Resolvent estimates for high-contrast elliptic problems with periodic coefficients

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1 Resolvent estimates for high-contrast elliptic problems with periodic coefficients K. D. Cherednichenko and S. Cooper July 16, 2015 Abstract We study the asymptotic behaviour of the resolvents A ε + I of elliptic second-order differential operators A ε in R d with periodic rapidly oscillating coefficients, as the period ε goes to zero. The class of operators covered by our analysis includes both the classical case of uniformly elliptic families where the ellipticity constant does not depend on ε and the double-porosity case of coefficients that take contrasting values of order one and of order ε 2 in different parts of the period cell. We provide a construction for the leading order term of the operator asymptotics of A ε + I in the sense of operator-norm convergence and prove order Oε remainder estimates. 1 Introduction The subject of the present article is the investigation of analytical properties of partial differential equations PDE of a special kind that emerge in the mathematical theory of homogenisation for periodic composites. The study of composite media has been attracting interest since the middle of the last century see e.g. 9 of the monograph [12], where some heuristic relationships for the overall properties of mixtures are discussed, although the question of averaging the microstructure in order to get intuitively expected macroscopic quantities goes back a few more decades still. In the early 1970 s a number of works have appeared concerning the analysis of PDE with periodic rapidly oscillating coefficients, which could be thought of as the simplest, yet already mathematically challenging, object representing the idea of a composite structure. For a classical overview of the related developments we refer the reader to the books [2], [9]. In the following years a large amount of literature followed, extending homogenisation theory in various directions. One of the central themes of this activity has been in understanding the relative strength of various notions of convergence in terms of characterising the homogenised medium. Unlike in the classical case of uniformly elliptic PDE, whose solutions are compact in the usual Sobolev spaces W l,p, non-uniformly elliptic problems offer a variety of descriptions for the homogenised medium that depend on the notion of convergence used. From the computational point of view, one is presented with the question of what approaches yield controlled error estimates for the difference between the original and homogenised solutions. A number of results have been obtained recently concerning the difference, in the operator norm, between the resolvent of the differential operator representing the original heterogeneous medium x div A u, u D ε L 2 Ω, ε > 0, 1.1 ε and the resolvent of the operator representing the homogenisation limit div A hom u, u D hom L 2 Ω. 1.2 Here Ω is an open connected subset of R d, the matrix function A is [0, 1 d -periodic, bounded and uniformly positive definite, the constant matrix A hom represents the homogenised medium, and D ε, D hom denote the domains of the corresponding operators. While a basic order O ε estimate for this setup has been known for a long time, see e.g. [9], one should in principle expect the better rate of convergence of order Oε suggested by the formal asymptotic analysis assuming that the domain Ω is sufficiently regular. The work [3] contains the related result for problems in the whole space Ω = R d, via a combination of spectral theoretic machinery based on the Bloch fibre decomposition of periodic PDE and asymptotic analysis. Earlier works [14], [4] used similar ideas to prove resolvent convergence, but they did not go as far as getting the order Oε operator norm estimates. The more recent papers [17], [11] use different techniques to show an improved rate of convergence of order Oε log ε σ, σ > 0, for problems in bounded domains: [17] for scalar Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom Laboratoire de Mécanique et Génie Civil de Montpellier, 860 Rue de Saint-Priest, 34095, Montpellier, France 1

2 equations and [11] for systems. Finally, the papers [6], [13] prove the expected order Oε convergence for such problems: [6] by using the method of periodic unfolding and only for scalar problems, and [13] by combining the earlier results of [3] with some elements of the approach of [17] and including the case of a system. The focus of the present paper is on obtaining operator-norm resolvent-type estimates for a class of nonuniformly elliptic problems of the double porosity type, where the matrix A = A ε takes values of order one and of order ε 2 in mutually complementary parts of the unit cell [0, 1 d. The presence of multiscale effects for such problems was first highlighted in the paper [1]. An analysis of the relation between these effects and the resolvent behaviour of double-porosity problems was carried out in [15]. The earlier results [3] concerning resolvent estimates for are based on the analysis of spectral projections of the associated operators in a neighbourhood of zero. This approach does not suffice in the double porosity case as all spectral projections provide a leading-order contribution to the behaviour of the resolvent as ε 0. Bearing this in mind, we analyse the asymptotic behaviour of the fibres of the operator provided by the Bloch decomposition. As was observed by [7], the pointwise limit of the fibres is insufficient for norm-resolvent estimates. We show that in fact the convergence of the individual fibre resolvents is non-uniform with respect to the quasimomentum κ [0, 2π d. This effect is due to the presence of a boundary layer in the neighbourhood of the origin κ = 0, where the asymptotics for each fixed κ fails to be valid. To obtain uniform estimates in this neighbourhood we study the asymptotics for the rescaled fibres parametrised by = κ/ε. The corresponding inner expansion is coupled to the pointwise outer expansion in a matching region where neither expansion is uniform. We briefly outline the structure of the paper. In Section 2 we introduce the sequence of problems we analyse. In Section 3 we recall the notions of the direct fibre decomposition and of the associated Gelfand transform. Section 4 contains the formulation of our main result using these notions. In Section 5 we describe the resolvent asymptotics in the inner region for relevant values of the quasimomentum ε [0, 2π d. In Section 6 we introduce spaces V κ H# 1, κ [0, 2πd, which play a key role in our construction. We also prove some lemmas used in the proof of the main result, namely a special Poincaré-type inequality for the projection on the space orthogonal to V κ with respect to the inner product of H# 1, as well as several elliptic estimates that are uniform in. Section 7 is devoted to the proof of our main result Theorem 4.1, which consists of two pieces of analysis, in the inner region 1 and in its complement 1. In Section 8 we discuss the outer region ε /2 and show that the inner and outer approximations jointly are only sufficient to obtain a norm-resolvent estimate of order Oε α, α 0, 1. In Section 9 we calculate the limit of the spectra of the operators div A ε /ε and explain its relation to an earlier study of [15]. Finally, in Section 10 we show that our main theorem contains as a particular case a result of [3], followed by a discussion of some key points of the work [15] and the relation of its result to our convergence statement. 2 Problem setup In what follows we study the problem div In the above equation A ε x ε u + u = f, f L 2 R d. 2.3 A ε = A 1 + ε 2 A 0, where A 0, A 1 are -periodic symmetric d d-matrix functions with entries in L. We assume that A 0 νi, ν > 0 and that A 1 νi on a Lipschitz open set 1 := [0, 1 d the stiff component of the composite with A 1 = 0 on the interior of \ 1 the soft component, which we denote by 0. We also assume that 0 0, 1 d, which implies, in particular, that the set n Z d 1 + n is connected in R d. We next recall the construction of the operator A ε associated with 2.3. The closed sesquilinear form a ε u, v = A ε x ux vxdx, u, v H 1 R d, R ε d is symmetric and non-negative in L 2 R d, hence it generates a self-adjoint operator A ε whose domain DA ε is dense in L 2 R d and whose action is described by the identity A ε u, v L 2 R d = a ε u, v for u DA ε, v H 1 R d. The solution u = u ε to 2.3 is understood as the result of applying the resolvent of A ε to f, i.e. u ε = A ε + I f. The last formula is well defined for any f L 2 R d : indeed, the operator A ε + I is clearly bounded below by I, hence it is injective, and the only element g L 2 R d orthogonal to the image of A ε + I is g = 0 by virtue of the fact that the form a ε u, v + u, v L 2 R d, u, v H 1 R d, is positive. The same fact implies that the resolvent A ε + I is a bounded operator. Throughout the text we denote by H 1 # the space of -periodic functions that belong to H1 loc Rd. We use the letter C for any positive constant whose exact value may vary from line to line. 2

3 3 Bloch formulation and Gelfand transform Using a procedure similar to the above definition of A ε + I, for each ε, where := [0, 2π d, we define u ε H1 # as the solution to ε 2 + iε A ε + iε u ε + u ε = F, F L 2, 3.4 In other words, for all ε one has u ε = B ε, + I F, where the operators B ε, are generated by the closed sesquilinear forms b ε, u, v = ε 2 A 1 + A 0 + iε u + iε v, u, v H 1 #. Lemma 3.1. For each ε > 0 there exists a unitary map U ε : L 2 R d L 2 ε such that U ε A ε + I U ε = ε B ε, + I d, i.e. for all f L 2 R d the formula A ε + I f = U ε g holds, where for each ε one has g, = B ε, + I U ε f,. Proof. For a given ε > 0 set U ε f, y := ε d n Z d f εy + n e iε y+n, ε, y. Note that for each ε the operator U ε is the composition T ε G ε of a scaled version of the usual Gelfand transform G ε : L 2 R d L 2 ε ε, given by G ε f, z := ε d/2 n Z d fz + εne i z+εn, ε, z ε, and the scaling transform T ε : L 2 ε ε L 2 ε given by The inverse U ε and the inverse of G ε given by T ε h, y := ε d/2 h, εy. is the composition Gε Tε of the inverse of T ε given by T ε h, x := ε d/2 h, x/ε, ε, x R d, G ε hx = ε d/2 ε h, xe i x d, x R d. The map U ε is unitary since the corresponding property clearly holds for T ε and is well known for G ε, see e.g. [2]. 4 Homogenised operator in -representation and the main convergence result First, we introduce a -parametrised operator family that plays a central role in our analysis of the operators A ε as ε 0. We denote H 0 := C H0 1 0, and for each ε > 0 and ε consider the sesquilinear form b hom ε, c, u, d, v := A hom cd + where A hom is the usual homogenised matrix A hom ξ ξ = min u H 1 # A 0 + iεu + iεv, c, u, d, v H 0, 4.5 A 1 ξ + u ξ + u, ξ R d. 4.6 Note that the matrix A hom is positive definite. Indeed, using the ellipticity assumption on A 1 one has, for ξ R d, A hom ξ ξ ν min ξ + u 2 = ν ξ 2 Mξ/ ξ, u H# 1 1 3

4 where the function Mη := has a positive minimum M min, hence A hom νm min. In what follows we also denote min u H# 1 η + u 2, 1 η = 1, L := {c + ũ : c C, ũ L 2, ũ 1 = 0} L 2, and use the invertible identification map I : C L 2 0 L that takes each pair c, u to the function c + ũ L with ũ = u on 0 and ũ = 0 on 1. We next define operators Bε, hom in the Hilbert space C L 2 0 equipped with the inner product c, u, d, v 0 = Ic, u, Id, v L 2. These operators are associated, for each value of ε, with the forms b hom ε, by means of the identity B hom ε, c, u, d, v 0 = bhom ε, c, u, d, v, d, v H0, where the pairs c, u are taken from the maximal possible domain D Bε, hom dense in H 0 and hence in C L 2 0., which can be shown to be The operators B hom 0, can be viewed, roughly speaking, as the -components of the Fourier transform of the two-scale homogenised operator, see Section 10 below, with respect to the macroscopic variable. However, as we also discuss in the same section, in order to obtain operator-norm resolvent estimates it is important to deal with a suitable truncation of this Fourier transform that restricts the Fourier variable to the set ε. From this perspective the analysis below can be viewed as a rigorous procedure for such a truncation. Note that in view of the non-uniform behaviour of these truncations as ε 0, as we discuss in Section 1 and in Section 8, the expression ε in 4.5 can not be set to zero in the region 1, hence the dependence of the operators B hom ε, on ε. We also denote by P the orthogonal projection of the Hilbert space L 2 ε onto its closed subspace { c + g : c L 2 ε, g L 2 ε, g, y = 0 a.e., y ε 1 }, and by P f its analogue on each fibre, the orthogonal projection of L 2 onto L. The main result of the present paper is as follows. Theorem 4.1. The resolvents of the operator family A ε are asymptotically close as ε 0 to the family R ε := U ε ε I B hom ε, + I I d PU ε, where the corresponding approximation error is of order Oε. More precisely, there exists a constant C > 0, independent of ε, such that A ε + I R ε L2Rd L2Rd Cε. 4.7 Note that the operator R ε can also be written as R ε = U ε ε I B hom ε, + I I P f d U ε, which follows from the definitions of the projection operators P and P f. 5 The inner expansion and principal term for Bε, hom region 1. in the inner In this section we provide an explicit representation for the behaviour in ε of the operators Bε, hom in the region 1. We refer to this expansion as the inner expansion and to its region of validity as the inner region. Let us consider an asymptotic expansion for solutions to 3.4 of the form u ε = n=0 ε n u n, u n H 1 #, n = 0, 1, 2, Substituting 5.8 into 3.4 and comparing the coefficients in front of ε 2 on both sides of the resulting equation we find A 1 u 0 = 0, 5.9 4

5 or, equivalently, u 0 V := { u H 1 # A1 u = 0 }, 5.10 a space that is naturally isometric to H 0 via the mapping I defined above: H 0 c, v u = c + ṽ V, where, as before, ṽ = v on 0 and ṽ = 0 on 1. This implies that u 0 = c 0 +v 0, where the pair c 0, v0 belongs to H 0. Further, comparing the coefficients in front of ε and using 5.10 yields Introducing unit-cell solutions N k, k = 1,..., d, that satisfy A 1 u 1 = i A 1 c d i A1 ij j N k = i,j=1 we note that, up to an arbitrary additive constant, one has u 1 = i d j=1 d j A 1 kj, 5.12 j=1 N j j c The concrete choice of the constant added to 5.13 plays an important role in the justification of the asymptotic expansion, which we discuss in Section 6.2 see proof of Lemma 6.3. Finally, comparing the coefficients in front of ε 0 yields an equation for u 2 as follows where A 1 u 2 = F, 5.14 F := F + i A 1 + A 1 u 1 + A 0 u 0 A 1 c 0 u Solvability of 5.14 requires that F, v = 0 for all v V. The formula 5.13 and the solvability condition for 5.14 imply that u 0 = c 0 + v 0, where the pair c 0, v0 H0 satisfies the identity 0 c d + ϕ = P f F d + ϕ d, ϕ H A hom c 0 d + A 0 v 0 ϕ + + v 0 Following the method outlined in Section 4 for the construction of B hom, we introduce the operator Bhom associated to the problem 5.16 such that c 0 B hom 0, is ε-close in norm to B hom, v0 = B hom ε, in the inner region of. ε, 0, + I I P f F. The next result shows that Lemma 5.1. There exists C > 0 such that the estimate I B0, hom + I I P f I Bε, hom + I I L P f Cε, 2 L 2 holds for all ε satisfying the inequality 1. Proof. For each value as in the lemma consider the pairs c, v = B0, hom + I I P f F and c ε, v ε = + I I P f F, that is B hom ε, and A hom c ε d + A hom cd + A 0 v ϕ + c + vd + ϕ = P f F d + ϕ d, ϕ H 0, 5.17 A 0 + iεv ε + iεϕ + cε + v ε d + ϕ = P f F d + ϕ d, ϕ H By setting d, ϕ = c ε, v ε in 5.18 and noting v ε H we arrive at the a priori bound for some constant C. v ε L2 0 C F L , 5

6 To prove the result we show that for u ε := Ic ε, v ε and u := Ic, v there exists a constant C > 0 independent of ε, such that u ε u H 1 Cε F L Subtracting 5.17 from 5.18 implies A hom c ε cd + A 0 v ε v ϕ + cε + v ε c v d + ϕ = A 0 v ε iεϕ A 0 iεv ε + iεϕ, d, ϕ H Setting d, ϕ = c ε c, 0 in 5.21 gives A hom + 1 c ε cc ε c = c ε c v ε v, hence c ε c C vε v L 2 0, since v ε, v H Setting d, ϕ = c ε c, v ε v in 5.21 gives vε v 2 L 2 C 0 A 0 v ε v v ε v C [ Taking into account 5.19 this implies 5.20, since A 0 v ε iεv ε v Cε v ε L2 0 v ε v L2. ] A 0 iεv ε + iεv ε v u ε u L 2 c ε c + v ε v L 2 0 C vε v L Auxiliary material 6.1 Cell problems One of the key elements in the proof of our main result is the analysis of the properties of the following family of auxiliary cell problems : A 1 w = G, G H κ := H 1 κ Here Hκ, 1 κ, is the space of κ quasi-periodic functions belonging to H 1, i.e. u Hκ 1 if, and only if, uy = expiκ yvy, y, where 1 v H# 1. Note that 5.9, 5.11, 5.14 all have the form 6.22 for κ = 0 with G = 0, G = i A 1 c 0, G = F, respectively. For a given matrix function A 1 we consider the space V κ := { v H 1 κ A1 v = 0 } Note that, for A 1 satisfying the assumptions prescribed in Section 2, we find { V for κ = 0, V κ = H0 1 0 for κ 0. A criterion for the existence of solutions to 6.22 is given below by a variant of the Lax-Milgram lemma. Lemma 6.1. For all κ, denote by V κ the orthogonal complement of V κ in H 1 κ. Then, for all values of κ: i There exists a constant C > 0 independent of κ such that 1 Hκ 1 coincides with H# 1 when κ = 0. P V κ w H 1 Cdκ A 1 w L 2 w H 1 κ,

7 where dκ = { 1 for κ = 0, κ for κ 0, and P V κ is the orthogonal projection of H 1 κ onto V κ. ii There exists a solution w H 1 κ to 6.22 if and only if G, ϕ = 0 for all ϕ V κ. iii Any solution to 6.22 is unique up to the addition of an element from V κ : if w satisfies 6.22 then w +v satisfies 6.22 for any v V κ, and if w 1, w 2 satisfy 6.22 then w 1 w 2 V κ. In particular, if w is a solution to 6.22 then P V κ w is the unique part in V κ of any solution to Proof. i The inequality 6.24 holds if there exists a constant C > 0 such that for all u Hκ 1 there exists v V κ such that u v H 1 Cdκ u 2 dy. 1 We shall now verify this for two distinct cases. Case 1: κ = 0. For fixed u H# 1, denote ũ H1 # to be an extension of u such that ũ L2 C u L2 1. Notice that such an extension exists for connected 1 cf. [9, Section 3.1]. Defining v := u ũ+ 1 1 ũ, we see that v V and u v H1 = ũ 1 1 C ũ 2 C u 2, H1 1 where the first inequality is a variant of the standard Poincaré inequality. Case 2: κ 0. For fixed u H 1 κ, we show there exists a v V κ such that 1 ũ u v 2 H 1 C κ 2 u 2 L 2 1. Denoting the map as above, we find u ũ =: v H = V κ and u v 2 H 1 = ũ 2 H 1 C κ 2 ũ 2 L 2 C κ 2 u 2 L 2, 1 which proves the result. Here we have used the following Poincaré type inequality u 2 L 2 C κ 2 u 2 L 2 u H 1 κ, which is true since κ 2 is the first eigenvalue of the Laplace operator with κ-quasiperiodic boundary conditions. ii Let w be a solution of 6.22 and let ϕ V κ. Then, using the symmetry of A 1 and 6.23, G, ϕ = A 1 w ϕ = w A 1 ϕ = which yields G, ϕ = 0 for all ϕ V κ. Conversely, suppose that G, ϕ = 0 for all ϕ V κ, and seek w Hκ 1 that satisfies By 6.25, the identity A 1 w ϕ = G, ϕ 6.26 holds automatically for all ϕ V κ, therefore it is sufficient to verify it for all ϕ V κ. Seeking w in V κ reduces the problem to showing that, in the Hilbert space H := V κ with the norm inherited from H 1, the problem 6.26 satisfies the conditions of the Lax-Milgram lemma see e.g. [9, Section 1.1]. As the bilinear form B[v, w] := A 1 v w is clearly bounded in H, i.e. for some C > 0 one has B[v, w] C v H 1 w H 1, in order to satisfy the conditions of the Lax-Milgram lemma it remains to be shown that the form B is coercive, i.e. for some ν > 0 the bound B[v, v] ν v 2 H 1 holds. To this end, note that the boundedness of A 1 and 6.24 imply B[v, v] := A 1 v v = A1 1/2 v 2 L 2 C A 1 v 2 L 2 ν v H 1. 7

8 Now by the Lax-Milgram lemma, there exists a unique solution w V κ to the problem B[w, ϕ] = G, ϕ ϕ V κ, and hence to iii If w satisfies 6.22 and v V κ then A 1 v = 0 and hence w + v also satisfies Assuming further that w 1 and w 2 both satisfy 6.22, notice that v = w 1 w 2 is a solution of 6.22 with G = 0. Finally, setting ϕ = v in 6.26 yields 0 = A 1 v v = A1 1/2 v 2 L 2, implying that A 1 1/2 v = 0 and hence A 1 v = 0, i.e. one has v V κ. Assuming now that the solutions w 1, w 2 are in V κ, the difference v = w 1 w 2 belongs to both V κ and V κ and is therefore zero. Corollary 6.1. For each k = 1,..., d, there exists a unique solution N k V to the unit-cell problem In particular, for all ε and c 0 C, there exists a unique solution u 1 V to the problem 5.11, for which the estimate u 1 H 1 N k H 1 k c holds. 6.2 Elliptic estimates In our proof of Theorem 4.1 we use the following two statements. Lemma 6.2. For each ε, let u 0 = Ic 0, v0, where c 0, v0 is the solution to 5.16 with F L 2, and let u 1 H# 1 be the solution 5.13 to the unit-cell problem Then the following estimates hold with some C > 0 : c 0 C F L 2, u C F H 1 L 2, u C H F L Proof. Setting d, ϕ = c 0 and 6.29 follows by the Cauchy-Schwarz inequality. Setting d, ϕ = c 0, 0 in 5.16 yields A hom c 2 + c 2 = Using the estimate P f F, v0 in 5.16, and dropping the scripts 0 and for convenience, yields A hom c 2 + A 0 u u + u 2 = P f F u, P f F c v c. 0 c v c F L 2 c + 0 1/2 v L 2 c 0 F L 2 c + 0 1/2 u L 2 c + 0 1/2 c 2, along with the positivity of A hom and the bound 6.29, we infer The estimate 6.30 is now a direct consequence of 6.28 and Lemma 6.3. For each ε, 1, let u 0 ε, = I c 0 ε,, v0 0 ε,, where the pair c ε,, ε, v0 H0 satisfies the identity c b hom 0 ε, ε,, v0 0 ε,, d, ϕ + c ε, + v0 ε, d + ϕ = P f F d + ϕ, d, ϕ H 0, 6.31 with F L 2. We denote by u 1 ε, a solution to the unit-cell problem A 1 u 1 ε, = i A 1c 0 ε, 8

9 such that Then the following estimates hold with some C > 0 : A 1 u 1 ε, = c ε, C F L2, 6.33 u 0 ε, C F L 2 L 2, iεv 0 ε, C F L 2 L 2, u H 1 C F L Proof. Taking the unique solution w ε, V to the problem ε, A 1 w ε, = i A 1 c 0 ε,, we find by Corollary 6.1 that w ε, H 1 C c0 ε,. Denoting u 1 ε, = w ε, A 1 A 1 w ε,, it is clear that 6.32 holds. By the properties of boundedness and ellipticity of A 1 we find that In particular, the estimate A 1 A 1 w ε, C w L2 ε,. 1 u H1 C c0 ε, holds. Inequalities are now shown by appropriately modifying the proof of Lemma 6.2. Lemma 6.4. Let ε, and let F be given by There exists a function R H# 1 satisfying A 1 R = F, ε,. such that R H1 C F L for some constant C > 0 independent of ε,. Proof. The functions u 0 and u 1 are chosen so that F satisfies the solvability condition for the equation 5.14, thus the existence of a solution u 2 is guaranteed by Lemma 6.1. Denoting by R to be the unique part in V of any such solution, i.e. letting R V be such that A 1 R ϕ = F, ϕ ϕ H#, we find, by choosing ϕ = R in 6.38 and using the assumptions on A 1, that A 1 R 2 L 2 A 1/2 1 2 L A 1/2 1 R 2 L 2 C F H # R H 1 #, where A 1/2 1 is the square root matrix of A 1. Due to Lemma 6.1i, it remains to show that F H # C F L 2 for some constant C. This can be seen by Lemma 6.2 and by noting, for ε, that F, ϕ = F ϕ ia 1 u 1 ϕ + i A 1 u 1 ϕ A 0 u 0 ϕ A 1 c 0 ϕ u0 ϕ C F L u + 0 H 1 u H1 c ϕ H 1 for all ϕ H 1 #. 9

10 Lemma 6.5. For each ε > 0, 0, consider H ε, H # such that H ε,, ϕ = 0 for all ϕ H Then there exists a solution R ε, H# 1 to the problem + iε A 1 + iε Rε, = H ε,, that satisfies the estimate [ Rε,H 1 1 C Hε, H ε,, 1 ε H # + 1 Hε, ε 2, 1 ] 6.39 for some constant C > 0 independent of ε,. Proof. For κ 0, V κ = H0 1 0, see By Lemma 6.1, the assumption H ε,, ϕ = 0 for all ϕ H0 1 0 implies there exists a unique weak solution w ε, V ε to the problem and A 1 w ε, y = expiε yh ε, y, y, w ε, H1 C A 1 w L ε, ε As w ε, Hε 1, the function r ε,y := exp iε yw ε, y is an element of H# 1 and satisfies the identity A 1 + iε rε, + iε ϕ = H ε,, ϕ ϕ H#, and by 6.40 we find that r ε, H 1 C ε 2 H ε, H # The estimate 6.39 can be viewed as a refined version of 6.42 by writing r ε, = s ε, + t ε,, where s ε,, t ε, are the solutions to 6.41 for the right-hand sides H ε, H ε,, 1 ψ and H ε,, 1 ψ respectively, for any chosen ψ L 2, ψ 0 = 0, ψ = 1. We then argue that in order to prove the theorem it is sufficient to ensure that there exists a function R ε, solving 6.41 which satisfies the bound R ε, H 1 C ε H ε, H #, 6.43 for the class of H ε, H # such that H ε,, 1 = 0. Indeed, one has Hε, H ε,, 1 ψ, ϕ = 0, Hε,, 1 ψ, ϕ = 0 ϕ H0 1 0, and Let w ε,, r ε, be as above for a given H ε, extension of r ε, such that Hε, H ε,, 1 ψ, 1 = H ε,, 1 H ε,, 1 ψ = 0. H # such that H ε,, 1 = 0. Denoting by R ε, an R ε, = r ε, in 1, 6.44 Rε,L 2 C rε,l 2, it is clear that R ε, also satisfies 6.41 with r ε, replaced by R ε,. We next show that R ε, satisfies the inequality Substituting ϕ 1 in 6.41, and recalling that H ε,, 1 = 0, we infer that and hence R ε, = ε 2 A 1 ε 2 A 1 R ε, R ε, + iε A 1 R ε,, R ε, C ε 2 ε 2 R ε, R ε, + ε Rε,L L In particular, by 6.45 and the standard Poincaré inequality, it follows that R ε, C rε,l2 ε

11 Therefore, by we find that R H ε, C 1 2 1/2 R ε, + R ε, 2 C r L2 ε, ε. 1 To prove 6.43 it now remains to show that r ε, L2 1 C H ε, H # 6.47 for some constant C > 0. By virtue of the inequality r ε, L 2 w ε, H 1 and 6.40 we find that r ε, L2 1 + iε r ε, L2 1 + ε r ε, L2 1 C + iε r ε, L Further, substituting ϕ = R ε, in 6.41 and recalling 6.44 yields A 1 + iε r ε, + iε r ε, = H ε,, R ε, = H ε,, R ε, = H ε,, R ε, R ε, + H ε,, R ε, R ε, C H ε, H # Rε, L 2 C Hε, H # rε, L The last equality above follows from the assumption that H ε,, 1 = 0. Finally, inequalities 6.48 and 6.49 imply Proof of the main result In terms of the notation introduced in Sections 3 and 4, proving Theorem 4.1 is equivalent to showing that there exists a constant C > 0 independent of and ε such that B ε, + 1 I B hom ε, + 1 I L2 P f Cε. L 2 This fact is a consequence of the following theorem. Theorem 7.1. For each ε > 0, ε, let u ε be the solution to 3.4 and let u0 ε, := c0 ε, + v0 ε, where the pair c 0 ε,, ε, v0 H0 satisfies the identity Then there exists a constant C > 0 independent of and ε such that u ε u 0 Cε F L 2 L 2. ε, Proof. To prove the result we consider ε in two regions. Case 1: 1. Let U 1 ε, = u0 + εu 1 + ε 2 R, where u 0, u1 and R are given by 5.16, 5.11 and Lemma 6.4, respectively. Due to the fact that the functions u 0 and u 0 ε, are ε-close in L2 uniformly in for 1 cf. Lemma 5.1, it is sufficient to prove that u ε U 1 Cε F L 2 L 2. ε, By direct calculation we find that the difference z 1 ε, := uε U 1 ε, is the H1 #-solution of the equation ε 2 + iε A ε + iε z 1 ε, + z1 ε, = F 1 ε,, 7.50 where the coefficients for the non-positive powers of ε have cancelled due to the construction of U 1 ε,. The right-hand side F 1 of 7.50 takes the form where ε, H # F 1 ε, := 4 n=1 ε n T n, T 1 := i A 1 + A 1 R + A 0 A 1 I u 1 + i A 0 + A 0 u 0, 7.51 T 2 := A 0 A 1 I R + i A 0 + A 0 u 1 A 0 u 0, 7.52 T 3 := i A 0 + A 0 R A 0 u 1, 7.53 T 4 := A 0 R

12 are elements of H #. A straightforward calculation shows that equations with the inequalities 6.29, 6.30 and 6.37 imply the bound Hence, the required inequality follows. Case 2: 1. Let U 2 ε, = u0 Lemma 6.5 for the right-hand side ε, + εu1 F 1 ε, H # Cε F L 2. z 1 ε, L 2 Cε F L 2. ε, + ε2 R ε,, where u 1 ε, is defined in Lemma 6.3 and R ε, is given by H ε, := F + i A 1 + A 1 u 1 ε, ε A 1u 1 ε, + + iε A 0 + iεv 0 ε, A 1c 0 ε, u0 ε,. Notice that the following inequalities hold H ε, H ε,, 1 H # C F L 2, H ε,, 1 C ε F L2, for some constant C > 0 independent of ε and. These follow from Lemma 6.3 and the estimates H ε,, ϕ = F ϕ ia 1 u 1 ε, ϕ + i A 1 u 1 ε, ϕ ε A 1u 1 ε, ϕ A 0 + iεv 0 ε, + iεϕ + A 1c 0 ε, ϕ + u0 ε, ϕ C F L u + 0 H1 + iεv L2 0 c 0 + u ε, L 2 ϕ H 1 and Hε,, 1 = ε, A 0 + iεv 0 ε, iε C ε 0 + iεv ε, ε, ε, L2 0 The assumptions of Lemma 6.5 hold for H ε, which implies, along with the above inequalities, that the existence of a function R ε, H# 1 is guaranteed such that R ε, H 1 C 1 ε F L By direct calculation we find that the error z 2 ε, := uε U 2 ε, is the H1 #-solution of the equation ε 2 + iε A ε + iε z 2 ε, + z2 ε, = F 2 ε,, where the coefficients for the non-positive powers of ε have cancelled due to the construction of U 2 ε,. In the above equation the right-hand side F 2 is given by where ε, H # F 2 ε, := 4 n=1 ε n S n, S 1 := A 0 I u 1 ε, + i A 0 + A 0 c 0 ε,, 7.56 S 2 := A 0 I R ε, + i A 0 + A 0 u 1 ε, A 0c 0 ε,, 7.57 S 3 := i A 0 + A 0 R ε, A 0 u 1 ε,, 7.58 S 4 := A 0 R ε, 7.59 are elements of H #. Equations together with inequalities 6.33, 6.36 and 7.55 imply that F 2 ε, H # Cε F L 2. Therefore, the bound holds, and the result follows. 2 z H 1 Cε F L 2 ε, 12

13 8 The outer expansion and principal term for Bε, hom region ε /2. in the outer For fixed κ 0 we shall study the asymptotics of the following problem: find w ε,κ H 1 κ such that ε 2 A 1 + A 0 wε,κ + w ε,κ = F, F L Let us consider an asymptotic expansion for the solution to the above problem of the form w ε,κ = n=0 ε 2n w n κ, w n κ H 1 κ, n = 0, 1, 2, Substituting 8.61 in 8.60 and comparing the coefficients in front of ε 2 on both sides of the resulting equation yields A 1 w κ 0 = 0, i.e. w κ 0 V κ or, equivalently, w κ 0 H0 1 0, see Further, comparing the coefficients in front of ε 0 yields A 1 w κ 1 = F + A 0 w κ 0 w κ The existence of a solution to 8.62 is guaranteed by Lemma 6.1 if, and only if, w κ 0 Existence and uniqueness of w κ 0 give the inequality satisfies the identity 0 A0 w 0 κ ϕ + w 0 κ ϕ = F, ϕ ϕ H is implied by the ellipticity of A 0 in 0 and standard ellipticity estimates w 0 κ H C F L Furthermore, by Lemma 6.1, 8.64 and 8.62 the unique part in V κ of such a solution satisfies the inequality P V κ w κ 1 C H1 κ 2 F L 2, 8.65 for some constant C independent of κ. Comparing the powers of ε 2n, for n 1, yields A 1 w n+1 κ = A 0 w n κ The existence of a solution to 8.66 is guaranteed by requiring that P V κ w n κ A0 P V κ w κ n 0 Equation 8.67 implies ϕ+p V κ w κ n ϕ = A0 P V κ w κ n 0 w n κ satisfies the identity ϕ+p V κ w n κ ϕ ϕ H PV κ w κ n C PV H 1 κ w κ n H for some constant C. Therefore, by Lemma 6.1, 8.68 and 8.66 there exists a constant C > 0 independent of κ such that P V κ w κ n+1 H1 C κ 2 P V κ w κ n H1. In particular, by recalling 8.65 we find that w n κ H 1 C κ 2n F L Now constructing the function we have the following result. U N ε,κ = N n=0 ε 2n w n κ H 1 κ, 13

14 Theorem 8.1. Let w ε,κ be the solution to Then for any positive integer N there exists a constant C N > 0 independent of κ and ε such that 2N+1 wε,κ U ε,κ N ε H1 C N F L κ In particular, Proof. Substituting U N+1 ε,κ ε 2 A 1 + A 0 wε,κ U N+1 ε,κ w ε.κ w κ 0 H1 C 1 2 ε F L2. κ in to 8.60 and equating powers of ε yields + wε,κ U N+1 ε,κ = ε 2N+1 A 0 w N+1 κ w κ N+1. Using 8.69 and the standard ellipticity estimates results in an estimate similar to 8.70 for the difference w ε,κ U ε,κ N+1. Finally, noticing that U ε,κ N = U ε,κ N+1 ε 2N+1 w κ N+1 and once again employing 8.69 yields the required estimate Denote by [g] the multiplication operator for a given function g, and denote by B 0 the operator associated with the problem 8.63 such that w κ 0 = B 0 + I P 0 F, where P 0 is the orthogonal projection of L 2 onto H Theorem 8.1 implies that that B 0 is ε-close to Bε, hom in the region ε /2, in the following sense. Corollary 8.1. There is a constant C > 0 such that the estimate [e iε ] B 0 + I P 0 [e iε ] IBε, hom + I I P L f 2 Cε, holds for all ε such that ε /2. Corollary 8.1 and Theorem 7.1 imply that in the region κ ε 1/2 the term w κ 0 is the principle term in the approximation to w ε,κ y = expiκ yu ε ε κ y, y, in the slow variable κ. Further, Lemma 5.1 states that in the region 1 the function u 0 = Ic 0, v0 is the principle term in the approximation to u ε in the fast variable = κ/ε. This leads to the presence of a boundary layer in the Bloch space in the region 1 ε /2, where neither the outer operator B 0 nor the inner operator B0, hom are suitable for order Oε estimates. This leads to an interpretation of Bε, hom as being the non-trivial matching of B 0 and B0, hom in the boundary layer necessary to achieve order Oε estimates. This interpretation is further supported by the following result, which states that by extending B 0 and B0, hom into the boundary layer one can only achieve Oε 2/3 estimates. Corollary 8.2. For all ε > 0, α 0, 1, denote by B ε α0 the set { : < ε α } and consider the operators S ε,α := Uε I B0, hom + I I P f d + [e iε ] B 0 + I P 0 [e iε ] d U ε B ε α 0 ε \B ε α 0 There exists a constant C = Cα independent of ε such that A ε + I S ε,α L2 R d L 2 R d C ε α + ε 21 α. In particular, minimising the estimates over α, the operators S ε,α are shown to be ε 2/3 -close in the operator norm to the resolvents A ε + I. The above result can be extended to get error estimates in terms of higher powers of ε, by including further correcting terms from the asymptotic expansions for the functions u ε Section 5 in the new inner region ε β and for the functions w ε,κ the present section above in the new outer region κ ε 1 γ equivalently, ε γ for β, γ > 0 such that β + γ 1. A direct construction of this kind yields operatornorm estimates for A ε + I of order O ε βn+1 + ε 2γN+1, which is optimised to O ε 2N+1/3 by setting β = 2/3, γ = 1/3. Further, using the fact that the two expansions coincide in the overlapping region ε γ ε β, when β + γ < 1, one can then construct a -uniform asymptotic expansion, whose N-th truncation i.e. replacing with N in 5.8 and 8.61 yields an error of order O ε σn+1 with 0 < σ 1. This asymptotic procedure is well known in the analysis of parameter-dependent functions of the spatial variables x, y, see e.g. [8], especially in the context spatial representation of solutions to PDE. Our approach exploits similar ideas in the dual formulation, with respect to the quasimomenta, κ, which in our view has a potential to yield powerful results for operator-norm resolvent estimates for a general class of parameter-dependent families of operators. In particular, the proof of our main result Theorem 4.1 can be viewed as a matching procedure that achieves σ = 1 for the case N = 0. 14

15 9 Spectra of the operators B hom ε, Using the definition of the form b hom ε,, see Section 4, we infer that a pair c, u H 0 is an eigenvector of the operator Bε, hom corresponding to an eigenvalue λ if and only if A hom cd + A 0 + iεu + iεv = λ c + ud + v d, v H Setting v = 0 in 9.71 with an arbitrary d C yields A hom c = λ c + Further, setting d = 0 in 9.71 with an arbitrary v H0 1 0 yields A 0 + iεu + iεv = λ c + uv, u from which we deduce that either λ S 0 := {λ j } j=0, the set of eigenvalues of the operator A 0 = A 0 in L 2, defined by the sesquilinear form a 0 u, v := A 0 u v, u, v H0 1 0, on the maximal possible domain DA 0, or λ / S 0 and u = λc j=0 λ j λ ϕ j ϕ j, where ϕ j y := ϕ jy expiε y, y, and ϕ j is the eigenfunction of A 0 corresponding to the eigenvalue λ j, j = 0, 1,... We assume that the eigenvalues are ordered in the order of magnitude λ 0 < λ 1 λ 2..., where multiple eigenvalues are appear the number of times equal to their multiplicity and that ϕ j, j = 0, 1, 2,..., are real-valued and linearly independent. In the former case one has c = 0 and 9.72 implies u = 0, while in the latter case c C is arbitrary and by substituting 9.73 into 9.72 one gets The expression A hom = λ 1 + λ βλ := λ 1 + λ λ j λ ϕ j j=0 λ j λ 2 ϕ j, 0 j=0 obtained by setting ε = 0 in the right-hand side of 9.74, appeared in the work [15], where the behaviour of the spectra of the operators A ε was analysed. In particular, our main theorem above Theorem 4.1 implies the result of [16] on convergence of the spectra of A ε, as follows. Theorem 9.1. The spectra of the operators A ε converge in the Hausdorff sense to the union of the set S 0 and the set { lim λ : βλ = A hom } = { λ : βλ 0 }. ε 0 ε 10 Two particular examples of the family A ε Here we discuss two model cases included in our analysis that have emerged in the literature Classical homogenisation: 0 = This is the case when V consists of constant functions on. The inequality 6.24 trivially holds for κ 0 and for κ = 0 takes the form of the usual Poincaré inequality for functions with zero mean over. Clearly, the space H 0 is isometric to C and the operator family R ε consists of just one element, the resolvent of the usual homogenised operator A hom v := A hom, 15

16 where the matrix A hom is given by 4.6. Indeed, in this example the operator family Bε, hom does not depend on ε and for each specific value of ε represents -components of the direct fibre decomposition of the operator A hom treated as an operator with ε-periodic coefficients, i.e. Uε A hom U ε = A hom d = ε IBε, hom ε I d. Hence in this case Theorem 4.1 recovers the result of Birman and Suslina [3] regarding the resolvent convergence estimates for classical homogenisation in R d The double porosity problem: 0, A 0 1 = 0 This was considered in the work by Zhikov [16], where the spectrum of double-porosity problems in R d was analysed, following an earlier work [15] concerning double-porosity models in bounded domains. The paper [16] contains a proof of the strong two-scale convergence of the sequence of solutions u = u ε to the problems 2.3 to the solution v 1, v 0 H dp := H 1 R d L 2 R d, H0 1 0, v 0 = v 0 x, y, of the problem a dp v 1, v 0, ϕ 1, ϕ 0 + v 1 + v 0 ϕ 1 + ϕ 0 = fϕ 1 + ϕ 0, R d R d where the form a dp, with Da dp = H dp, is given by a dp v 1, v 0, ϕ 1, ϕ 0 := A hom v 1 ϕ 1 + R d R d A 0 y v 0 y ϕ 0. The author of [16] refers to the operator A dp generated by a db as the homogenised operator for the family A ε and proves that the spectra of A ε converge to the spectrum of A dp as ε 0. For continuous right-hand sides f the strong two-scale convergence result of [15] implies that u ε v 1 x ṽ 0 x, x L2Rd < Cε, ε where ṽ 0 is the -periodic extension of the function v 0 = v 0 x, y after setting it to zero for y 1. In the estimate the constant C = Cf > 0 is independent of ε, but it can not be replaced by C f L2 R d with a constant C that is independent of both ε and f. In other words, there are sequences f ε that are bounded in L 2 R d and are such that Cf ε as ε 0. The estimate can also be written in the form A ε + I f S ε A dp + I f < Cfε, L 2 R d where in the expression A dp +I f the function f is treated as an element of L 2 R d, and the operator S ε : L 2 R d L 2 R d is defined by S ε ux = ux, x/ε, x R d. The inequality 10.76, however, can not be upgraded to an operator-norm resolvent type statement, in view of the fact that the difference of the corresponding spectral projections on a neighbourhood of any point of the form λ + I, where λ is such that βλ as λ λ, does not go to zero in the operator norm as ε 0. Such points λ are the eigenvalues of the operator A 0 that have at least one eigenfunction with non-zero integral over. Our estimate 4.7 therefore rectifies this drawback and captures the operator-norm resolvent asymptotic behaviour of the sequence A ε. Acknowledgements This work was carried out under the financial support of the Leverhulme Trust Grant RPG 167 Dissipative and non-self-adjoint problems and the Engineering and Physical Sciences Research Council Grant EP/L018802/1 Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory. We are also grateful to the anonymous reviewer for their insightful comments. References [1] Allaire, G Homogenization and two-scale convergence, SIAM J. Math. Anal. 23, [2] Bensoussan, A., Lions, J.-L., and Papanicolaou, G. C., Asymptotic Analysis for Periodic Structures, North-Holland 16

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