Homogenization with doubly high contrasts

Homogenization of spectral problems in bounded domains with doubly high contrasts

Abstract.

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 proved.

2000 Mathematics Subject Classification. Primary: 35B27; Secondary: 34E

Key words. Homogenization, periodic media, high-contrasts, eigenvalue asymptotics

1. 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 is critically scaled again the periodicity size , , 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 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)

with Dirichlet boundary conditions on the exterior boundary, most generally, both and are -periodic, in the connected matrix and , 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 and . For , it is not hard to see that it is when the spectral problems at the macro and micro-scales are coupled in a non-trivial way. To explore this, we choose and 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 in the matrix and a function of both and the fast variable in the inclusions, the limit equations themselves are quite different. We show that there exist asymptotic series of eigenvalues with 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 in the matrix. In fact, is either a solution of , where is a function introduced by Zhikov [19], or is an eigenvalue of the Dirichlet Laplacian in the inclusion with a zero mean eigenfunction. In the matrix, , where is an eigenfunction of the homogenized operator in , whose eigenvalue determines the second term in the asymptotics of , see (57). This is first derived via formal asymptotic expansions, but then we prove a non-standard error bound:

see Theorems 86 & 4.7. The proof employs a combination of a high contrast boundary layer analysis with maximum principle and estimates in Hilbert spaces with -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 , 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.

2. Problem statement and notations

We consider a model of eigenvibrations for a body occupying a bounded domain in () containing a periodic array of small inclusions, see Figure 1. The size of inclusions is controlled by a small positive parameter , . First we introduce necessary notation.

Figure 1. The geometry and the periodicity cell

Let be a reference periodicity cell in . Let be a periodic set of “inclusions”, i.e. , , and is a reference inclusion lying inside with -smooth boundary , see Figure 1. Let , , . Introducing we refer to as to a fast variable, as opposes to the slow variable . In the -variable the periodicity cell is . If then , . We denote , , , see Figure 1. The trace on of function is denoted by . Let be the outer unit normal to on its boundary and let denote the similar normal on .

Let stiffness and density be as follows

with a small positive .

We study the asymptotic behaviour of self-adjoint spectral problem

(2)

as . If and are smooth enough then variational problem (2) can be equivalently represented in a classical formulation

(3)
(4)

implying that at the interfaces the transmission conditions are satisfied

(5)

3. Formal asymptotic expansions

We seek formal asymptotic expansions for the eigenvalues and eigenfunctions in the form

(6)
(7)

Here all the functions , , , are required to be periodic in the “fast” variable ; and are not simultaneously identically zero

(8)

In a standard way, the ansatz (6), (7) is then formally substituted into (3)–(5). In particular, from (3), for , we obtain

(9)
(10)
(11)

(with and denoting the Laplace operators in and , respectively, and summation henceforth implied with respect to repeated indices), and for we have

(12)
(13)
(14)

Further, the first of conditions (5) transforms to

(15)

Similarly, the other transmission condition (5) yield

(16)
(17)
(18)

The above has employed the identity

(19)

where , , with and standing for gradients in and , respectively.

Finally, (4) suggests

(20)

(The boundary layer problem does not generally permit satisfying (4) by and for , as also clarified later.)

Combining (9) and (16), together with the periodicity conditions in , implies that is a constant with respect to , i.e.

Then, (10) and (17) form the following boundary value problem for

(21)

The latter is solvable if and only if

(22)

Considering next (12) and (15) gives

(23)

Since

condition (22) is equivalent to

(24)

where

We notice that (23)–(24) together with (8) constitutes restrictions on possible values of . Those are described by Theorem 3.1 below. Before, let us consider an auxiliary Dirichlet problem

(25)

Let be eigenvalues for (25), labelled in the ascending order counting for the multiplicities, and let be the corresponding eigenfunctions, orthonormal in , i.e.

where is Kronecker’s delta. Denote by the spectrum of (25): .

We additionally introduce the following auxiliary problem:

(26)

Notice that (26) is solvable if and only if or with all the associated eigenfunctions having zero mean, 1. In the former case is determined uniquely and (23) implies . In the latter case is determined up to an arbitrary eigenfunction associated with , however is determined uniquely.

By direct inspection, (23), (24) has a non-trivial solution , i.e. with (8) holding, if and only if is an eigenvalue of following problem:

(27)
Theorem 3.1.

The problem (27) is equivalent to an eigenvalue problem for a self-adjoint operator in 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 and those corresponding to different mutually orthogonal.

The spectrum consists of all the eigenvalues of problem (25) with a zero mean eigenfunction and all the solutions of the equation

(28)

(which are hence all real non-negative). In (28) the summation is with respect to only those for which there exists an eigenfunction with a non-zero mean.

The associated eigenfunctions are either proportional to 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

(29)

with domain

(30)

To see this, in the weak formulation of the eigenvalue problem associated with (29)–(30)

(31)

we first set to be an arbitrary function from which implies in , and then set yielding . Further, since the resolvent is obviously compact, each eigenvalue has a finite multiplicity, the set of all eigenfunctions is complete in and those corresponding to different are mutually orthogonal.

Obviously, the spectrum of (27) includes those and only those eigenvalues of (25) which have an eigenfunction with zero mean. In this case corresponding eigenfunctions of (27) are given by , . If does not have a zero-mean eigenfunction, then the solvability of (27) requires implying . Considering other possibilities, fix outside and let be the unique solution of (26). Then is an eigenvalue of (27) if and only if

(32)

with corresponding eigenfunction given by , .

Via the spectral decomposition, the solution to (26) is found to be, cf. [19]:

(33)

Substituting (33) further into (32) yields (28). ∎

The formula (28) can be transformed to read

(34)

where function has been introduced by Zhikov [19]:

(35)

see Figure 2. This implies that is either a solution to the nonlinear equation

(36)

as visualized on Figure 2, or is an eigenvalue of (25) with a zero mean eigenfunction.

Figure 2. The limit eigenvalues
Remark 1.

If is a ball of radius , i.e , then we have an explicit representation for . Indeed, for the solution of (26) is radially symmetric and (placing the origin in the ball’s centre) reads where is Bessel function. Further, we have Using (35), (33) we obtain In particular, for we have,

We next explore in detail the further steps in the method of asymptotic expansions, to determine , etc. Let us consider a -dimensional eigenspace () for a given eigenvalue of (27), and let be associated linearly independent eigenfunctions. Then, (23) and (24) imply

(37)

Following Theorem 3.1 we distinguish two cases:

  • . In this case (26) and (23) suggest

    (38)

    and (8) implies .

  • . The latter means for some . This includes two further possibilities:

    • The eigenspace of (25) has an eigenfunction with a non-zero mean. Since the solvability conditions for (23) include

      (39)

      necessarily . Moreover, with denoting the multiplicity of as of the eigenvalue of the Dirichlet problem (25), necessarily : if then and thus (24) implies and contradicting to (8). Hence is given by (37) with , with , being linearly independent eigenfunctions of (25) with zero mean (such eigenfunctions exist).

    • All of the eigenfunctions corresponding have a zero mean. In this case is again given by (37), with if i.e. and if with where is any solution of (26).

3.1. Case (a):

In this case are solutions of (36). There is a countable set of , as Figure 2 illustrates. Note that this includes . Function blows up at the points , which are eigenvalues of (25) having an eigenfunction with a non-zero mean, monotonically increasing between such points. It also directly follows from (35) that for , implying . Let satisfying (36) be fixed.

We consider problem (21) taking into account (38), i.e.

(40)

where solves (26) and is given by (33). Hence is a solution to a problem depending linearly on and , implying

(41)

with an arbitrary function . The choice of does not affect the subsequent constructions, so we set for simplicity . In (41) functions and are solutions to the problems

(42)

and

(43)

Solvability of (43) requires

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 , which from (13) and (15) combined with (38) reads

(44)
(45)

Since the problem depends linearly on , and , the solution admits representation

(46)

where functions , and are solutions to the problems

(47)
(48)

and

(49)

Since by the assumption , all the problems (47) – (49) are uniquely solvable.

The problem for is in turn given by (11) and (18), whose solvability condition hence reads

(50)

with functions , and given by (41), (46) and (38) respectively.

Appendix A provides a detailed calculation showing that the above yields the following equations for :

(51)
(52)

Here is the classical homogenized matrix for periodic perforated domains, see e.g. [11]

(53)
(54)

where

(55)

Note that the problem (51)–(52) involves as a spectral parameter.

The spectrum of (51)–(52) consists of a countable set of eigenvalues

(56)

Corresponding eigenfunctions form an orthonormal basis in ,

Fixing an eigenvalue of (51), (52) with corresponding eigenfunction of unit norm in , according to (54) we find

(57)

The following diagram summarizes the algorithm for constructing the first terms of the asymptotic expansions (for the case )

We can additionally construct from (14) and (15), whose unique solution exists for any choice of . For purposes of the justification of the first two terms in the asymptotics (the next section) it is sufficient to set and fix the corresponding solution .

This completes constructing a formal asymptotic approximation, which we now summarize. We introduce an approximate eigenvalue

(58)

and corresponding approximate eigenfunction

(59)

The essence of the above formal asymptotic construction is that the action of differential operator on defined by

(60)

produces a small right-hand side in both and , and on the interface in the following sense.

Lemma 3.2.

Proof.

Since the function is two-scale by the construction, in

(61)

Since is a solution to (40), the coefficient of vanishes. The same is with the coefficient of since satisfies (11). Functions , and are solutions of elliptic problems with smooth enough coefficients to guarantee belonging solutions to . Thus, maxima for coefficients of , and in (61) exist.

Similarly, in

Since is chosen according to (26), the coefficient of vanishes. The coefficient of vanishes due to (44). Further, satisfies (14) with and thus the coefficient of is zero as well. Since and are solutions of elliptic problems with smooth enough coefficients, the maxima of the coefficients of and exist.

Using (19), we obtain