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Norm-resolvent convergence in perforated domains

Abstract

For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator Δ in the perforated domain Ωi2εZdBaε(i), aεε, to the limit operator Δ+μι on L2(Ω), where μιC 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.

1.Introduction

In this article we study the following homogenisation problems labelled by ι{D,N,α} (“D” for Dirichlet, “N” for Neumann, and “α” for Robin). Let ΩRd, d2 be open (bounded or unbounded) with C2 boundary. For unbounded domains Ω we assume translation invariance, i.e., Ω+z=Ω for any zZd. Let αC{0},Re(α)0 and denote Ωε:=ΩiLεBaε(i) where ε(0,1), Baε(i) is the ball of radius

(1.1)aεD=εd/(d2),d3,e1/ε2,d=2,aεN=o(ε)(ε0),aεα=εd/(d1)
centered at the point iLε, and
(1.2)Lε:={i2εZd:dist(i,Ω)>ε}.

Consider the boundary value problems (cf. Figure 1)

(Dir)(Δ+1)uε=fin Ωε,uε=0on Ωε,(Neu)(Δ+1)uε=fin Ωε,νuε=0on Ωε,(Rob)(Δ+1)uε=fin Ωε,νuε+αu=0on Ωε,
i.e. the resolvent problem for the Laplacian, subject to the Dirichlet, Neumann and Robin boundary conditions, respectively. It is easy to see, using the Lax–Milgram theorem, that for all ε(0,1) each of these problems has a unique weak solution uε. It is a classical question, which we refer to as the homogenisation problem, whether the family of solutions to (Dir), (Neu), (Rob), obtained by varying the parameter ε, converges in the sense of the L2-norm to a function uL2(Ω) as ε0 and whether the limit function u solves, in a reasonable sense, some PDE whose form is independent of the right-hand side datum f.

Fig. 1.

Sketch of the perforated domain.

Sketch of the perforated domain.

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 (Dir), (Rob), (Neu) converge strongly in L2(Ω) to the solution uH1(Ω) of (Δ+1+μι)u=f, where

(1.3)μι=π2,ι=D,d=2,(d2)Sd2d,ι=D,d3,0,ι=N,αSd2d,ι=α,
where Sd denotes the surface area of the unit ball in Rd.

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 (Dir), (Neu), (Rob) as ε0.

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 d2, and let

Tε:=iLεTiε,Tiε:=Baει(i),iLε,
where aει, Lε as in (1.1), (1.2). Denote Ωε:=ΩTε. We also denote Biε:=Bε(i) and Piε:=ε[1,1]d+i for iLε. Constants independent of ε will be denoted C and may change from line to line. Note that our assumptions on Ω ensure that the set {ϕ|Ω:ϕC0(Rd)} is dense in H1(Ω) (cf. [1, Cor. 9.8]) in the cases ι=N,α.

Moreover, since we are dealing with varying spaces L2(Ωε), it is convenient to define the identification operators

(2.1)Jε:L2(Ωε)L2(Ω),Jεf(x)=f(x),xΩε,0,xΩΩε(2.2)Iε:L2(Ωε)L2(Ω),Iεg(x)=g|Ωε(2.3)Tε:H1(Ωε)H1(Ω),Tεu=u in Ωε,v in Tε,
where v is the harmonic extension of u into the holes, i.e.
(2.4)Δv=0in Tε,v=uon Tε.

Lemma 2.1.

For Iε,Jε as above, one has

(2.5)IεJε=idL2(Ωε)(2.6)JεIεidL2(Ω)L(H1(Ω),L2(Ω))0.
Moreover, IεL(L2(Ωε),L2(Ω)),JεL(L2(Ω),L2(Ωε)) are uniformly bounded in ε.

Proof.

The only nontrivial statement is (2.6). To prove this, let fH1(Ωε). Then fJεIεfL2(Ω)=fL2(Tε). To show that this quantity converges to 0 uniformly in f, denote Qk:=[0,1)d+k for kZd a cube shifted by k, so that Rd=kZdQk. Then we have

fL2(Tε)2=kZdfL2(QkTε)2kZd1L2p(QkTε)2fL2q(QkTε)2
for p,q>1 with p1+q1=1, by Hölder’s inequality. Since fH1(Ω), we can use the Gagliardo–Sobolev–Nierenberg inequality to conclude (for suitable q) that
fL2(Tε)21L2p(Q0Tε)2kZdfL2q(QkTε)21L2p(Q0Tε)2kZdCfH1(QkTε)2=|Q0Tε|1/pCfH1(Tε)2
with some suitable p>0. Since |Q0Tε|0 as ε0 (cf. the definition of aει, (1.1)), the desired convergence follows. □

Lemma 2.2.

In the cases ι{N,α} the harmonic extension operator Tε satisfies

  • (i) lim supε0TεH1(Ωε)H1(Ω)<.

  • (ii) There exists C>0 such that TεwH1(Piε)CwH1(Piε) for all wH1(Ωε) and iLε.

  • (iii) For any sequence wε such that lim supε0wεH1(Ωε)< one has TεwεJεwεL2(Ω)0.

Proof.

See [5], [11, p. 40]. □

In the above geometric setting, we will study the linear operators Aει, ι=D,N,α in L2(Ωε), defined by the differential expression Δ+1, with (dense) domains

D(AεD)=H01(Ωε)H2(Ωε),D(AεN)={uH2(Ωε):νu=0 on Ωε},D(Aεα)={uH2(Ωε):νu+αu=0 on Ωε},
respectively, and the linear operators Aι in L2(Ωε) defined by the expression Δ+1+μι, with domains
D(AD)=H01(Ω)H2(Ω),D(AN)={uH2(Ω):νu=0 on Ω},D(Aα)={uH2(Ω):νu+αu=0 on Ω},
respectively, where μι, ι=D,N,α, are defined in (1.3).

Remark 2.3.

In the case when d3 one has the characterisation

(2.7)μD=12dmin{RdB1(0)|u|2,uH1(Rd),u=1 on B1(0)}.
Note that the factor 1/2d arises from the fact that the unit cell is of size 2ε.

Using the notation above, we recall the following classical results.

Theorem 2.4

Theorem 2.4([2]).

Let ΩRd be open (bounded or unbounded). Suppose that fL2(Ω), and let uε and u˜ be the solutions to

(Δ+1)uε=f,uεH01(Ωε),(Δ+1+μD)u˜=f,u˜H01(Ω).
Then Jεuεε0u˜ in H01(Ω).

Theorem 2.5

Theorem 2.5([5]).

Let ΩRd be open (bounded or unbounded), and suppose that Ω is smooth. Suppose also that fL2(Ω), and let uε and u˜ be the solutions to

(Δ+1)uε=f,uεD(Aεα,N),(Δ+1+μα,N)u˜=f,u˜D(Aα,N).
Then one has
Jεuεε0u˜in H1(Ω).

Proof of Theorems 2.4and 2.5.

The results are obtained by following the proofs of [2, Thm 2.2], [5, Thm 2]. Note that the weak convergence in H1(Ω) is immediately obtained also for unbounded domains (and complex α). □

An important ingredient in the proofs are auxiliary functions wϵιW1,(Rd) defined, for each ε(0,1), as the solution to the problems

(2.8)wεN1,wεD=0in Tiε,ΔwεD=0in BiεTiε,wεD=1in PiεBiε,wεDcontinuous,νwεα+αwεα=0on Tiε,Δwεα=0in BiεTiε,wεα=1in PiεBiε,wεαcontinuous,
used as a test function in the weak formulation of the problems (Dir), (Neu), (Rob).

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 wεD is nothing but the potential for the capacity cap(Bε(i);BaεD(i)). It can be shown that one has the convergences

asy-110-asy1481-g002.jpg

as ε0, where δTε denotes the Dirac measure on the boundary of the holes (for a proof of the above facts, see [2, Lemma 2.3] and [5, Section 3]).

3.Main results

In what follows we prove the following claim.

Theorem 3.1.

Let Jε,Aει,Aι be defined as in the previous section. Then for ι{D,N,α} one has

(3.1)Jε(Aει)1(Aι)1JεL(L2(Ωε),L2(Ω))0(ε0),
that is, the operator sequence Aει converges to Aι in the norm-resolvent sense.

Corollary 3.2.

If Aε,A are as in Theorem 3.1, then

(3.2)(Aει)1IεIε(Aι)1L(L2(Ω),L2(Ωε))0,
where Iε is as in (2.2).

Proof.

For notational convenience, denote Rε:=(Aει)1 and R:=(Aι)1. A quick calculation shows that

RεIεIεR=Iε(JεRεRJε)Iε(IεJε1)RεIε=Iε(JεRεRJε)Iε,
since IεJε=idL2(Ωε). Hence
RεIεIεRL(L2(Ω),L2(Ωε))IεL(L2(Ω),L2(Ωε)2JεRεRJεL(L2(Ωε),L2(Ω))0
as ε0, by (3.1) and uniform boundedness of IεL(L2(Ωε),L2(Ω)). □

We note an important consequence of the above theorem.

Corollary 3.3.

For all compact KC, one has σ(Aει)Kε0σ(Aι)K in the Hausdorff sense.1

Proof.

First, note that the spectra of Aει converge to that of Aι, in the sense that for each compact Kρ(Aι) there exists ε0>0 such that Kρ(Aει) for all ε(0,ε0). The proof of this is obtained by combining the proofs of Lemma 3.11, Theorem 3.12 and Corollary 3.14 in [8]. On the other hand, an analogous proof using (3.2) and (2.6) shows that if Kρ(Aει) for almost all ε>0, then Kρ(Aι). Together these two facts imply Hausdorff convergence. □

In particular, this corollary shows that (if Re(μι)>0) a spectral gap opens for Aει between 0 and Re(μι).

Remark 3.4.

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 div(A) with continuous coefficients A (cf. [2]).

4.Uniformity with respect to the right-hand side

In this section we prove that the result of Theorems 2.4, 2.5 hold in a strengthened form, namely, uniformly with respect to the right-hand side f. More precisely, the following holds.

Theorem 4.1.

Suppose that εn0, fnL2(Ωεn), nN, with fnL2(Ωε)1, and let unι and u˜nι be the solutions to the problems (ι{D,N,α})

(4.1)(Δ+1)unι=fn,unιD(Aεnι),(4.2)(Δ+1+μι)u˜nι=Jεnfn,u˜nιD(Aι).
Then for every bounded, open KΩ one has
Jεnunιu˜nι0strongly in L2(K),Jεnunιu˜nι0weakly in L2(K),
for ι{D,N,α}.

Proof.

We have the following a priori estimates (note Lemma 2.2):

Tεnunα,NH1(Ω)CJεnfnL2(Ω),JεnunDH1(Ω)CJεnfnL2(Ω),u˜nιH1(Ω)CJεnfnL2(Ω)ι{D,N,α}.
Thus, there exists a subsequence (still indexed by n) and uι,u˜ιH1(Ω) such that
asy-110-asy1481-e001.jpg

Note that that for every bounded KΩ the convergence statements (4.3) are strong in L2(K). In particular, employing Lemma 2.2(i), (iii) and the Rellich Theorem we immediately obtain

Jεnunιuιstrongly in L2(K),Jεnunιuιweakly in L2(K)(4.4)u˜nιu˜ιstrongly in L2(K),(4.5)u˜nιu˜ιweakly in L2(K).
for all ι{D,N,α}. Next, choose a further subsequence (still indexed by n) such that also Jεnfnnf weakly in L2(Ω), where the limit f may depend on the choice of subsequence. Now, consider the weak formulations of the problem (4.2), i.e.
Ωu˜nιϕ+(1+μι)Ωu˜nιϕ=Ωfnϕ,
where ϕC0(Ω) for ι=D and ϕC0(Rd) for ι=α,N. Letting n and using the convergencies (4.4), (4.5) (with K=Ωsuppϕ) we obtain
Ωu˜ιϕ+(1+μι)Ωu˜ιϕ=Ωfϕ.
Next consider the weak formulation of (4.1),where we choose the test function wεnιϕ:
Ωεnunι(wεnιϕ)+Ωεnunιwεnιϕ=Ωεnfnwεnιϕ,
where again ϕC0(Ω) for ι=D and ϕC0(Rd) for ι=α,N. It follows from the results of [2,5] that the left and right-hand side of this equation converge to
Ω(uιϕ+(1+μι)uιϕ)andΩfϕ,
respectively. Thus, we obtain
Ω(uιϕ+(1+μι)uιϕ)=Ωfϕ,
and hence uι and u˜ι are weak solutions to the same equation. Uniqueness of solutions (for all ι{D,N,α}) implies u˜ι=uι, which shows the assertion for the chosen subsequence.

Finally, applying the above reasoning to every subsequence of (Jεnunιu˜nι) yields the result for the whole sequence. □

Corollary 4.2.

If the domain Ω is bounded, one has

Jε(Aει)1(Aι)1JεL(L2(Ωε),L2(Ω))0(ε0)
for ι{D,N,α}, i.e., Theorem 3.1 holds in that case of bounded Ω.

Proof.

Since Ω is bounded, the embedding of H1(Ω) in L2(Ω) is compact, thus the sequence Jεnunιu˜nι from the previous proof has a subsequence converging to 0 strongly in L2(Ω). Since this can be done for every subsequence of (Jεnunιu˜nι), the whole sequence converges to 0.

Now, choose a sequence fnL2(Ωεn),fnL2(Ωε)1, such that

supfL2(Ωεn)f1(Jεn(Aει)1(Aι)1Jεn)fL2(Ωε)1n<(Jεn(Aεnι)1(Aι)1Jεn)fnL2(Ωεn).
By the above, the right-hand side of this inequality converges to zero, which implies the claim. □

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 (Δ+1)u=f 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 URd open satisfying the strong local Lipschitz condition, λ>12 and consider the problems (cf. (Dir), (Neu), (Rob))

(5.1)(Δ+λ)uα=fin U,νuα+αuα=0on U;(5.2)(Δ+λ)uN=fin U,νuN=0on U;(5.3)(Δ+λ)uD=fin U,uD=0on U.
Let x0Rd, and define the function ω(x)=cosh(|xx0|). Then the following statement holds.

Proposition 5.1.

Let fL2(U), supp(f) compact. Then each of the problems (5.1)–(5.3) has a unique weak solution uιH1(U) satisfying

(5.4)U|uι|2ωdxMU|f|2ωdx(5.5)U|uι|2ωdxMU|f|2ωdx,
where M:=max{2,(λ12)1}.

We postpone the proof, in order to introduce some notation and prove auxiliary results. First, let us denote dμ:=ωdx and introduce the weighted Sobolev spaces H:=W1,2(U;ω), H0:=W01,2(U;ω) with scalar product

u,vH=Uuvdμ+Uu·vdμ.
Moreover, let λ>12 and define the sesquilinear forms
(5.6)aα(u,v):=U(u·v+λuv)dμ+Uvu·ωωdμ+αUuvωdSon H,(5.7)aN(u,v):=U(u·v+λuv)dμ+Uvu·ωωdμon H,(5.8)aD(u,v):=U(u·v+λuv)dμ+Uvu·ωωdμon H0.
Lemma 5.2.

For λ>12 and ι{D,N,α}, the form aι is continuous and coercive on H (on H0 in the case ι=D).

Proof.

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 |ω|ω. By Hölder’s inequality with respect to μ one has

|I|ωω1uL2(μ)vL2(μ)12uL2(μ)2+12vL2(μ)2,
and thus
|a(u,u)|uL2(μ)2+λuL2(μ)2+|α|ω1/2uL2(U)2+IuL2(μ)2+λuL2(μ)212uL2(μ)212uL2(μ)2=12uL2(μ)2+(λ12)uL2(μ)2,
which shows coercivity in H. Continuity follows by estimating the boundary term. By the trace theorem [3, Prop. IX.18.1] we have, for each δ>0,
(5.9)U|u|2ωdx2δ(ω1/2u)L2(U)2+Cδω1/2uL2(U)2.
The first term can be estimated using the special choice of ω:
(ω1/2u)L2(U)2=U|ω1/2u+12uωω1/2|2dx2Uω|u|2dx+12U|u|2|ω|2ωdx2uL2(μ)+2ωω2U|u|2ωdx(5.10)2uH1(μ)2.
The desired continuity now follows immediately by combining (5.9) and (5.10). □

Lemma 5.3.

Let fL2(U), ι{D,N,α}, and suppose that supp(f) is compact. Then the problem

(5.11)aι(u,v)=UfvdμvH
has a solution in H.

Proof.

By Hölder inequality, one has

|Ufvdμ|fL2(μ)vL2(μ)ωL(suppf)fL2(U)vL2(μ),
so fH. The assertion now follows from Lemma 5.2 and the Lax–Milgram theorem for complex, non-symmetric sesquilinear forms [13, Thm. VI.1.4]. □

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 uH1(U), since HH1(U). Moreover, let ϕC0(Rd) be arbitrary and decompose it as ϕ=ωψ. Then ψC0(Rd)H and one has

Uu·ϕdx+λUuϕdx+αUuϕdS=Uu·(ωψ+ψω)dx+λUuψωdx+αUuψωdS=aα(u,ψ)=Ufψdμ=Ufϕdx.
Thus, the function u solves the problem
(5.12)Uu·ϕdx+λUuϕdx+αUuϕdS=UfϕdxϕC0(Rd).
Uniqueness of solutions and density of C0(Rd) in H1(U) implies that u is the weak solution in H1(U) to the Robin problem (5.1).

The estimates (5.4), (5.5) follow from the coercivity of aι. □

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 Qi, writing f=ifχQi and then applying the above results to each term fχQi. The following lemma shows uniform convergence with respect to the position of the cubes.

Lemma 6.1.

Let εn0 and fnL2(Ω), nN, be such that fnL2(Ω)1 and supp(fn)Qin, where Qin=[0,1]d+in with inZd. Let unι,u˜nι be the solutions to the problems

(6.1)Aεnιunι=fn|Ωεn,Aιu˜nι=fn,nN,ι{D,N,α}.
Then Jεnunιu˜nιL2(Ω)0 for all ι{D,N,α}.

Proof.

The idea of the proof is to use translation invariance, in order to shift supp(fn) back near zero for every n, and then use the Fréchet–Kolmogorov compactness criterion to obtain a convergent subsequence of (Jεnunιu˜nι); 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 nN,

un(x):=un(x+in),u˜n(x):=u˜n(x+in),fn(x):=fn(x+in).

These functions still solve the problems (6.1) with fn replaced by fn and Ω replaced by Ωin. The new sequence fn has the nice property that supp(fn)[0,1]d for all n. In the following we consider Jεnun,u˜n,fn as elements of L2(Rd) that are zero outside Ωin. We will now show that u˜nJεnun converges to zero in L2(Rd). To this end, consider the bounded set

(6.2)F:={u˜nJεnun:nN}L2(Rd).

Claim: F is precompact in L2(Rd).

We postpone the proof of this claim to Lemma 6.2. We immediately obtain that (u˜nJεnun) has a convergent subsequence in L2(Rd). Furthermore, Theorem 4.1 shows that u˜nJεnunL2(K)0 for every bounded KRd which identifies the limit of the subsequence as zero.

Arguing as above for all subsequences of (u˜nJεnun), we conclude that u˜nJεnun0 in L2(Rd). Since the shift uu(·+in) is an isometry in L2(Rd), this implies that u˜nJεnun0 in L2(Ω). □

Lemma 6.2.

The set F defined in (6.2) is precompact in L2(Rd).

Proof.

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

supnτh(u˜nJεnun)(u˜nJεnun)L2(Rd)0as h0nN,
where τh denotes the operator of translation by h. Indeed, the standard regularity theory implies
τh(u˜nJεnun)(u˜nJεnun)L2(Rd)(u˜nJεnun)L2(Rd)|h|CfnL2(Ω)|h|.

Step 2: Notice that

supnu˜nJεnunL2(RdBR(0))0as R,
due to the following estimate in which we set ω0(x):=cosh(|x|).
u˜nJεnunL2(RdBR(0))22u˜nω0ω01L2(ΩBR(0))2+2Jεunω0ω01L2((RdBR(0))24Mfnω0L2(Rd)2ω01L(RdBR(0))2Prop. 5.1CfnL2(Ω)2exp(R)
which completes Step 2. Applying the Fréchet–Kolmogorov theorem yields the precompactness of F.

Neumann and Robin case. Here the strategy is the same, but matters are complicated by the fact that Jεnun is not in H1(Rd). To show that F is precompact, we decompose elements in F as

u˜nJεnun=(u˜nTεnun)+(TεnJεn)un,
define F1:={u˜nTεnun:nN}, F2:={(TεnJεn)un:nN} and show that F1 and F2 are precompact in L2(Rd). We will begin by showing that F1 is precompact. To this end, denote by E:H1(Ω)H1(Rd) an extension operator satisfying Eu|Ω=u and EuH1(Rd)CuH1(Ω) for all uH1(Ω) (cf. Prop. VII.19.1 and Remark VII.19.2 in [3]). Note that by translation invariance one has Eτh=τhE and (Eun)=E(un). We start by proving that
supnτhE(u˜nTεnun)E(u˜nTεnun)20as h0
This readily follows from the estimate
τhE(u˜nTεnun)E(u˜nTεnun)L2(Rd)E(u˜nTεnun)L2(Rd)|h|Cu˜nTεnunH1(Ω+in)|h|CfnL2(Ω+in)|h|C|h|.
Next we prove that
supnE(u˜nTεnun)L2(RdBR(0))0as R.
Indeed, notice first that
E(u˜nTεnun)L2(RdBR(0))2C(u˜nL2((Ωin)BR(0))2+TεnunL2((Ωεnin)BR(0))2)(6.3)=C(u˜nL2(ΩBR(in))2+TεnunL2((Ωεn)BR(in))2),
To treat the two terms on the right-hand side we apply Lemma 2.2(ii) and Proposition 5.1 with ωin(x)=cosh(|x+in|) as follows. For the second term in (6.3), we obtain
TεnunL2(ΩεnBR(in))C(unL2(ΩBR/2(in))+unL2(ΩBR/2(in)))ωin1/2ωin1/2unL2(ΩBR/2(in))+ωin1/2ωin1/2unL2(ΩBR/2(in))C(ωin1/2unL2(ΩBR/2(in))+ωn1/2unL2(ΩBR/2(in)))ωin1/2L(ΩBR/2(in))CMfnωin1/2L2(Ω)exp(R/3)2CMexp(R/3),
where we use the fact that ωin is bounded by 2 on suppfn. With an analogous calculation for the first term in (6.3), we finally find
(6.4)E(u˜nTεnun)L2(RdBR(0))Cexp(R/3),
with C independent of n. Applying the Fréchet–Kolmogorov theorem yields the precompactness of the set {E(u˜nTεnun):nN}. Finally, noting that F1={E(u˜nTεnun):nN}χΩ and that multiplication by χΩ is a bounded operator on L2(Rd) we obtain precompactness of F1.

To prove precompactness of F2, first note that by Lemma 2.2(iii) for any δ>0 there exists a n0 such that

(JεnTεn)un2<δn>n0.
Let us fix arbitrary δ>0 and n0 as above. It remains to estimate the terms
τh(JεnTεn)un(JεnTεn)unL2(Rd),nn0,
but these are only finitely many, which clearly converge to zero individually, and hence
supnn0τh(JεnTεn)un(JεnTεn)un20as h0
Altogether we have shown that
supnτh(JεnTεn)un(JεnTεn)unL2(Rd)=max{supnn0τh(JεnTεn)un(JεnTεn)un2,2δ}h02δ.
Since δ>0 was arbitrary we finally get
limh0supnNτh(JεnTεn)un(JεnTεn)unL2(Rd)=0.
This completes the first Fréchet–Kolmogorov-condition. The proof of the second condition
supn(JεnTεn)unL2(RdBR(0))0as R
is analogous to the case of F1. Applying the Fréchet–Kolmogorov theorem yields precompactness of F1 and completes the proof. □

Corollary 6.3.

There exists δε with δεε00 such that

(Jε(Aι)1(Aει)1Jε)(fχQiΩε)L2(Ω)δεfχQiL2(Ω)
for all fL2(Ω) and iZd.

Proof.

We argue by contradiction. Suppose that there is no such function δε. Then there exist sequences εn,fn,in with fnL2(Ω)=1 such that (Jε(Aι)1(Aεnι)1Jε)(fnχQinΩεn)L2(Ω) 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.

Lemma 6.4.

Suppose that fL2(Ω), and denote

ui:=(Jε(Aι)1(Aει)1Jε)(fχQiΩε),iZd.
Then one has
(6.5)|ui,ujL2(Ω)|CfχQiL2(Ω)fχQjL2(Ω)exp(|ij|/2)
for all i,jZd with ij, where ·,·L2(Ω) denotes the standard inner product in L2(Ω).

Proof.

For convenience we write fi:=fχQi, iZd. Denote ωi(x)=cosh(|xi|) and note that by Proposition 5.1 we have ωi1/2uiL2(Ω)Cfiωi1/2L2(Ω). The statement of the lemma is a consequence of the following estimate:

|ui,ujL2(Ω)|Ω|ui(x)||uj(x)|dx=Ω(|ui(x)|ωi1/2)(|uj(x)|ωj1/2)ωi1/2ωj1/2dxuiωi1/2L2(Ω)ujωj1/2L2(Ω)ωi1/2ωj1/2L(Ω)Cfiωi1/2L2(Ω)fjωj1/2L2(Ω)ω01/2ωji1/2L(Ω)CfiL2(Ω)fjL2(Ω)exp(|ij|/2),
where we use the fact that supp(fi)Qi and ωi|Qi2. □

Lemma 6.5.

Suppose that fC0(Ωε) and define ui:=(Jε(Aει)1(Aι)1Jε)(fχQi), iZd. Then for every n>1 one has the inequality

(6.6)m=1NuimL2(Ω)2C(n3m=1NuimL2(Ω)2+fL2(Ωε)exp(n/3)),
where N is the number of cubes such that Qiksupp(f), and C,n do not depend on N.

Proof.

m=1NuimL2(Ω)2m,p=1Nuim,ujpL2(Ω)=k=0(|ij|[k,k+1)ui,ujL2(Ω))k=0n(|ij|[k,k+1)uiL2(Ω)ujL2(Ω))+k=n(|ij|[k,k+1)ui,ujL2(Ω))k=0n|ij|[k,k+1)(uiL2(Ω)22+ujL2(Ω)22)+k=n(|ij|[k,k+1)ui,ujL2(Ω))k=0nm=1N({j:|imj|[k,k+1)}uimL2(Ω)2)+k=n(|ij|[k,k+1)ui,ujL2(Ω))Ck=1nk2m=1NuimL2(Ω)2+k=n(|ij|[k,k+1)ui,ujL2(Ω))(6.7)Cn3m=1NuimL2(Ω)2+k=n(|ij|[k,k+1)ui,ujL2(Ω)).

We now study the last term of (6.7). It follows from Lemma 6.4 that

|ui,ujL2(Ω)|CfiL2(Ω)fjL2(Ω)e12|ij|.
Using this fact and fixing k for the moment, we obtain
||ij|[k,k+1)ui,ujL2(Ω)|C|ij|[k,k+1)fiL2(Ω)fjL2(Ω)exp(|ij|/2)C|ij|[k,k+1)(fiL2(Ω)22+fjL2(Ω)22)exp(|ij|/2)Cm=1NfimL2(Ω)2k2exp(k/2)=CfL2(Ω)2k2exp(k/2)CfL2(Ω)2exp(k/2).
Summing this inequality from k=n to infinity concludes the proof. □

Combining the above lemmas, we have the following quantitative statement.

Proposition 6.6.

Suppose that fC0(Ωε). Then for every nN,

(Jε(Aει)1(Aι)1Jε)fL2(Ω)2C(n3δε2+exp(n/3))fL2(Ω)2
for some C>0, where δε was defined in Corollary 6.3.

Proof.

We denote uiε:=(Jε(Aει)1(Aι)1Jε)(fχQi), iRd, and estimate

(Jε(Aει)1(Aι)1Jε)fL2(Ω)2=m=1NuimεL2(Ω)2Lemma 6.5C(n3m=1NuimεL2(Ω)2+exp(n/3)fL2(Ωε))Cor. 6.3C(n3δε2m=1NfimL2(Ωε)2+exp(n/3)fL2(Ωε))=C(n3δε2+exp(n/3))fL2(Ω)2.

Proof of Theorem 3.1.

Let gL2(Ωε) with gL2(Ωε)1. Fix δ>0 and choose fC0(Ωε) such that gfL2(Ωε)2<δ and choose nN such that exp(n/3)δ. Now compute

(Jε(Aει)1(Aι)1Jε)gL2(Ω)22(Jε(Aει)1(Aι)1Jε)fL2(Ω)2+2(Jε(Aει)1(Aι)1Jε)(gf)L2(Ω)2C((n3δε2+exp(n/3))fL2(Ωε)2+Jε(Aει)1(Aι)1Jε2boundedgfL2(Ωε)2)C(n3δε2+δ)gL2(Ωε)2+Cδ,
hence
supgL2(Ωε)1(Jε(Aει)1(Aι)1Jε)gL2(Ω)2Cn3δε2+Cδ+Cδ,
and therefore
lim supε0(Jε(Aει)1(Aι)1Jε)L(L2(Ωε),L2(Ω))2Cδ.
Since δ>0 is arbitrary, the result follows. □

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 Aα 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 L2 (operator-) norm and scalar product, respectively, and the symbol Σθ denotes a sector of half-angle θ in the complex plane.

7.1.Decay of et(AαId)

Let αC and assume Reα>0. We want to study the decay properties of the heat semigroup et(Δμα). To this end, let us denote by Bα:=AαId the Robin Laplacian on Ω. It is our goal to derive estimates on the numerical range of Bα. Let uD(Bα)=D(Aα) and assume that uL2(Ω)=1. Notice that

Bαu,u=Ω|u|2dx+μαΩ|u|2dx+αΩ|u|2dS=u2+μα+αuL2(Ω)2,
and therefore
ReBαu,uReμα+ReαuL2(Ω)2,|ImBαu,u||Imμα|+|Imα|uL2(Ω)2.
Now, let λ(0,Reμα) and compute
|Im(Bαλ)u,u||Imμα|+|Imα|uL2(Ω)2(7.1)=|Imμα|ReμαReμα+|Imα|ReαReαuL2(Ω)2.
Recall from Section 1 that μα=αSd/2d and hence |Imμα|/Reμα=|Imα|/Reα. Combining this with (7.1), we obtain (cf. Figure 2)
|Im(Bαλ)u,u||Imα|Reα(Reμα+ReαuL2(Ω)2)|Imα|Reα(Re(Bαλ)u,u+λ)|Imα|Reαλ2dSdRe(Bαλ)u,u.

Fig. 2.

The sector of decay and angle θλ for Bα.

The sector of decay and angle θλ for Bα.

Using standard generation theorems about analytic semigroups, the next statement follows.

Proposition 7.1.

The operator (Bαλ) generates a bounded analytic semigroup in the sector Σπ2θλ, where

θλ=arctan(|Imα|Reαλ2dSd).
Equivalently, Bα generates an analytic semigroup with
exp(zBα)exp(λz)zΣπ2θλ.

Proof.

See [6, Ch. IX.1.6]. □

7.2.Decay of et(AεαId)

In this section we denote Bεα:=AεαId. By calculations analogous to the above, we have

|ImBεαu,u||Imα|ReαReBεαu,u,
that is, Bεα is sectorial with sector Σθ0, where θ0=arctan(|Imα|/Reα), and hence generates a bounded analytic semigroup in the sector Σπ2θ0. In this subsection we improve this a priori result using spectral convergence. To this end, let δ>0 and define the compact set
Kδ:={x+iy:x[0,Reμα],y[|Imμα|,|Imμα|]}.
Note that then Σθ0{RezReμαδ}Kδ. By [4, Th. III.2.3] one has Kδρ(Bα) for every δ>0. Applying Corollary 3.3 we see that for every δ>0 there exists a ε0>0 such that Kδρ(Bεα) for all ε<ε0.

In particular we have shown that the resolvent norm (Bεαz)1 is bounded on Σθ0{RezReμαδ}. By a trivial calculation analogous to the previous subsection this leads to the following statement.

Lemma 7.2.

For every λ(0,Reμαδ) one has

σ(Bεαλ)Σθλδ,θλδ=arctan(|Imμα|Reμαλδ).

Furthermore, we obtain the following lemma.

Lemma 7.3.

For every λ(0,Reμαδ) one has CΣθλδρ(Bεαλ) and there exists a M=M(λ,δ)>0 such that

(Bεαλz)1M|z|zCΣθλδ.

Proof.

This is obtained by combining Lemma 7.2 with the following two facts:

|ImBεαu,u||Imα|ReαReBεαu,u,(Bεαz)1Con Kδ.

By the theory of analytic semigroups (cf. [6, Ch. IX.1.6]), we immediately obtain the following corollary.

Corollary 7.4.

For all λ(0,Reμαδ), the operator Bεαλ generates a bounded analytic semigroup in the sector Σπ2θλδ.

This yields the main result of this section, as follows.

Theorem 7.5.

For every δ>0 there exists ε0>0 such that for every λ(0,Reμδ) there exists M>0 such that

exp(zBεα)Mexp(λRez)zΣθλδ,ε(0,ε0).

Remark 7.6.

It is straightforward to repeat the above proof for the case of Dirichlet boundary conditions to obtain an analogous result for exp(t(ADId)). Here, the selfadjointness of AD allows us to choose the half-angle θ arbitrarily close to π/2.

8.Conclusion

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 JεAε1A1JεL(L2(Ωε),L2(Ω)) 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 [10]. Another interesting question is whether in the case Ω=Rd there exist gaps in the spectrum of Aε and how these depend on ε. The existence of spectral gaps has been confirmed in two dimensions [9], but to the authors’ knowledge the higher-dimensional case is still open.

Notes

1 For the definition of Hausdorff convergence, see e.g. [12].

Acknowledgements

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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