ABSTRACT
Using a suitable stochastic version of the compactness argument of [Zhikov VV. On an extension of the method of two-scale convergence and its applications. Sb Math. 2000;191(7–8):973–1014], we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled ‘macroscopic’ and ‘microscopic’ components of the field, giving rise to an analogue of the ‘Zhikov function’, which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit.
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1. Introduction
Asymptotic analysis of differential equations with rapidly oscillating coefficients has featured prominently among the interests of the applied analysis community during the last half a century. The problem of understanding and quantifying the overall behaviour of heterogeneous media has emerged as a natural step within the general progress of material science, wave propagation and mathematical physics. In this period several frameworks have been developed for the analysis of families of differential operators, functionals and random processes describing multiscale media, all of which have benefitted from the invariably deep insight and mathematical elegance of the work of V. V. Zhikov. In the present paper, we touch upon two subjects in which his contributions have inspired generations of followers: the stochastic approach to homogenisation, in particular through his collaboration with S. M. Kozlov during the 1980s, and the analysis of differential operators describing periodic composites with high contrast, which started with his fundamental contribution [Citation1].
Our present interest in the context of stochastic homogenisation of high-contrast composites stems from the relationships that have recently been indicated between media with negative material properties (‘metamaterials’), and more generally time-dispersive media, and ‘degenerate’ families of differential operators, where e.g. loss of uniform ellipticity of the symbol is known to lead to non-classical dispersion relations in the limit of vanishing ratio ϵ of the microscopic (l) and macroscopic (L) lengths: . The work [Citation1] has provided an example, in the periodic context, of what one should expect in the limit as in terms of the two-scale structure of the solution as well as the spectrum of the related differential operator, in the case when the metamaterial is modelled by disjoint ‘soft’ inclusions with low, order values of the material parameters (say, elastic constants in the context of linearised elasticity), embedded in a connected ‘stiff’ material with material constants of order . In mathematical terms, the coefficients of the corresponding differential expression alternate between values of different orders in ϵ, where the contrast increases as ϵ gets smaller.
In the present article, we introduce a stochastic framework for the analysis of homogenisation problems with soft inclusions and explore the question on what version of the results of [Citation1] can be achieved in this new framework. In particular, we are interested in the equations that describe the stochastic two-scale limit, in an appropriate sense, of the sequence of solutions to the probabilistic version of a Dirichlet problem in a bounded domain of . Furthermore, we show that the spectra of such problems converge, in the Hausdorff sense, to the spectrum of the limit problem, which we analyse in a setting that models distributions of soft inclusions whose shapes are taken from a certain finite set and whose sizes vary over an interval. To our knowledge, the present manuscript is the first work containing an analysis of random heterogeneous media with high contrast that results in a ‘complete’ Hausdorff-type convergence statement for the spectra of the corresponding differential operators. Various aspects of multiscale analysis of high-contrast media in the stochastic context have been looked at in a handful of papers, e.g. [Citation2–4].
While in the periodic context norm-resolvent convergence results been obtained for high-contrast media, see [Citation5,Citation6], the stochastic case remains open to developments of a similar nature. It is anticipated that the operator-theoretic approach to problems of the kind we discuss in the present article will provide a general description of the types of spectral behaviour that can occur in the real-world applications where it is difficult to enforce periodicity of the microstructure. On the other hand, as we show in the present work, new wave phenomena should be expected in the stochastic setting (e.g. a non-trivial continuous component of the spectral measure of the homogenised operator for a bounded-domain problem), which makes the related future developments even more exciting.
Next, we outline the structure of the paper. In Section 2 we recall the notion of stochastic two-scale convergence, which we use, in Sections 3 and 4, to pass to the limit, as in a family of homogenisation problems with random soft inclusions. In Section 3 we give a formulation of the high-contrast problem we study and provide some auxiliary statements. In Section 4 we describe the limit problem and prove the strong resolvent convergence of the ϵ-dependent family to the limit system of equations. In Section 5 we provide a link between the spectra of the Laplacian operator on realisations of the inclusions and of the corresponding stochastic Laplacian. In Section 6 we prove that sequences of normalised eigenfunctions of the ϵ-dependent problems are compact in the sense of strong stochastic two-scale convergence. Finally, in Section 7 we discuss two examples of the general stochastic setting and describe the structure of the corresponding limit spectrum.
In conclusion of this section, we introduce some notation used throughout the paper. For a Banach space X and its dual we denote by the corresponding duality. For a Hilbert space H the inner product of is also denoted by and, if by . For a set we denote by its characteristic function, which takes value one on the set and zero on the complement to in the appropriate ambient space. For we denote by its closure and by its Lebesgue measure. Further, we use the notation for the ball in of radius r with the centre at the origin; Y denotes the cube with torus topology, where the opposite faces are identified; and . For an operator on some Hilbert space, we denote by its spectrum. Finally, for a Lipschitz open set we denote by the (positive) Laplace operator with the Dirichlet boundary condition on . For , we denote by the element of which satisfies . For , by we denote the kth coordinate vectors.
2. Stochastic two-scale convergence
2.1. Probability framework
Let be a complete probability space. We assume that is countably generated, which implies that the spaces , are separable. For a function we will sometimes write for .
Definition 2.1
A family of measurable bijective mappings on a probability space is called a dynamical system on Ω with respect to P if:
;
, ;
is measurable (for the standard σ-algebra on the product space, where on we take the Lebesgue σ-algebra).
We next define the notion of ergodicity for dynamical systems introduced above.
Definition 2.2
A dynamical system is called ergodic if one of the following equivalent conditions is fulfilled:
f measurable, .
.
Henceforth we assume that the dynamical system is ergodic.
Remark 2.1
Note that for the condition (b) the implication has to hold, if the symmetric difference between and B is a null set. It can be shown (see, e.g. [Citation7]) that ergodicity is also equivalent to an a priori weaker implication
For we write , defining the realisation . There is a natural unitary action on associated with (1) (1) It can be shown that the conditions of Definition 2.1 imply that this is a strongly continuous group (see [Citation8]). It is often necessary that the set of full measure be invariant in the sense that together with the point ω it contains the whole ‘trajectory’ . This requirement can always be met on the basis of the following simple lemma (see [Citation8, Lemma 7.1]).
Lemma 2.1
Let be a set of full measure in Ω. Then there exists a set of full measure such that , and for a given we have for almost all .
For each we define the infinitesimal generator of the unitary group by the formula (2) (2) where the limit is taken in . Notice that are commuting, self-adjoint, closed and densely defined linear operators on the separable Hilbert space . The domain of such an operator is given by the set of -functions for which the limit (Equation2(2) (2) ) exists. We consider the set (3) (3) and similarly It is shown by the standard semigroup property that is dense in . We also define the space By the smoothening procedure discussed in [Citation8, p.232] (see also the text before Lemma 3.1 below), it is shown that is dense in for all as well as in for all k. Furthermore, it is shown that is separable. Notice that, due to the infinitesimal generator being closed, can be equivalently defined as the function that satisfies the property (4) (4) If we may also define for all . It can be shown that the following identity holds (see [Citation9]): (5) (5) Moreover, for a.e. the function is the distributional derivative of a proof of this fact can be found in [Citation9, Lemma A.7].
Following [Citation10], we denote by the seminorm on given by By we denote the completion of with respect to the seminorm . The gradient operator and the operator are extended uniquely by continuity to mappings from to and from to respectively. Finally, by a density argument, it is easily seen that is also the completion of with respect to .
2.2. Definition and basic properties
The key property of ergodic systems is the following theorem, due to Birkhoff (for a more general approach, see [Citation11]).
Theorem 2.1
‘Ergodic Theorem’
Let be a probability space with an ergodic dynamical system on Ω. Let , and let be a bounded open set. Then for P-a.e. one has (6) (6) Furthermore, for all and a.e. the function satisfies . For one has weakly in as .
The elements ω such that (Equation6(6) (6) ) holds for every and bounded open are refereed to as typical elements, while the corresponding sets are called typical trajectories. Note that the separability of implies that almost every is typical, and in what follows we only work with such ω.
For vector spaces we denote by by their usual tensor product. We define the following notion of stochastic two-scale convergence, which is a slight variation of the definition given in [Citation12]. In [13], the authors average over the probability space and do not use the Birkhoff Ergodic Theorem. As a consequence, they do not obtain convergence almost everywhere but only in mean, which results in a weaker notion of stochastic two-scale convergence than the one introduced in [12]. In the context of calculus of variations, the first results are obtained in [14,15]. The authors of these papers do not use stochastic two-scale convergence at all, as this was introduced later on, but rely on a formula for non-periodic homogenisation for a.e. ω∈Ω as well as on the ergodic theorem. We shall stay in the Hilbert setting (p=2), as it suffices for our analysis. Finally, we denote by S a bounded open LIpschitz set in .
Definition 2.3
Let be a typical trajectory and a bounded sequence in . We say that weakly stochastically two-scale converges to and write if (7) (7)
If additionally we say that strongly stochastically two-scale converges to u and write .
Remark 2.2
The convergence of is defined along a fixed typical trajectory and a priori the limit depends on this trajectory. In applications, such as the analysis of the PDE family in Section 4, it often turns out that the limit does not depend on the trajectory chosen. For this reason, and to simplify notation, in what follows we often do not indicate this dependence explicitly.
Note also that, by density, the set of admissible test functions g in (Equation7(7) (7) ) can be extended to .
In the next proposition, we collect the properties of stochastic two-scale convergence that we use in the present work.
Proposition 2.1
The following properties of stochastic two-scale convergence hold.
Let be a bounded sequence in . Then there exists a subsequence not relabelled and such that .
If then .
If is a bounded sequence with in for some then .
If is uniformly bounded and strongly in for some and is bounded in with for some then .
Let be a bounded sequence in . Then on a subsequence (not relabelled) in and there exists such that
Let be a bounded sequence in such that is bounded in . Then there exists such that on a subsequence (8) (8)
Proof.
In view of analogies with the periodic case, we just give a sketch of the proof. A proof of (a) can be found in [Citation12, Lemma 5.1]. For the proof of (b), we take an arbitrary and calculate We obtain the claim by approximating with functions . The proof of (c), (d) is straightforward. The proof of (e) goes in the same way as in the periodic case, by the duality argument.
First, one proves that if is such that then there exists such that . One then proceeds in the same way as in the periodic case (see [Citation16]). In order to show the claim (f), take the subsequence such that , where and , where . We choose the test functions of the form , where and , and using integration by parts we conclude from which the claim follows by a density argument, in view of the property (Equation4(4) (4) ).
3. Problem formulation and auxiliary statements
Let be a bounded open Lipschitz set. We take such that and for each consider its ‘realisation’ We assume that the following conditions are satisfied.
Assumption 3.1
For a.e. one has (9) (9) where:
are open connected sets with Lipschitz boundary;
One has
There exist such that
There exists a sequence of disjoint bounded domains such that and such that for all the following extension property holds: for all there exists satisfying
It is easily seen that Assumption 3.1 holds for the examples given in Section 7.1. Denote by Λ the set of typical elements satisfying the conditions listed in Assumption 3.1, and for all define as the union of all components that are subsets of S and stay sufficiently far from its boundary, in the sense that there exists such that (10) (10) We denote the complement of the set by and the corresponding set indicator functions by and .
For each we consider the following Dirichlet problem in S: for and find such that (11) (11) where with a symmetric and positive-definite matrix .
For all we also define the Dirichlet operator in corresponding to the differential expression e.g. by considering the bilinear form
It is well known that the spectrum of is discrete. The following subspace of will play an essential role in our analysis: Notice that as a consequence of Ergodic Theorem (Theorem 2.1) one has i.e. consists of -functions that vanish on . Henceforth we assume that without mentioning it explicitly.
The next two lemmas use a standard smoothening (or ‘mollification’) procedure, which we now describe. We take and (cf. [Citation8, p.232]) choose a nonnegative even function with and write for all . For each we define the mollification of g by Notice that (12) (12) from which we infer that Arguing by induction, we show that and if then . Before we state and prove the lemmas, we introduce additional notation. We define the space as well as the sets Also, for all we define the set By using the density of in it can be seen that for all the set is measurable. Notice that for each fixed where m is large enough, there exist constants such that (13) (13) In the next lemma we assume that a relaxed version of the right inequality in (Equation13(13) (13) ) holds uniformly in ω.
Lemma 3.1
Suppose that for a.e. there exists a sequence of positive values converging to zero, such that Then the set is dense in .
Proof.
Using Ergodic Theorem and the assumption of the lemma, it can be shown that as . To prove the density, it suffices to approximate , where by a function from for which we use the above mollification procedure. Notice that for small enough, one has . It remains to check as , but this follows from the strong continuity of the group see (Equation1(1) (1) ): as required.
Notice that, by the standard Poincaré inequality, for each there exists C>0 such that (14) (14) In the following lemma we impose this condition uniformly.
Lemma 3.2
Assume that for a.e. there exists a constant C>0 such that (15) (15) and that (Equation14(14) (14) ) is satisfied for all k and large enough m. Then the set is dense in .
Proof.
We take and define . Notice that as a consequence of (Equation5(5) (5) ), and . It is easily seen that in as . Thus we can assume, without loss of generality, that . We define . It can be seen from the proof of Lemma 3.1 that in as . Notice that for a.e. we have, for , and therefore
Notice also that for a.e. there exist such that for all where m is sufficiently large, we have where we have used (Equation12(12) (12) ), (Equation14(14) (14) ), (Equation15(15) (15) ) and Young's inequality. Using the Ergodic Theorem we conclude that there exists C>0 such that from which the claim follows.
4. Limit equations and two-scale resolvent convergence
We define the quadratic form and denote by the completion of with respect to the seminorm . The proof of the following lemma is straightforward.
Lemma 4.1
For each there exists such that or, equivalently, (16) (16) In particular, one has .
Remark 4.1
It follows from the observations in [Citation12, p.265–266] that if the following extension property is satisfied for a.e. : for all there exists and a sequence of functions such that where is a constant independent of u and ϵ, then the matrix is positive definite.
Notice that under Assumption 3.1, the extension property in Remark 4.1 is satisfied. We define the space which is clearly a direct sum, naturally embedded in . Before stating the next theorem we prove a simple lemma that implies that gives norm bounds for each component of H by the norm in .
Lemma 4.2
Let and such that on . Then there exists a constant C>0 such that where we use the natural embedding .
Proof.
By Cauchy-Schwartz inequality we have and hence (17) (17) It remains to bound by , which is done by the triangle inequality:
By we denote the orthogonal projection. For we have
Theorem 4.1
Under Assumption 3.1, let and suppose that is a bounded sequence in such that . For each consider the solution to (Equation11(11) (11) ). Then for a.e. one has where satisfy (18) (18) (19) (19)
Remark 4.2
The system (Equation18(18) (18) ) and (Equation19(19) (19) ) is understood in the weak sense: (20) (20) Noting that is a closed subspace of and bearing in mind Lemma 4.2, it follows by the Lax-Milgram lemma that for all the problem (Equation20(20) (20) ) has a unique solution in . Its solutions for the right-hand sides and coincide. The solution of the Equation (Equation19(19) (19) ) has the form (21) (21) where is the solution of (Equation19(19) (19) ) obtained by setting and is the solution of (Equation19(19) (19) ) obtained by setting and f=0. Substituting (Equation21(21) (21) ) into (Equation18(18) (18) ), we obtain an equation on .
Proof.
The proof follows a standard argument. First, by (Equation11(11) (11) ), there exists a constant C>0 such that (22) (22) For each we extend using Assumption 3.1, to a sequence which is bounded in . From Proposition 2.1 we infer that there exist , , such that on a subsequence we have (23) (23) To obtain the Equation (Equation18(18) (18) ), we take test functions of the form in (Equation11(11) (11) ), where , and . In the limit as we obtain (24) (24) Setting it follows that and the characterisation (Equation16(16) (16) ) yields a.e. . Taking arbitrary in (Equation24(24) (24) ), we obtain the ‘macroscopic’ part (Equation18(18) (18) ) of the limit problem. The ‘microscopic’ part (Equation19(19) (19) ) is obtained by taking test functions of the form in (Equation11(11) (11) ), where , . The convergence of the whole sequence can be deduced by uniqueness of the solution of the system (Equation18(18) (18) )–(Equation19(19) (19) ).
Remark 4.3
The following observation was made in [Citation1] in the periodic setting. The formulation (Equation20(20) (20) ) can be interpreted from the operator-theoretic point of view. Namely, we define a positive-definite operator on a dense linear subset of (which is a dense subset of H under the condition of Lemma 3.1), as follows. One takes and defines the domain as the set of solutions of (Equation20(20) (20) ) obtained for varying . To see that is dense in H, take the solutions for respectively. Setting , as the test function in the equation for and , as the test function in the equation for yields Thus, if then necessarily which implies g=0. The operator defined by is a bounded linear mapping between Hilbert spaces, where the norm on is given by
We shall need the following statement for the convergence of spectra of the operators associated with (Equation11(11) (11) ). It is proved in the same way as the previous theorem, and we omit the proof.
Proposition 4.1
Under Assumption 3.1, let and suppose that is a bounded sequence in such that . For each let be the solution of (25) (25) Then for a.e. one has where is the solution of the problem (26) (26)
Remark 4.4
Theorem 4.1 and Proposition 4.1 are still valid if, instead of a fixed we take a sequence such that and , for Theorem 4.1, i.e. for Proposition 4.1, where . Notice that splits into the spectra of scaled Laplace operators on each inclusion contained in (27) (27) where is defined in (Equation10(10) (10) ). Notice that there exists C>0 such that for all the solution of (Equation11(11) (11) ) satisfies (28) (28) and similarly the solution of (Equation25(25) (25) ) satisfies (29) (29)
In what follows we denote by the operator generated by the bilinear form (30) (30) As a consequence of Proposition 4.1 and Remark 4.4, we have the following statement.
Corollary 4.1
Assume that Assumption 3.1 holds. Then
Proof.
Take and and define . As a consequence of Remark 4.4, the sequence of solutions of (Equation25(25) (25) ) converges weakly two-scale to the solution of (Equation26(26) (26) ), which is a resolvent equation. Moreover, (Equation27(27) (27) ) and (Equation29(29) (29) ) imply the existence of C>0 such that and therefore .
5. Spectral completeness for inclusions
Next we prove that We shall use the assumptions of Lemma 3.2 as well as assume that for each there exists such that for a.e. the following implication holds: (31) (31) Notice that, by regularity theory, the above condition is satisfied for a fixed and whenever the boundary is sufficiently regular. In what follows we use a sequence that is dense in where the constant is defined in Assumption 3.1.
We will now define a sequence of random variables that is invariant for all whose realisation is such that the shape that contains the origin is the same. For define the set
Lemma 5.1
For every is measurable.
Proof.
Notice that (32) (32) Since for each fixed there is a countable set of lines satisfying the property (Equation32(32) (32) ), the set is measurable.
We define the random variables Notice that whenever and also, due to the assumption, for a.e. . We denote by F the random vector (33) (33) For a.e. , we define the set Furthermore, we introduce the set , which is a translation of the set containing the origin: Finally, we define a characteristic function of the translation of the set and a measurable function of ω taking values in : (34) (34) Notice that for a.e. one has .
Lemma 5.2
For every the function taking values in is measurable with respect to the Borel σ-algebra on .
Proof.
Firstly notice that (35) (35) is a measurable mapping taking values in the set , with Borel σ-algebra. To check this notice that for each the set is measurable: the related proof is similar to that of Lemma 5.1. Further, for the norm is written as a limit of Riemann sums, and each Riemann sum can be written in terms of a finite number of and values of function . Thus is measurable. Since the topology in is generated by the balls of the form , where and we have that the mapping given by (Equation35(35) (35) ) is measurable. The claim follows by using the fact that the convolution is a continuous (and thus measurable) operator from to .
Notice that by construction is a dense subset of for a.e. (see also the proof of Lemma 3.2). For we introduce the following subset of (36) (36) For and a.e. we also define as follows: Finally, for every and we define the random variable (37) (37)
Lemma 5.3
For every and , the function is measurable.
Proof.
We use Lemma 5.2 and the fact that is a continuous map from to and is a measurable function, since
Lemma 5.4
For the set is measurable.
Proof.
The claim follows by observing that
Now we are going to define a measurable mapping from to the subspace . We set it to be an -projection onto of a specially chosen function of x and ω. We need the following measurability lemma.
Lemma 5.5
Assume that is a measurable function taking values in for a.e. . Then the -distance is a measurable map.
Proof.
The claim follows from the formula
For and we define a measurable map as follows: where (38) (38) and is the minimal value of m in (Equation38(38) (38) ) Notice that in this way for a.e. the -projection of on is not zero. We also define the random variable in the following way: By invoking the measurability of see Lemma 5.1, it is easily seen that R is indeed measurable. Next, for , l>0 we define the random variable by where, for all is the corresponding minimal valueFootnote1 of m, and is the ball with the centre at D and radius see (5.1). We also define Notice that in this way is the value at the origin (taking into account for the relative position of the origin with respect to the shape) of the (unique) -projection of onto . As a consequence of (Equation31(31) (31) ), we have Notice that by construction if . We are ready for the proof of main statement.
Theorem 5.1
Under Assumption 3.1, the assumption of Lemma 3.2 and (Equation31(31) (31) ), one has
Proof.
We take . There are two possibilities:
There exists such that has zero probability. In this case we denote
For all the set has positive probability.
In the case (a), by the continuity of probability, we conclude that where (cf. (Equation36(36) (36) )) By Lemma 2.1 and Corollary 4.1 we infer that In particular, we conclude that .
In the case (b) we construct a Weyl sequence showing that . To this end, we define Then, by the above construction and using Ergodic Theorem, one has It follows from the above that consists of exactly those that satisfy the property (b). The set is closed, hence its complement is a countable union of open disjoint intervals. Every element of such an interval satisfies the property (a) with and therefore . Using Lemma 2.1, we obtain The claim follows since there is only countable number of such intervals.
6. Convergence of spectrum
In our analysis we keep in mind the examples set in Section 7, for which it is shown that . In the present section we assume that this holds, as well as the conclusion of Theorem 5.1, i.e. (39) (39) We are interested in approximating the spectra of the operators (see Section 3) by the spectrum of the limit operator. We claim that for a.e. where the convergence is understood in the Hausdorff sense:
For all there are such that .
If and , then .
We prove this claim by adapting the argument of [Citation17]. First, we introduce the notion of strong resolvent convergence.
Definition 6.1
Let and be the operators acting on and on respectively. We say that strongly two-scale resolvent converge to and write if
It can be shown that the property (a) is satisfied if we have strong two-scale resolvent convergence (see the proof of [Citation17, Proposition 2.2]). Theorem 4.1 shows that the following implication holds: It can be shown that this is equivalent to strong two-scale resolvent convergence (see [Citation17, Proposition 2.8]) and thus the property (a) is satisfied.
In order to prove (b), we start from the eigenvalue problem for the operator (it has a compact resolvent and its spectrum is discrete), i.e. (40) (40) If we have that and , then we would also have . However, the problem would be if u=0, because then . The next lemma tells us if then necessarily the sequence of eigenvalues is compact with respect to the strong two-scale convergence and thus s belongs to the point spectrum of the operator , since then necessarily .
Theorem 6.1
Suppose that (Equation39(39) (39) ) holds and that for each satisfy (Equation40(40) (40) ). If then for a.e. the sequence is compact in the sense of strong two-scale convergence.
Proof.
satisfies We use Assumption 3.1 and for each ϵ extend , denoting the extensions by . Notice that there exists C>0 such that (41) (41) The difference satisfies: (42) (42) From the estimate (Equation41(41) (41) ) we see that is weakly compact in and thus there exists such that which immediately implies . Furthermore, as a consequence of (Equation27(27) (27) ), (Equation29(29) (29) ) and (Equation39(39) (39) ), the following estimate holds for some Therefore, from Proposition 4.1 and Remark 4.4 we conclude that , where the limit z satisfies (43) (43) We also consider the problem (44) (44) In the same way as before we conclude that for some C>0: Analogously, we conclude that which satisfies (45) (45) By testing (Equation42(42) (42) ) with and (Equation44(44) (44) ) with we conclude Finally, by testing (Equation43(43) (43) ) with and (Equation45(45) (45) ) with and integrating over S we conclude which completes the proof.
7. Spectrum of the limit operator: examples
This section is devoted to the description of the spectrum of the limit operator. Since it crucially depends on the intrinsic properties of the microscopic part of the operator and the properties of the probability space, it does not seem feasible (at least at the current stage of research in this area) to provide a characterisation of the spectrum in a general setting. We shall consider several interesting, from the point of view of applications, examples of probability spaces and configurations of soft inclusions. The general example of a finite number of shapes of randomly varying size is described in Section 7.1. Then we consider the case of a single shape of fixed size in Section 7.2, and the case of a single shape of randomly varying size in Section 7.3, for which we provide the full description of the spectrum of the limit operator with the proofs. The characterisation of the spectrum in the general case of Section 7.1 is analogous to the case of a single shape considered in Section 7.3.
7.1. The setting of finite number of shapes of varying size
Let be a sequence of independent and identically distributed random vectors taking values in where and is an appropriate probability space. We also assume that we have a finite number of shapes that represent the inclusions, where the first and the second components of model the shape and the size, respectively. We also set . On there is a natural shift , which is ergodic. We next state the discrete analogue of Lemma 2.1.
Lemma 7.1
Assume that is a set of full measure. Then there exists a subset of full measure such that for each we have .
We treat as a probability space with Lebesgue measure dy and the standard algebra of Lebesgue measurable sets, and define On Ω we define a dynamical system . We consider the set . It is easily seen that is measurable. For a fixed the realisation consists of the inclusions . Next, we describe the generators in the present example. Taking and using the above lemma, note that there exists a subset of full measure such that for all and we have . It is clear that for one has Using this fact and the statement following (Equation5(5) (5) ), we infer that and (46) (46)
7.2. Simple example
In this section we set so that and, by a standard procedure, see e.g. [Citation18], identify the elements of the probability space with sequences whose components take values in the two-element set . Let be an open subset of Y whose closure is contained in Y (‘soft inclusion’). The value 0 or 1 of corresponds to the absence or the presence of the inclusion in the ‘shifted cell’ respectively. We also set Then, for a given the realisation is the union of the sets (‘inclusions’) over all such that . For this example the space consists of all functions of the form (47) (47) It is also important to understand how one applies the stochastic gradient. For a function we have (see (Equation46(46) (46) )) (48) (48) Consider formally the spectral problem for the limit operator: (49) (49) (50) (50) We write the solution to the ‘microscopic’ Equation (Equation50(50) (50) ) in the form , where (recall Remark 4.2) (51) (51) In other words, v is given by (Equation47(47) (47) ) with satisfying (52) (52) whenever such that and otherwise.
We label the eigenvalues of the operator in (Equation52(52) (52) ) in the increasing order, where we repeat multiple eigenvalues, so that and are, respectively, the eigenvalues whose eigenfunctions have non-zero integral over and the eigenvalues whose eigenfunctions have zero integral over . Following [Citation1], we write the solution to (Equation52(52) (52) ) via the spectral decomposition (53) (53) and thereby (54) (54) Substituting the obtained representation for into the ‘macroscopic’ equation (Equation49(49) (49) ) yields (55) (55) where (56) (56) is a stochastic version of the ‘Zhikov function’ β in [Citation1]. Assume for the moment that . Then the intervals where are the ‘spectral bands’ of and additionally a Bloch-type spectrum is given by . The set corresponds to the gaps in the spectrum of .
In the setting of this paper, namely, for a bounded domain instead of each spectral band lying to the left of we have a ‘band’ of discrete spectrum: a countable set of eigenvalues (57) (57) with the accumulation point at the right end of each band, where are the eigenvalues of the operator defined by the form (58) (58) The Bloch-type spectrum of consists of eigenvalues of infinite multiplicity with eigenfunctions of the form with and (59) (59) Summarising, the spectrum of is given by (60) (60)
7.3. More advanced example
Here we allow the inclusions to randomly change size, so that . By analogy with the previous section, we assume that consists of sequences such that . We also assume that the restriction to to of the probability measure on is absolutely continuous with respect to Lebesgue measure. As before, consider and denote by , where is the centre of Y, the ‘scaled inclusion’, requiring that , in order for the extension property in Assumption 3.1 to hold. The values 0 or of correspond to the absence of an inclusion or the presence of the inclusion in the cell respectively. Furthermore, define . Then a realisation is the union of the inclusions for all where in the case we set . The space consists of functions of the form (61) (61) Consider the spectral problem for namely and separate the variables, as in Section 7.2: where the function v satisfies (62) (62) The stochastic gradient is given by and therefore the problem (Equation62(62) (62) ) is equivalent to (63) (63) For each the eigenvalues and (orthonormal) eigenfunctions of the operator acting in are obtained by scaling the eigenvalues and eigenfunctions of acting in , in particular, , . Therefore, the formula (Equation53(53) (53) ) with replaced by gives the solution to (64) (64) If and the set has gaps, then for the solution to (Equation63(63) (63) ) is given by (Equation61(61) (61) ), where the functions solve (Equation64(64) (64) ) with . Substituting it into the spectral problem for (Equation49(49) (49) ) yields the problem (Equation55(55) (55) ) with the Zhikov-type function β given by (cf. (Equation56(56) (56) )) (65) (65) The integral in (Equation65(65) (65) ) is well defined for and the description of the spectrum on the intervals where follows Section 7.2.
Theorem 7.1
Under the assumptions of the current subsection, the spectrum of is given by where for each k, the values are solutions to see (Equation57(57) (57) ). The point spectrum of the operator is given by .
It is clear that if the set has gaps, then also has gaps. We are going to prove the theorem in several steps formulated in the following lemmas. We begin by studying the spectrum of the ‘microscopic’ part of the limit operator.
Lemma 7.2
The spectrum of the operator (see (Equation30(30) (30) )) is given by and does not contain eigenvalues of .
Proof.
Let for some and assume that is an eigenfunction corresponding to λ, i.e. . (For argument is similar.) Then v is of the form (Equation61(61) (61) ), where in whenever . But λ is only an eigenvalue of the operator acting in if , hence (66) (66) It remains to observe that is a set of measure zero in Ω and hence . The second claim of the lemma follows.
Now we show that by constructing a Weyl sequence. Without loss of generality we can assume that . For small enough we choose an -function such that and e.g. we can choose to be equal to a constant proportional to on . Consider the sequence We have and hence It follows that is a Weyl sequence for .
It remains to prove that λ is in the resolvent set whenever . Assume the contrary and let then the resolvent equation has a unique solution given by (Equation61(61) (61) ) with solving . Moreover, since we have and it follows immediately that which concludes the proof.
Next, we focus on the spectrum of .
Lemma 7.3
The inclusion holds.
Proof.
The proof of the inclusion repeats the related part of the proof of the Lemma 7.2. Namely, for , , we define a Weyl sequence where and is given by with an arbitrary fixed and defined as in Lemma 7.2. In order to show that suppose that for some and . Assume, to the contrary, that there exists a bounded resolvent i.e. the system (Equation18(18) (18) )–(Equation19(19) (19) ) has a unique solution for all . For the second equation reads Then must necessary be of the form , where v is of the form (Equation61(61) (61) ) and in i.e. (67) (67) which clearly blows up as . We show that the corresponding v is not an element of leading to a contradiction. Indeed, using the identity one has where the last integral diverges.
Lemma 7.4
If for some then λ is an eigenvalue of .
Proof.
Let , and denote by be the corresponding eigenfunction of . Since the problem has a solution given by (Equation61(61) (61) ), (Equation67(67) (67) ). Therefore is the eigenfunction of corresponding to λ.
Lemma 7.5
A point λ belongs to the resolvent set of if and i.e. or and .
Proof.
We claim that the problem (Equation18(18) (18) ) and (Equation19(19) (19) ) has bounded resolvent. Indeed, suppose that and write (Equation19(19) (19) ) in the form Since λ is not in the spectrum of , the latter has a bounded resolvent at λ and , where is as in (Equation61(61) (61) ) and . In particular, Substituting the expression for in (Equation18(18) (18) ) we obtain For the operator is invertible and from which the claim follows.
Proposition 7.1
The set does not contain eigenvalues of the operator .
Proof.
Assume that , for some and , is an eigenvalue of , i.e. there exists such that (68) (68) Suppose that for some then , where v solves (69) (69) Arguing as for the second inclusion of Lemma 7.3, we see that (Equation69(69) (69) ) has no -solution for the given λ. It follows that and therefore is an eigenfunction of which cannot be true by Lemma 7.2.
Now we assume that . Arguing as above, for we have , where v solves (Equation69(69) (69) ). The solution exists and is given by (Equation61(61) (61) ), (Equation67(67) (67) ). Substituting into (Equation68(68) (68) ) we see that must satisfy which cannot be true since .
Finally, if then we argue as above for the case again arriving at a contradiction.
This completes the proof of Theorem 7.1.
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Notes
1. The function is an ‘approximate eigenfunction’ for see (Equation37(37) (37) ).
References
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