For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator in the perforated domain , , to the limit operator on , where is a constant depending on the choice of boundary conditions.
This is an improvement of previous results [Progress in Nonlinear Differential Equations and Their Applications 31 (1997), 45–93; in: Proc. Japan Acad., 1985], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem.
In this article we study the following homogenisation problems labelled by (“D” for Dirichlet, “N” for Neumann, and “α” for Robin). Let , be open (bounded or unbounded) with boundary. For unbounded domains Ω we assume translation invariance, i.e., for any . Let and denote where , is the ball of radius
Consider the boundary value problems (cf. Figure 1)
Homogenisation problems of this type have been studied extensively for a long time [2,5,7,11]. For example, results by Cioranescu & Murat and Kaizu give a positive answer to the previous question for all three choices of boundary conditions at least in the case of bounded domains. In fact, they showed that the solutions of (), (), () converge strongly in to the solution of , where
In this article we attempt to improve this result in two directions. First, we show the above convergence not only in the strong sense, but in the norm-resolvent sense (that is, the right-hand side f is allowed to depend on ε). Second, our result is then extended to the case of unbounded domains. As a corollary, we obtain a statement about the convergence of the spectra of the perforated domain problems (), (), () as .
The paper is organised as follows. In Section 2 we will briefly give a more precise formulation of the problem and include previous results. In Section 3 we will state our main result and its implications. Sections 4, 5 and 6 contain the proof of the main theorem and in Section 7 we consider implications of our main theorem for the semigroup generated by the Robin Laplacian. Section 8 contains a brief conclusion and discusses open problems.
2.Geometric setting and previous results
As above, assume , and let
Moreover, since we are dealing with varying spaces , it is convenient to define the identification operators
For as above, one has
The only nontrivial statement is (2.6). To prove this, let . Then . To show that this quantity converges to 0 uniformly in f, denote for a cube shifted by k, so that . Then we have
In the cases the harmonic extension operator satisfies
(ii) There exists such that for all and .
(iii) For any sequence such that one has .
In the above geometric setting, we will study the linear operators , in , defined by the differential expression , with (dense) domains
In the case when one has the characterisation
Using the notation above, we recall the following classical results.
Let be open (bounded or unbounded). Suppose that , and let and be the solutions to
Let be open (bounded or unbounded), and suppose that is smooth. Suppose also that , and let and be the solutions to
Proof of Theorems 2.4and 2.5.
An important ingredient in the proofs are auxiliary functions defined, for each , as the solution to the problems
These functions were used in [2,5] as test functions to prove strong convergence of solutions. They are “optimal” in the sense that they minimise the energy in annular regions around the holes. In the Dirichlet case, the function is nothing but the potential for the capacity . It can be shown that one has the convergences
In what follows we prove the following claim.
Let be defined as in the previous section. Then for one has
For notational convenience, denote and . A quick calculation shows that
We note an important consequence of the above theorem.
For all compact , one has in the Hausdorff sense.1
First, note that the spectra of converge to that of , in the sense that for each compact there exists such that for all . The proof of this is obtained by combining the proofs of Lemma 3.11, Theorem 3.12 and Corollary 3.14 in . On the other hand, an analogous proof using (3.2) and (2.6) shows that if for almost all , then . Together these two facts imply Hausdorff convergence. □
In particular, this corollary shows that (if ) a spectral gap opens for between 0 and .
We note that our assumption on the spherical shape of the holes was made for the sake of definiteness, however, our results easily generalise to more general geometries as detailed in [2, Th. 2.7]. Moreover, our results are also valid for more general elliptic operators with continuous coefficients A (cf. ).
4.Uniformity with respect to the right-hand side
Suppose that , , , with , and let and be the solutions to the problems ()
We have the following a priori estimates (note Lemma 2.2):
Finally, applying the above reasoning to every subsequence of yields the result for the whole sequence. □
If the domain Ω is bounded, one has
Since Ω is bounded, the embedding of in is compact, thus the sequence from the previous proof has a subsequence converging to 0 strongly in . Since this can be done for every subsequence of , the whole sequence converges to 0.
Now, choose a sequence , such that
Treating unbounded domains requires further effort. Since we lack compact embeddings in this case, we will have to take advantage of the sufficiently rapid decay of solutions to and a decomposition of the right hand side with a bound on the interactions.
5.Exponential decay of solutions
We begin with a general result which we assume is classical, but include for the sake of completeness. Let open satisfying the strong local Lipschitz condition, and consider the problems (cf. (), (), ())
We postpone the proof, in order to introduce some notation and prove auxiliary results. First, let us denote and introduce the weighted Sobolev spaces , with scalar product
For and , the form is continuous and coercive on (on in the case ).
We will only treat the Robin case here, the other cases being analogous. Denote by I the second term in (5.6) and note that ω was chosen so that . By Hölder’s inequality with respect to μ one has
Let , , and suppose that is compact. Then the problem
By Hölder inequality, one has
Proof of Proposition 5.1.
Again we focus on the Robin case, the other cases being analogous. Denote by u the solution obtained from Lemma 5.3. Then , since . Moreover, let be arbitrary and decompose it as . Then and one has
6.Decomposition of the right-hand side
In this section we consider the case of unbounded Ω. We conclude the proof of Theorem 3.1 by decomposing the domain into cubes , writing and then applying the above results to each term . The following lemma shows uniform convergence with respect to the position of the cubes.
Let and , , be such that and , where with . Let be the solutions to the problems
The idea of the proof is to use translation invariance, in order to shift back near zero for every n, and then use the Fréchet–Kolmogorov compactness criterion to obtain a convergent subsequence of ; Theorem 4.1 will identify its limit as zero. Since the following analysis is independent of the choice of boundary conditions, we henceforth omit ι to simplify notation.
We now carry out the outlined strategy. We set, for all ,
These functions still solve the problems (6.1) with replaced by and Ω replaced by . The new sequence has the nice property that for all n. In the following we consider as elements of that are zero outside . We will now show that converges to zero in . To this end, consider the bounded set
Claim: is precompact in .
We postpone the proof of this claim to Lemma 6.2. We immediately obtain that has a convergent subsequence in . Furthermore, Theorem 4.1 shows that for every bounded which identifies the limit of the subsequence as zero.
Arguing as above for all subsequences of , we conclude that in . Since the shift is an isometry in , this implies that in . □
The set defined in (6.2) is precompact in .
We will use the notation and conventions from the previous proof and distinguish between the Dirichlet case and the Robin/Neumann cases.
Dirichlet case. Step 1: We have
Step 2: Notice that
Neumann and Robin case. Here the strategy is the same, but matters are complicated by the fact that is not in . To show that is precompact, we decompose elements in as
To prove precompactness of , first note that by Lemma 2.2(iii) for any there exists a such that
There exists with such that
We argue by contradiction. Suppose that there is no such function . Then there exist sequences with such that does not converge to zero, which is a contradiction to Lemma 6.1. □
In order to finalise the decomposition, we require he following two lemmas.
Suppose that , and denote
For convenience we write , . Denote and note that by Proposition 5.1 we have . The statement of the lemma is a consequence of the following estimate:
Suppose that and define , . Then for every one has the inequality
Combining the above lemmas, we have the following quantitative statement.
Suppose that . Then for every ,
We denote , , and estimate
Proof of Theorem 3.1.
Let with . Fix and choose such that and choose such that . Now compute
7.Behaviour of the semigroup
In this section we want to give an application of Theorem 3.1. In particular, we focus on the non-selfadjoint operator and study the large-time behaviour of its semigroup. In order to do this, we shall first study the numerical range of the Robin Laplacians more closely. In the remainder of this section, unless otherwise stated, the symbols and will denote the (operator-) norm and scalar product, respectively, and the symbol denotes a sector of half-angle θ in the complex plane.
Let and assume . We want to study the decay properties of the heat semigroup . To this end, let us denote by the Robin Laplacian on Ω. It is our goal to derive estimates on the numerical range of . Let and assume that . Notice that
Using standard generation theorems about analytic semigroups, the next statement follows.
The operator generates a bounded analytic semigroup in the sector , where
See [6, Ch. IX.1.6]. □
In this section we denote . By calculations analogous to the above, we have
In particular we have shown that the resolvent norm is bounded on . By a trivial calculation analogous to the previous subsection this leads to the following statement.
For every one has
Furthermore, we obtain the following lemma.
For every one has and there exists a such that
This is obtained by combining Lemma 7.2 with the following two facts:
By the theory of analytic semigroups (cf. [6, Ch. IX.1.6]), we immediately obtain the following corollary.
For all , the operator generates a bounded analytic semigroup in the sector .
This yields the main result of this section, as follows.
For every there exists such that for every there exists such that
It is straightforward to repeat the above proof for the case of Dirichlet boundary conditions to obtain an analogous result for . Here, the selfadjointness of allows us to choose the half-angle θ arbitrarily close to .
We have shown norm-resolvent convergence in the classical perforated domain problem with Dirichlet boundary conditions which has the interesting implication of spectral convergence (Cor. 3.3). Some questions remain open and will be addressed in the future. While the norm converges to 0, it is not clear from our method of proof how fast this happens. It would be desirable to obtain a precise convergence rate. In the case of Dirichlet boundary conditions a explicit convergence rate has been found by . Another interesting question is whether in the case there exist gaps in the spectrum of and how these depend on ε. The existence of spectral gaps has been confirmed in two dimensions , but to the authors’ knowledge the higher-dimensional case is still open.
1 For the definition of Hausdorff convergence, see e.g. .
KC is grateful for the financial support of the Engineering and Physical Sciences Research Council (UK): Grant EP/L018802/2 “Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory”.
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