Operator estimates for the crushed ice problem

Operator estimates for the crushed ice problem

Abstract.

Let be the Dirichlet Laplacian in the domain . Here and is a family of tiny identical holes (“ice pieces”) distributed periodically in with period . We denote by the capacity of a single hole. It was known for a long time that converges to the operator in strong resolvent sense provided the limit exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded ) an estimate for the difference of the -th eigenvalue of and . Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.

Keywords: crushed ice problem; homogenization; norm resolvent convergence; operator estimates; varying Hilbert spaces

2010 Mathematics Subject Classification:
Primary 58J50; Secondary 35B27, 35P15, 47A10

1. Introduction

In the current work we revisit one of the classical problems in homogenization theory – homogenization of the Dirichlet Laplacian in a domain with a lot of tiny holes. It is also known as crushed ice problem. Below, we briefly recall the setting of this problem and the main result.

Let be an open domain in () and be a family of small holes. The holes are identical (up to a rigid motion) and are distributed evenly in along the -periodic cubic lattice – see Figure 1. We set

 Ωε:=Ω∖(⋃i¯Diε).

The domain is depicted in Figure 1. More precise description of this domain will be given in the next section.

In we study the following problem:

 −ΔΩεuε+uε=f↾Ωε,

where is the Dirichlet Laplacian in , is a given function, is the restriction of to . The goal is to describe the behaviour of the solution to this problem as .

It turns out that the result depends on the limit being finite or infinite (here is the capacity of a single hole, see (8) for details). Namely, if then as . Otherwise, if , as , where is the solution to the problem

 −ΔΩu+qu+u=f.

This result was proven independently by V.A. Marchenko, E.Ya. Khruslov [MK64] (the case ), J. Rauch, M. Taylor [RT75] (the cases and ) and D. Cioranescu, F. Murat [CM82] (all scenario) by using different tools — potential theory, capacitary methods and variational approach (the so-called Tartar’s energy method), respectively. J. Rauch and M. Taylor also treated the case of randomly distributed holes under assumptions resembling the case in a deterministic case; the pioneer result in this direction was obtained by M. Kac in [Kac74], who investigated the case of uniformly distributed holes.

Note, that this result remains valid if on the external boundary (i.e. on ) one imposes Neumann, Robin, mixed or any other -independent boundary conditions (then is the Laplace operator subject to these conditions on ).

Besides the resolvent convergence one can study the convergence of spectrum or the convergence of the semi-group . In the later case the name crushed ice problem is indeed reasonable1. Also domains with a lot of Dirichlet holes have interesting scattering properties (fading/solidifying obstacles, cf. [R75b, RT75]).

For more details on the topic we refer also to articles [Ba88, Be95, Kh72, Oza83, R75a, PV80], as well as to the monographs [Br02, Cha84, CPS07, MK74, MK06, S79].

In what follows, we focus on the case .

In the language of operator theory one can reformulate the above result as follows: the operator converges to the operator in strong resolvent sense. Strictly speaking, we are not able to treat the classical resolvent convergence (since the underlying operators act in different Hilbert spaces), but we have its natural analogue for varying domains with :

 ∀f∈L2(Ω): ∥(−ΔΩε+I)−1Jεf−Jε(−ΔΩ+q+I)−1f∥L2(Ωε)→0 as ε→0, (1)

where .

In the recent preprint [DCR17] the authors improved (1) by proving (a kind of) norm resolvent convergence, namely

 (2)

where is the operator of extension by zero. The authors assumed that are balls, distributed -periodically in . For bounded their proof resembles the variational approach developed in [CM82], for unbounded they also utilize a rapid decay of the Green’s function of .

In the current work we extend the result of [DCR17] providing an estimate for the rate of convergence in (2) (see Theorem 2.5 below). We also improve (1) (see Theorem 2.3) deriving the operator estimate

 ∥(−ΔΩε+I)−1Jε−Jε(−ΔΩ+q+I)−1∥L(L2(Ω),L2(Ωε))≤4δε,

where with depending on the dimension (for the “physical” cases and one has and , respectively).

As a consequence of our main results, we establish uniform convergence of the corresponding semi-groups and (for bounded ) an estimate for the difference between the -th eigenvalue of and  — see Theorems 2.62.7.

Let us stress that in all our results (except Theorem 2.7) we do not assume that the domain is bounded.

Our proofs are based on the abstract scheme for studying the convergence of operators in varying Hilbert spaces which was developed by the second author of the present article in [P06] and in more detail in the monograph [P12].

Before proceeding to the main part of the work let us mention several related results:

• Some estimates for the rate of convergence in (1) were obtained in [CPS07, §16]. Namely, assuming that , are balls of radius (that is ) distributed -periodically, and the function belongs to the Hölder class , the authors derived the estimates

 ∥(−ΔΩε+I)−1Jεf−Jε(−ΔΩ+q+I)−1f∥L2(Ωε)≤Cε∥f∥C0,a(¯¯¯Ω). ∥(−ΔΩε+I)−1Jεf−Jεφε(−ΔΩ+q+I)−1f∥H1(Ωε)≤Cε∥f∥C0,a(¯¯¯Ω), (3)

where is the operator of multiplication by a certain cut-off function.

• (3)-like estimates were also obtained in [Be95]. In this work the holes are distributed -periodically in a bounded domain (), no special assumptions on the geometry of holes are imposed. Let . Then one has the estimate

 ∥(−ΔΩε)−1Jεf−Jεψε(−ΔΩ+q)−1f∥H1(Ωε)≤Λε∥f∥L∞(Ω), limε→0Λε=0,

where the small factor is expressed in terms of the first eigenvalue of the Laplace operator on a period cell subject to the Dirichlet conditions on the hole boundary and the periodic conditions on the external part of the period cell boundary; the function is built on the basis of the corresponding eigenfunction.

• One can also study a surface distribution of holes, i.e. holes being located near some hypersurface intersecting . This problem was first considered in [MK64]; it was proved that the limit operator is . Here is a positive function, and is a delta-distribution supported on . For the case , the norm resolvent convergence with estimates on the rate of convergence were obtained in [BCD16], where even more general elliptic operators were treated. The proofs in [BCD16] rely on variational formulations for the pre-limit and the homogenized resolvent equations (the key object of their analysis is a certain integral identity associated with the difference of the resolvents). Note that the method we use in the current works allows to treat surface distributions of holes as well. Nevertheless, to simplify the presentation, we focus on the bulk distribution of holes only.

• Operator estimates in homogenization theory is a rather young topic. The classical homogenization problem concerning elliptic operators of the form

 Aε=−div(A(.ε)∇),where A(⋅) is a Zn% -periodic function,

was first treated in [BS04, BS07, Gr04, Gr06, Zh05a, Zh05b, ZhP05], see the recent papers [ZhP16, S16] for further references. In particular, the article [Zh05b] deals with a perturbation which is defined by rescaling an abstract periodic measure. The technique developed in [Zh05b] can be applied for deriving operator estimates is the case of periodically perforated domains provided the sizes of holes and distances between them are of the same smallness order (evidently, this does not hold for the problem we study in the current paper). Operator estimates were also obtained for elliptic operators with frequently alternating boundary conditions (see, e.g., [BBC10]), for problems in domains with oscillating boundary [BCFP13], or for the “double-porosity” model in [CK17]. For more results we refer to the paper [BCD16] containing a comprehensive overview on operator estimates in homogenization theory.

• In [AP17] we treat (possibly non-compact) manifolds with an increasing (even infinite) number of balls removed (similarly as in [RT75]), and show operator estimates using similar methods as in this article.

Acknowledgements

This research was carried on when the first author was a postdoctoral researcher in Karlsruhe Institute of Technology. He gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173 “Wave phenomena: analysis and numerics”.

2. Setting of the problem and main results

Let and let be a domain (not necessarily bounded) with -boundary . We also assume that there exists a constant such that the following map is injective on provided :

 (x,t)↦x+tν(x) (4)

where the unit inward-pointing normal vector field on .

Additionaly, we require to be uniformly regular in the sense of Browder [Br59]. This requirement is automatically fulfilled, for example, for domains with compact smooth boundaries or for compact, smooth perturbations of half-spaces. Under this assumption the Dirichlet Laplacian in defined via

 dom(ΔΩ)=H2(Ω)∩H10(Ω),ΔΩf:=Δf

is a self-adjoint operator (see, e.g., the recent paper [BLLR17] for more details and references on this issue).

We note, that our results remain valid under less restrictive assumptions on , see Remark 4.8 below.

In what follows we denote by , etc. generic constants depending only on the dimension .

We set .

Now we describe a family of holes in (see Figure 2). Let be a Lipschitz domain in depending on a small parameter . We denote by the radius of the smallest ball containing . It is assumed that

 (dε)n−2≤Cεnas n≥3,|lndε|−1≤Cε2as n=2,or (5) dε≤{C1εn/(n−2),as n≥3,exp(−1/(Cε2)),as n=2

(hence, in particular, ). For , let be a set enjoying the following properties:

 Diε coinsides with Dε up % to a rigid motion, B(Diε)⊂□iε:=ε(□+i), dist(B(Diε),∂□iε)≥κε for some κ>0, (6)

where is the smallest ball containing (the radius of this ball is ).

Finally, we set

 Ωε:=Ω∖(⋃i∈Iε¯¯¯¯¯¯¯¯Diε),

where

 Iε:={i∈Zn: □iε⊂Ω},

i.e. the set of those indices for which the rescaled unit cell is entirely in (with positive distance to ). The domain is depicted in Figure 1.

By we denote the Dirichlet Laplacian on , i.e. the operator acting in the Hilbert space associated with the closed densely defined positive sesquilinear form

 aε[u,v]:=∫Ωε∇u⋅∇¯vdx,dom(aε):=H10(Ωε).

Our goal is to describe the behaviour of the resolvent as under the assumption that the following limit exists and is finite:

 q=limε→0cap(Dε)εn, (7)

where is the capacity of the set . Recall (see, e.g., [T11]), that for the capacity of a set is defined via

 cap(D)=∫Rn|∇H(x)|2dx, (8)

where is a solution to the problem

 ⎧⎪⎨⎪⎩ΔH(x)=0,x∈Rn∖¯¯¯¯¯D,H(x)=1,x∈∂D,H(x)→0,|x|→∞. (9)

One has also the following variational characterization of the capacity, namely

 cap(D)=min∫Rn|∇u(x)|2dx, (10)

where the minimum is taken over being equal to on a neighbourhood of .

For the right-hand-side of (10) is zero for an arbitrary domain , hence we need a modified definition. It is as follows:

 cap(D)=∫B1|∇H(x)|2dx, (11)

where is the unit ball concentric with – the smallest ball containing (here we assume that the set is small enough so that ), solves the problem

 ⎧⎪⎨⎪⎩ΔH(x)=0,x∈B1∖¯¯¯¯¯D,H(x)=1,x∈∂D,H(x)=0,x∈∂B1. (12)

Further, proving the main results, we will use the following pointwice estimates for the functions at some positive distance from , see [MK06, Lemma 2.4].

Lemma 2.1.

Let . We denote by the distance from to , and by the radius of . One has:

 |H(x)|≤C⋅dn−2(ρ(x))n−2,|∇H(x)|≤C⋅dn−2(ρ(x))n−1as n≥3,|H(x)|≤C|lnd|−1|lnρ(x)|−1,|∇H(x)|≤C|lnd|−1ρ(x)as n=2

provided as or as ,  for some .

Remark 2.2.

Due to (7) one has

 cap(Dε)ε−n=O(1). (13)

In fact, this condition also follows directly from (5). Indeed, using the monotonicity of the capacity, we get , where is ball of radius containing . For this ball the function can be computed explicitly:

 H(x)=(dε)n−2|x|n−2 as n≥3,H(x)=lnxlndε as n=2,

hence as and as , hence, due to (5), we get (13).

Finally, we introduce the limiting operator . It acts in and is defined by

 A:=−ΔΩ+qI,dom(A):=dom(ΔΩ)=H2(Ω)∩H10(Ω).

By we denote the associated form:

 a[u,v]:=∫Ω(∇u⋅∇¯v+qu¯v)dx,dom(a):=H10(Ω).

The operators and act in different Hilbert spaces, namely and , respectively. Therefore we are not able to apply the usual notion of resolvent convergence and thus a suitable modification is needed. There are many ways how to do this in a “smart” way. For example (cf. [IOS89, Vai05]), one can treat the behaviour of the operator

 (Aε+I)−1Jε−Jε(A+I)−1∈L(H,Hε),

where is a suitable bounded linear operator satisfying

 ∀f∈H: ∥Jεf∥Hε→∥f∥H as ε→0. (14)

It is natural to choose the operator as the operator of restriction to , i.e.

 (Jεf)(x)=⎧⎨⎩f(x),x∈Ωε,0,x∈⋃i∈IεDiε. (15)

Due to (5) one has for each compact set

 ∑i:Diε⊂K|Diε|≤αε|K|, αε=o(1) % does not depend on K, (16)

where stands for the Lebesgue measure of . Hence, evidently, (14) holds. The results of [CM82, MK64, RT75] can be reformulated as follows:

 ∀f∈H:limε→0∥∥(Aε+I)−1Jεf−Jε(A+I)−1f∥∥Hε→0,

i.e. one has a kind of strong resolvent convergence.

Now, we can state our main result.

Theorem 2.3.

One has

 ∥∥(Aε+I)−1Jε−Jε(A+I)−1∥∥L(H,Hε)≤4δε,

where is defined by

 δε=|cap(Dε)ε−n−q|+CΩ,κ,β⋅⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩ε|lnε|,n=2,ε,n=3ε1−β, β>0,n=4,max{ε;dεε−1},n≥5, (17)

and the constant depends on the domain , the relative distance of the obstacles from the period cell boundary (see (6)), and, in the case , on .

Remark 2.4.

Via the same arguments as in Remark 2.2 one gets provided , hence, using the definition of , we obtain

 q>0, n≥5:δε=|cap(Dε)ε−n−q|+CΩ,κε2/(n−2).

Let be the operator of extension by zero:

 (J′εu)(x)=⎧⎨⎩u(x),x∈Ωε,0,x∈⋃i∈IεDiε. (18)

Then the main result of [DCR17] is equivalent to

 ∥∥J′ε(Aε+I)−1−(A+I)−1J′ε∥∥L(Hε,H)→0, ε→0.

The next theorem gives an improvement of this statement.

Theorem 2.5.

One has

where is defined in (17). Moreover,

 ∥∥(Aε+I)−1−Jε(A+I)−1J′ε∥∥L(Hε,Hε)≤13δε.

One important applications of the norm resolvent convergence is the uniform convergence of semi-groups generated by and . Namely, we can approximate in terms of simpler operators , and :

Theorem 2.6.

One has for each :

 ∥exp(−Aεt)−Jεexp(−At)J′ε∥L(Hε,Hε)≤ctδε,

where is defined in (17), and the constant depends only on .

Another important application is the Hausdorff convergence of spectra, see [DCR17]. Using Theorem 2.3 we are able to extend this result by obtaining an estimate for the difference between the corresponding eigenvalues. Namely, let the domain be bounded. We denote by and the sequences of the eigenvalues of and , respectively, arranged in the ascending order and repeated according to their multiplicities.

Theorem 2.7.

For each one has

 limε→0λk,ε=λk, (19)

moreover

 |λk,ε−λk|≤4Cε(λk,ε+1)(λk+1)δε, (20)

where is defined in (17), and , .

In the next section we introduce an abstract scheme, which then will be applied for the proof of the above theorems.

3. Abstract framework

In this section we present an abstract scheme for studying the convergence of operators in varying Hilbert spaces. It was developed by the second author of the present article in [P06] and in more detail in the monograph [P12] (see also the later work [MNP13], where non-self-adjoint operators were treated).

Let and be two separable Hilbert spaces. Note, that within this section is just a notation for some Hilbert space, which (in general) differs from the space , i.e. the sub-index does not mean that this space depends on a small parameter. Of course, further we will use the results of this section for -dependent space .

Let and be closed, densely defined, non-negative sesquilinear forms in and , respectively. We denote by and the non-negative, self-adjoint operators associated with and , respectively.

Associated with the operator , we can introduce a natural scale of Hilbert spaces defined via the abstract Sobolev norm:

 Hk=domAk/2,∥f∥Hk\coloneqq∥f∥k\coloneqq∥(A+I)k/2f∥H.

In particular, we have with , with , and with .

Similarly, we denote by the scale of Hilbert spaces associated with . The corresponding norms will be denoted by .

We now need pairs of so-called identification or transplantation operators acting on the Hilbert spaces and later also pairs of identification operators acting on the form domains.

Definition 3.1 ((see [P06, App.] or [P12, Ch. 4])).

Let and . Moreover, let and be linear bounded operators. In addition, let and be linear bounded operators on the form domains. We say that and are -close of order with respect to the operators , , , , if the following conditions hold:

 ∥Jεf−J1εf∥Hε≤δε∥f∥H1, ∀f∈H1, (C1a) ∥J′εu−J1′εu∥H≤δε∥u∥H1ε, ∀u∈H1ε, (C1b) |(Jεf,u)Hε−(f,J′εu)H|≤δε∥f∥H∥u∥Hε, ∀f∈H,u∈Hε, (C2) ∥Jεf∥Hε≤(1+δε)∥f∥H, ∀f∈H, (C3a) ∥J′εu∥H≤(1+δε)∥u∥Hε, ∀u∈Hε, (C3b) ∥f−J′εJεf∥H≤δε∥f∥H1, ∀f∈H1, (C4a) ∥u−JεJ′εu∥Hε≤δε∥u∥H1ε, ∀u∈H1ε, (C4b) ∣∣aε(J1εf,u)−a(f,J1′εu)∣∣≤δε∥f∥Hk∥u∥H1ε,∀f∈Hk,u∈H1ε. (C5)
Remark 3.2.

For the definition above implies that the operators and are unitary equivalent. Indeed, (C)–(C) assure that the operator is unitary with the inverse ; due to (C)–(C) and are the restrictions of and onto and , respectively. Hence, in view of (C), realises the unitary equivalence of and .

Now, we present the main implications of the definition of -closeness.

Theorem 3.3 ([P06, Th. A.5]).

One has

 ∥∥(Aε+I)−1Jε−Jε(A+I)−1∥∥L(H,Hε)≤4δε,

provided conditions (C), (C), (C), and (C) hold with .

Remark 3.4.

Let (), be non-negative self-adjoint operators in the same Hilbert space , and let and be the corresponding sesquilinear forms. We assume that and

 |aε(f,u)−a(f,u)|≤δε√a[f,f]+∥f∥H√aε[u,u]+∥u∥H, ∀f,u∈dom(a), (21)

where as . Due to (21) and are -close of order with respect to the identity maps (on ) and (on ). Then by Theorem 3.3

 ∥∥(Aε+I)−1−(A+I)−1∥∥L(H)→0. (22)

In fact, it would suffice for (22) if (21) is satisfied whenever , see Theorem VI.3.6 in T. Kato’s monograph [Kat66]. In this sense, Theorem 3.3 can be regarded as a generalization of this classical result to the setting of varying spaces.

Theorem 3.5 ([P06, Th. A.8]).

Let be an open set containing either or . Let be a bounded measurable function, continuous on and such that the limit exists.

Then there exists with as such that

 ∥ψ(Aε)Jε−Jεψ(A)∥L(H,Hε)≤ηψ(δε) (23)

for all pairs and , which are -close of order .

Remark 3.6.

The important example of the function satisfying the requirements of the above theorem is , is a parameter. Another important example is the function – the characteristic function of the interval with or . In this case Theorem 3.5 gives the closeness of the spectral projections.

Theorem 3.7 ([P06, Th. A.10]).

Let for some function the estimate (23) be valid. Then

 ∥J′εψ(Aε)−ψ(A)J′ε∥L(Hε,H)≤ηψ(δε)+2∥ψ∥∞δε, ∥J′εψ(Aε)Jε−ψ(A)∥L(H,H)≤Cψδε+2ηψ(δε), ∥ψ(Aε)−Jεψ(A)J′ε∥L(Hε,Hε)≤5Cψδε+2ηψ(δε)

provided (C)–(C) hold true. Here comes from (23), stands for the -norm, and is a constant satisfying for all .

For one has (see Theorem 3.3), , and hence we immediately get the following corollary from Theorem 3.7.

Corollary 3.8.

One has

 ∥∥(Aε+I)−1−Jε(A+I)−1J′ε∥∥L(Hε,Hε)≤13δε

provided and are -close of order .

For “good enough” functions the last statement of Theorem 3.7 can be improved. Evidently the function () satisfies the requirements of the theorem below.

Theorem 3.9 ([Mnp13, Th. 3.7]2).

Let with and be a holomorphic function satisfying for some . Let and be -close of order . Then

 ∥ψ(Aε)−Jεψ(A)J′ε∥L(Hε,Hε)≤cψδε, (24)

where is a constant depending on .

Remark 3.10.

In fact, (24) is valid even for less regular functions. For instance, it holds for as in Remark 3.6, see [P12, Sec. 4.5, Cor. 4.5.15].

The last result concerns the convergence of spectra in general. For two compact sets we denote by the Hausdorff distance between these sets, i.e.

 distH(X,Y)=max{supx∈Xdist(x,Y);supy∈Ydist(y,X)},

where .

Remark 3.11.

Let be a family of compact domains and

 distH(Xε,X)→0 as ε→0 (25)

for some compact domain . It is easy to prove (see, e.g., [P12, Proposition A.1.6]) that (25) holds iff the following two conditions are fulfilled:

• Let . Then there exists such that .

• Let . Then there exists a family with such that .

Theorem 3.12 ([P06, Th. A.13]).

There exists with as such that

 distH(11+σ(A),11+σ(Aε))≤η(δε)

for all pairs and which are -close of some order .

4. Proof of the main results

For an open subset () we denote by the mean value of over , i.e.

 ⟨f⟩M:=1