let {λ D j } ∞ j=1 be eigenvalues for (25), labelled in the ascending order counting for the multiplicities, and let {φ j } ∞ j=1 be the corresponding eigenfunctions, orthonormal in where δ jk is Kronecker's delta. Denote by σ D the spectrum of (25): σ D = ∞ j=1 λ D j . We additionally introduce the following auxiliary problem: Notice that (26) is solvable if and only if λ 0 ∈ σ D or λ 0 = λ D j with all the associated eigenfunctions φ j having zero mean, φ j = 0 1 . In the former case η is determined uniquely and (23) implies w 0 (x, y) = v 0 (x)η(y). In the latter case η is determined up to an arbitrary eigenfunction φ j associated with λ D j , however η is determined uniquely.
By direct inspection, (23), (24) has a non-trivial solution (v 0 , w 0 ), i.e. with (8) holding, if and only if λ 0 is an eigenvalue of following problem: Theorem 3.1. The problem (27) is equivalent to an eigenvalue problem for a selfadjoint operator in L 2 (Q 0 ) with a compact resolvent. Therefore the spectrum of (27) is a countable set of real non-negative eigenvalues (of finite multiplicity) with the only accumulation point at +∞, with the eigenfunctions complete in L 2 (Q 0 ) and those corresponding to different λ 0 mutually orthogonal. The spectrum consists of all the eigenvalues λ D of problem (25) with a zero mean eigenfunction and all the solutions of the equation (which are hence all real non-negative). In (28) the summation is with respect to only those λ D j for which there exists an eigenfunction with a non-zero mean. The associated eigenfunctions ζ are either proportional to η as in (26) or are eigenfunctions of (25) with zero mean.
Proof. We claim that (27) corresponds to a self-adjoint operator associated with the (symmetric, closed, densely defined, bounded from below) Dirichlet form To see this, in the weak formulation of the eigenvalue problem associated with (29)-(30) we first set h to be an arbitrary function from C ∞ 0 (Q 0 ) which implies −∆ y ζ = λ 0 ζ in Q 0 , and then set h ≡ 1 yielding λ 0 ζ = 0. Further, since the resolvent is obviously compact, each eigenvalue has a finite multiplicity, the set of all eigenfunctions ζ is complete in L 2 (Q 0 ) and those corresponding to different λ 0 are mutually orthogonal.
Obviously, the spectrum of (27) includes those and only those eigenvalues of (25) which have an eigenfunction φ j with zero mean. In this case corresponding eigenfunctions of (27) are given by ζ j = Cφ j , C = 0. If λ D j does not have a zeromean eigenfunction, then the solvability of (27) requires ζ y∈Γ = 0 implying ζ ≡ 0.
The formula (28) can be transformed to read where function β(λ) has been introduced by Zhikov [19]: see Figure 2. This implies that λ 0 is either a solution to the nonlinear equation as visualized on Figure 2, or is an eigenvalue of (25) with a zero mean eigenfunction.

Remark 1.
If Q 0 is a ball of radius 0 < a < 1/2, i.e Q 0 = B a = {y : |y| < a} + y 0 , then we have an explicit representation for β(λ). Indeed, for λ 0 ∈ σ D the solution of (26) is radially symmetric and (placing the origin in the ball's centre) reads In particular, for n = 3 we have, We next explore in detail the further steps in the method of asymptotic expansions, to determine v 0 , etc. Let us consider a K-dimensional eigenspace (K ≥ 1) for a given eigenvalue λ 0 of (27), and let ζ 1 , . . . , ζ K be associated linearly independent eigenfunctions. Then, (23) and (24) imply Following Theorem 3.1 we distinguish two cases: (a) λ 0 ∈ σ D . In this case (26) and (23) suggest The latter means λ 0 = λ D j for some j. This includes two further possibilities: (i) The eigenspace of (25) has an eigenfunction φ * j with a non-zero mean. Since the solvability conditions for (23) include if K D = 1 then w 0 = C(x)φ * j and thus (24) implies C(x) ≡ 0 and w 0 ≡ 0 contradicting to (8). Hence w 0 is given by (37) with K = K D − 1, with ζ k , k = 1, ..., K being linearly independent eigenfunctions of (25) with zero mean (such K eigenfunctions exist).
(ii) All of the eigenfunctions corresponding λ D j have a zero mean. In this case w 0 is again given by (37) In this case λ 0 are solutions of (36). There is a countable set of λ 0 = µ j , j = 1, 2, . . . as Figure 2 illustrates. Note that this includes λ 0 = 0. Function β blows up at the points λ D j , which are eigenvalues of (25) having an eigenvalue with a non-zero mean, monotonically increasing between such points. It also directly follows from (35) Let λ 0 satisfying (36) be fixed. We consider problem (21) taking into account (38), i.e.
where η(y) solves (26) and is given by (33). Hence v 1 is a solution to a problem depending linearly on v 0 and ∇ x v 0 , implying with an arbitrary function v * 1 (x). The choice of v * 1 does not affect the subsequent constructions, so we set for simplicity v * 1 ≡ 0. In (41) functions N j and N are solutions to the problems Solvability of (43) requires Γ ∂η ∂n y dy = 0, which is equivalent to (32) and is hence already assured. Since the solutions of (42) and (43) are unique up to an arbitrary constant, we fix those by choosing We next consider the problem for w 1 , which from (13) and (15) combined with (38) reads Since the problem depends linearly on v 0 , λ 1 v 0 and ∂v 0 ∂xj , the solution admits representation where functions M j , P and R are solutions to the problems Since by the assumption λ 0 ∈ σ D , all the problems (47) -(49) are uniquely solvable. The problem for v 2 is in turn given by (11) and (18), whose solvability condition hence reads with functions v 1 , w 1 and w 0 given by (41), (46) and (38) respectively. Appendix A provides a detailed calculation showing that the above yields the following equations for v 0 : is the classical homogenized matrix for periodic perforated domains, see e.g. [11] (53) Note that the problem (51)-(52) involves ν = ν(λ 1 ) as a spectral parameter. The spectrum of (51)-(52) consists of a countable set of eigenvalues (56) 0 < ν 1 < ν 2 ≤ · · · ≤ ν n ≤ · · · → +∞.
Corresponding eigenfunctions v n form an orthonormal basis in L 2 (Ω) , Fixing an eigenvalue ν of (51), (52) with corresponding eigenfunction v 0 of unit norm in L 2 (Ω), according to (54) we find (57) The following diagram summarizes the algorithm for constructing the first terms of the asymptotic expansions (for the case λ 0 ∈ σ D ) → v 2 .
We can additionally construct w 2 from (14) and (15), whose unique solution exists for any choice of λ 2 . For purposes of the justification of the first two terms in the asymptotics (the next section) it is sufficient to set λ 2 = 0 and fix the corresponding solution w 2 .
This completes constructing a formal asymptotic approximation, which we now summarize. We introduce an approximate eigenvalue and corresponding approximate eigenfunction The essence of the above formal asymptotic construction is that the action of differential operator A ε on W ε defined by (60) A ε W ε := div (a ε ∇W ε ) + Λ ε ρ ε W ε produces a small right-hand side in both Ω ε 1 and Ω ε 0 , and on the interface Γ ε in the following sense.
Proof. (i) Since the function W ε is two-scale by the construction, in Ω ε Since v 1 is a solution to (40), the coefficient of ε −1 vanishes. The same is with the coefficient of ε 0 since v 2 satisfies (11). Functions v 0 , v 1 and v 2 are solutions of elliptic problems with smooth enough coefficients to guarantee belonging solutions to C 2 . Thus, maxima for coefficients of ε 1 , ε 2 and ε 3 in (61) exist.
(iii) Using (19), we obtain The coefficients of ε 0 and ε 1 vanish because of (21) and (18) respectively. The rest of the coefficients are smooth enough to guarantee that their maxima for x ∈ Γ ε and y ∈ Γ exist.

Case
For simplicity, we consider here only the case of eigenvalues of multiplicity K = 1 with zero mean eigenfunction (φ = φ j ), assuming additionally λ 0 is not a solution of (34). All other degenerate cases, see page 7, could be considered similarly.
In this case we can introduce a refined approximation for the eigenfunction where (64) w 0 (x, y) = c(x)φ(y).

Justification of asymptotics
4.1. Operator formulation. We use a standard notation for Lebesgue and Sobolev spaces: L 2 p (Ω) is a p-weighted L 2 -space of square-integrable functions in Ω. Notation (·, ·) H is used for a scalar product in a Hilbert space H.
Let L ε = L 2 ρε (Ω) and H ε be H 1 0 (Ω) Sobolev space with a scalar product Following a standard procedure, see e.g. [11], we introduce a bounded operator Note that operator B ε is positive, self-adjoint and compact for any fixed ε > 0 (since its image is in H ε ). Eigenvalue problem (2) is equivalent to Hence the spectrum of the problem consists of a countable set of eigenvalues 0 < λ ε 1 < λ ε 2 ≤ · · · ≤ λ ε k ≤ · · · → +∞, with the only accumulation point at +∞. Moreover, the set of corresponding eigenfunctions is complete in L ε .

Case (a).
In this Section we justify the leading terms of asymptotic expansions constructed above in case λ 0 ∈ σ D and thus v 0 ≡ 0, see Section 3.1. Let λ 0 be a solution to equation (36). All the functions (η, N j , N , M, P, R, w 0 , w 1 , v 1 and w 2 , v 2 ) are as defined in Section 3.1. We also fix λ 1 according to (57). The approximate eigenvalue Λ ε and eigenfunction W ε are given by (58) and (59) respectively.
Notice that although W ε ∈ H 1 (Ω) since W ε 1 = W ε 0 , it does not satisfy the zero Dirichlet boundary conditions on ∂Ω. To fix this we introduce the following boundary-layer corrector to our approximation.
Proof. Clearly such solution of (70), (71) does exist. On each of the subsets Ω ε 1 and Ω ε 0 the coefficients of (70) are smooth. Then the function V ε can reach its positive maximum or negative minimum only at the boundaries Γ ε or ∂Ω. Let us prove that this cannot be Γ ε . Suppose to the contrary the existence of x * ∈ Γ ε such that max Ω |V ε | = |V ε (x * )|. The strong maximum principle yields that there is no more point inside Ω ε 1 or Ω ε 0 where the maximum is reached. Without loss of generality we assume V ε (x * ) > V ε (x) for any x ∈ Ω\Γ ε and V ε (x * ) ≥ 0 (otherwise the point would be a positive maximum for −V ε and we would then consider −V ε ). Then by the virtue of Hopf's Lemma [8, p.330] applied in the relevant component of Ω ε From transmission conditions (71) we have that the normal derivative on the Ω ε 1 side of domain is also positive. Therefore the value of V ε increases from the point x * inside Ω ε 1 in the n-direction and hence x * is not a point of maximum of V ε in Ω ε 1 . The contradiction proves that |V ε | reaches it's maximum at ∂Ω. Then, from boundary conditions (71), Obviously U ε = W ε +V ε satisfies zero boundary condition on ∂Ω and thus belongs to H 1 0 (Ω).
Remark 2. Notice that (86) implies weaker but more transparent interpretations on the approximate eigenfunctions. For example, introducing we claim that with appropriate d j (ε). Note that u(·, · ε ) L 2 (Ω) ≥ C 0 > 0. Then (89) follows from (86) by splitting its left hand side into the parts corresponding to Ω ε 1 and Ω ε 0 , removing the weight, retaining only the main-order terms and then adding the inequalities up.
We also remark that, in principle, the result (86) on the convergence of eigenfunctions could be further sharpened, e.g. using the technique of two-scale convergence, cf. Section 5 below and [5].

Case (b).
In this section we assume that λ 0 = λ D j for some j, its multiplicity is equal to 1 and the corresponding eigenfunction φ has zero mean, i.e. φ = 0, see Section 3.2.
Proof. Proof of this theorem literally follows the proof of Theorem 4.6 with reference to Lemma 3.3.
A direct analogue of Remark 2 also holds.

On the eigenfunction convergence
In this section we give a brief sketch of further refinement of the presented results using the technique of two-scale convergence, [15,1,19].
First, the inclusions intersecting or touching the boundary are "excluded", e.g. by re-defining a ε and ρ ε there as in the matrix phase (a ε (x) = ρ ε (x) = 1). Denoting now via ε → 0 an appropriate subsequence in ε, without relabelling, let u ε and λ ε be eigenfunctions and eigenvalues of the original problem, with normalization The boundedness of u ε in L 2 (Ω) is then implied by (92) e.g. via the uniform positivity of the double-porosity operator whose form is given by the left hand side of (92), [19,Thm 8.1]. This implies that, up to a subsequence, u ε 2 ⇀ u(x, y) and ε∇u ε 2 ⇀ ∇ y u(x, y), where u ∈ L 2 (Ω, H 1 per ) and 2 ⇀ denotes weak two-scale convergence. Additionally, since (92) implies ε ∇u ε L 2 (Ω ε 1 ) → 0, [19,Thm 4.1] assures that the two-scale limit is independent of y in the matrix, i.e. is exactly in the form (88). Further, by [19,Thm 4.2], v 0 ∈ H 1 0 (Ω) and where p ∈ L 2 (Ω, V pot ) with θ ε 1 and θ 1 (y) denoting the characteristic functions of Ω ε 1 and Q 1 , respectively, and V pot denoting the space of potential vector fields on Q 1 , i.e. with respect to the Lebesgue measure supported on Q 1 , cf. [19, §3.2]. Let λ ε → λ 0 and (λ ε − λ 0 )/ε → λ 1 . Selecting then in (2) appropriate oscillating test functions φ = φ ε one can pass to the limit recovering the weak forms of the equations derived in Section 3. For example, selecting φ ε (x) = εψ( This can be seen to be a weak form of (23) and (21). Selecting further φ ε (x) = ψ(x) can be seen, after some careful technical analysis, to recover (51), (52) and (54). The above implies that as long as (v 0 ) 2 + w 2 0 ≡ 0, λ 0 , λ 1 , v 0 and w 0 can only be those constructed in Section 3. This does not however rule out the possibility that v 0 and w 0 are both trivial (equivalently, the two-scale limit u(x, y) is identically zero). Therefore additional two-scale compactness type arguments are required, cf. [19,Lemma 8.2]. In fact, following literally the argument of Zhikov one observes that the two-scale compactness of the eigenfunctions does hold, i.e. u ε 2 → u(x, y), where 2 → denotes strong two-scale convergence, in particular there is a convergence of norms: However, this in turn does not rule out the possibility of u ε → 0 with the normalization (92), which requires a separate analysis. We announce here a partial result with this effect, postponing detailed discussions for future.
We remark that the above statement does not provide a full analogue of Hausdorff convergence of the spectra as in the double porosity case [19,Thm 8.1]. It does ensure however the existence of the gaps (on Figure 2, (λ D j , µ j+1 ), j ≥ 1) and of the spectrum accumulation near the left ends µ j , j ≥ 1, of the "bands" [µ j , λ D j ]. However it does not clarify whether the "rests" of the bands, (µ j , λ D j ] could be accumulation points. We conjecture that they could. For a chosen λ 0 = µ j there exist infinitely many λ 1 = λ (n) 1 according to (57), (56), and λ (n) 1 → +∞ as n → ∞. On any band, for any small enough ε there exists a finite but infinitely increasing number N (ε) of eigenvalues according to (85). The issue is hence, in a sense, whether ελ (n) 1 may become of order one for large n (n ∼ N (ε)). For λ (n) 1 ∼ ε −1 , according to (57) ν ∼ ε −1 , and hence, formally, the solutions v 0 of the homogenized equation (51) becomes oscillatory on the scale x/ε 1/2 . One can attempt deriving asymptotic expansions similarly to those in Section 3, involving this new scale. A preliminary analysis has shown that those have formal solutions near every point inside the band. More detailed analysis is beyond the scope of the present work.
Appendix A. Derivation of the limit equation for v 0 .
Proof. We estimate the linear coefficient Note that where (49) and (26)  = P k on Γ.
A solution to this problem exists since φ = 0 and we can present it as: (B.13) where < V 1 >= 0 and A is a constant which will be determined later.
Equating to zero the term of order ε 0 in (B.6), and using (B.8), (B.9) we obtain problems for Z The problem for P k has the form: (B.22) ∆ y P k (y) = −2 ∂V 1 ∂y k in Q 1 , ∂P k ∂n y = ∂Z