Resolvent estimates for high-contrast elliptic problems with periodic coefficients

We study the asymptotic behaviour of the resolvents $({\mathcal A}^\varepsilon+I)^{-1}$ of elliptic second-order differential operators ${\mathcal A}^\varepsilon$ in ${\mathbb R}^d$ with periodic rapidly oscillating coefficients, as the period $\varepsilon$ 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 $\varepsilon$) and the"double-porosity"case of coefficients that take contrasting values of order one and of order $\varepsilon^2$ in different parts of the period cell. We provide a construction for the leading order term of the"operator asymptotics"of $({\mathcal A}^\varepsilon+I)^{-1}$ in the sense of operator-norm convergence and prove order $O(\varepsilon)$ remainder estimates.


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 [10], 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], [7].
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 − div A x ε ∇u , u ∈ D ε ⊂ L 2 (Ω), ε > 0, (1.1) 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. [7], one should in principle expect the better rate of convergence of order O(ε) 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 [12], [4] used similar ideas to prove resolvent convergence, but they did not go as far as getting the order O(ε) operator norm estimates. The more recent papers [15], [9] use different techniques to show an improved rate of convergence of order O(ε| log ε| σ ), σ > 0, for problems in bounded domains. Finally, the paper [11] combines the earlier results of [3] with some elements of the approach of [15], for proving the "expected" order O(ε) convergence for such problems. The focus of the present paper is on obtaining operator-norm resolvent-type estimates for a class of non-uniformly 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 [13].
The earlier results ( [3]) concerning resolvent estimates for (1.1)-(1.2) 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 leadingorder 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 [6], 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 θ = 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 θ ∈ ε −1 [0, 2π) d . In Section 6 we introduce spaces V (κ) ⊂ H 1 # (Q), κ ∈ [0, 2π) d , 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 # (Q), as well as several elliptic estimates that are uniform in θ. 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 |θ| ≤ 1 and in its complement |θ| ≥ 1. In Section 8 we discuss the "outer" region |θ| ≥ ε −1/2 and show that the inner and outer approximations jointly are only sufficient to obtain a norm-resolvent estimate of order O(ε 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 [13]. 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 [13] and the relation of its result to our convergence statement.

Problem setup
In what follows we study the problem In the above equation where A 0 , A 1 are Q-periodic symmetric (d × d)-matrix functions with entries in L ∞ (Q). We assume that A 0 ≥ νI, ν > 0 and that A 1 ≥ νI on an open set Q 1 ⊂ Q := [0, 1) d (the "stiff" component of the composite) with A 1 = 0 on the interior of Q \ Q 1 (the "soft" component), which we denote by Q 0 . We also assume that Q 0 ⊂ (0, 1) d , which implies, in particular, that the set ∪ n∈Z d Q 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 is symmetric and non-negative in L 2 (R d ), hence it generates a self-adjoint operator A ε whose domain D(A ε ) is dense in L 2 (R d ) and whose action is described by the identity 3) is understood as the result of applying the resolvent of A ε to f, i.e. u ε = (A ε + I) −1 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) , is positive. The same fact implies that the resolvent (A ε + I) −1 is a bounded operator.
Throughout the text we denote by H 1 # (Q) the space of Q-periodic functions that belong to H 1 loc (R d ). We use the letter C for any positive constant whose exact value may vary from line to line.

Bloch formulation and Gelfand transform
Using a procedure similar to the above definition of (A ε + I) −1 , for each θ ∈ ε −1 Q ′ , where In other words, for all θ ∈ ε −1 Q ′ one has u ε θ = (B ε,θ + I) −1 F, where the operators B ε,θ are generated by the closed sesquilinear forms Lemma 3.1. For each ε > 0 there exists a unitary map U ε : Proof. For a given ε > 0 set 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 −1 Q ′ × εQ), given by and the scaling transform T ε : of the inverse of G ε given by and the inverse of T ε given by 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 θ-representation and the main convergence result First, we introduce a θ-parametrised operator family that plays a central role in our analysis of the operators A ε as ε → 0. We denote H 0 := C × H 1 0 (Q 0 ), and for each ε > 0 and θ ∈ ε −1 Q ′ consider the sesquilinear form where A hom is the usual homogenised matrix Note that the matrix A hom is positive definite. Indeed, using the ellipticity assumption on A 1 one has, for ξ ∈ R d , where the function M (η) := min In what follows we also denote and use the invertible "identification" map I : C × L 2 (Q 0 ) → L that takes each pair (c, u) to the function c + u ∈ L with u = u on Q 0 and u = 0 on Q 1 . We next define operators B hom ε,θ in the Hilbert space . These operators are associated, for each value of θ ∈ ε −1 Q ′ , with the forms b hom ε,θ by means of the identity where the pairs (c, u) are taken from the maximal possible domain D B hom ε,θ , which can be shown to be dense in H 0 and hence in C × L 2 (Q 0 ).
The operators B hom 0,θ can be viewed, roughly speaking, as the θ-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 θ to the set ε −1 Q ′ . 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 εθ in (4.5) can not be set to zero in the region |θ| ≥ 1, hence the dependence of the operators B hom ε,θ on ε. We also denote by P the orthogonal projection of the Hilbert space and by P f its analogue on each "fibre", the orthogonal projection of L 2 (Q) 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 where the corresponding approximation error is of order O(ε). More precisely, there exists a constant C > 0, independent of ε, such that Note that the operator R ε can also be written as which follows from the definitions of the projection operators P and P f .

The inner expansion and principal term for B hom
ε,θ in the inner region |θ| ≤ 1.
In this section we provide an explicit representation for the behaviour in ε of the operators B hom ε,θ in the region |θ| ≤ 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 Substituting (5.8) into (3.4) and comparing the coefficients in front of ε −2 on both sides of the resulting equation we find or, equivalently, u a space that is naturally isometric to H 0 via the mapping I defined above: where, as before, v = v on Q 0 and v = 0 on Q 1 . This implies that u Further, comparing the coefficients in front of ε −1 and using (5.10) yields Introducing "unit-cell solutions" N k , k = 1, ..., d, that satisfy we note that, up to an arbitrary additive constant, one has 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) θ as follows where Solvability of (5.14) requires that F θ , v = 0 for all v ∈ V. The formula (5.13) and the solvability condition for (5.14) imply that u Following the method outlined in Section 4 for the construction of B hom ε,θ , we introduce the operator B hom 0,θ associated to the problem (5.16) such that (c The next result shows that B hom 0,θ is ε-close in norm to B hom ε,θ in the inner region of θ. Lemma 5.1. There exists C > 0 such that the estimate holds for all θ ∈ ε −1 Q ′ satisfying the inequality |θ| ≤ 1.
Proof. For each value θ as in the lemma consider the pairs ( and (5.18) By setting (d, ϕ) = (c ε , v ε ) in (5.18) and noting v ε ∈ H 1 0 (Q 0 ) we arrive at the a priori bound for some constant C.
To prove the result we show that for u ε := I(c ε , v ε ) and u := I(c, v) there exists a constant C > 0 independent of ε, θ such that (5.20) Taking into account (5.19) this implies (5.20), since 6 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": Here H 1 κ (Q), κ ∈ Q ′ , is the space of κ quasi-periodic functions belonging to H 1 (Q), i.e. u ∈ H 1 κ (Q) if, and only if, u(y) = exp(iκ · y)v(y), y ∈ Q, where 1 v ∈ H 1 # (Q). Note that (5.9), (5.11), (5.14) all have the form (6.22) for κ = 0 with G = 0, G = i∇ · A 1 θc For a given matrix function A 1 we consider the space Note that, for A 1 satisfying the assumptions prescribed in Section 2, we find A criterion for the existence of solutions to (6.22) is given below by a variant of the Lax-Milgram lemma.
Then, for all values of κ: (i) There exists a constant C > 0 independent of κ such that Proof. (i) The inequality (6.24) holds if there exists a constant C > 0 such that for all u ∈ We shall now verify this for two distinct cases. Case 1: κ = 0. For fixed u ∈ H 1 # (Q), denote u ∈ H 1 # (Q) to be an extension of u such that Notice that such an extension exists for connected Q 1 (cf. [7, Section 3.1]).
where the first inequality is a variant of the standard Poincaré inequality.
Denoting the map as above, we find u − u =: v ∈ H 1 0 (Q 0 ) = V (κ) and which proves the result. Here we have used the following Poincaré type inequality 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), which yields G, ϕ = 0 for all ϕ ∈ V (κ). Conversely, suppose that G, ϕ = 0 for all ϕ ∈ V (κ), and seek w ∈ H 1 κ (Q) that satisfies (6.22). By (6.25), the identity Q 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 (Q), the problem (  (iii) If w satisfies (6.22) and v ∈ V (κ) then A 1 ∇v = 0 and hence w + v also satisfies (6.22). 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 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 θ ∈ ε −1 Q ′ and k = 1, ..., d, there exists a unique solution N k ∈ V ⊥ to the unit-cell problem (5.12). In particular, for any value c (0) ∈ C, there exists a unique solution u (1) ∈ V ⊥ to the problem (5.11), for which the estimate holds.

Elliptic estimates
In our proof of Theorem 4.1 we use the following two statements.
is the solution to (5.16) with F ∈ L 2 (Q), and let u (1) θ ∈ H 1 # (Q) be the solution (5.13) to the unit-cell problem (5.11). Then the following estimates hold with some C > 0 : θ ) in (5.16), and dropping the scripts "(0)" and "θ" for convenience, yields and (6.29) follows by the Cauchy-Schwarz inequality. Setting Using the estimate along with the positivity of A hom and the bound (6.29), we infer (6.28). The estimate (6.30) is now a direct consequence of (6.28) and (6.27).
with F ∈ L 2 (Q). We denote by u ε,θ a solution to the unit-cell problem Then the following estimates hold with some C > 0 : Proof. Taking the unique solution w ε,θ ∈ V ⊥ to the problem we find by Corollary 6.1 that it is clear that (6.32) holds. By the properties of boundedness and ellipticity of A 1 we find that In particular, the estimate u Inequalities (6.33)-(6.36) are now shown by appropriately modifying the proof of Lemma 6.2.
Proof. The functions u (0) and u (1) are chosen so that F θ 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 θ to be the unique part in V ⊥ of any such solution, i.e. letting R θ ∈ V ⊥ be such that we find, by choosing ϕ = R θ in (6.38) and using the assumptions on A 1 , that is the square root matrix of A 1 . Due to Lemma 6.1(i), it remains to show that for some constant C. This can be seen by Lemma 6.2 and by noting, for θ ∈ ε −1 Q ′ , that for all ϕ ∈ H 1 # (Q).

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 θ and ε such that This fact is a consequence of the following theorem.
θ and R θ are given by (5.16), (5.11) and Lemma 6.4, respectively. Due to the fact that the functions u ε,θ are ε-close in L 2 (Q) uniformly in θ for |θ| ≤ 1 (cf. Lemma 5.1), it is sufficient to prove that By direct calculation we find that the difference z where the coefficients for the non-positive powers of ε have cancelled due to the construction of U ε,θ . The right-hand side F ε,θ ∈ H −1 # (Q) of (7.48) takes the form are elements of H −1 # (Q). A straightforward calculation shows that equations (7.49)-(7.52) with the inequalities (6.29), (6.30) and (6.37) imply the bound Hence, the required inequality ε,θ is defined in Lemma 6.3 and R ε,θ is given by Lemma 6.5 for the right-hand side Notice that the following inequalities hold for some constant C > 0 independent of ε and θ. These follow from Lemma 6.3 and the estimates and H ε,θ , 1 = The assumptions of Lemma 6.5 hold for H ε,θ which implies, along with the above inequalities, that the existence of a function R ε,θ ∈ H 1 # (Q) is guaranteed such that By direct calculation we find that the "error" z where the coefficients for the non-positive powers of ε have cancelled due to the construction of U (2) ε,θ . In the above equation the right-hand side F (2) ε,θ ∈ H −1 # (Q) is given by are elements of H −1 # (Q). Equations (7.54)-(7.57) together with inequalities (6.33), (6.36) and (7.53) imply that F . Therefore, the bound z (2) ε,θ H 1 (Q) ≤ Cε F L 2 (Q) holds, and the result follows. For fixed κ = 0 we shall study the asymptotics of the following problem: find w ε,κ ∈ H 1 Let us consider an asymptotic expansion for the solution to the above problem of the form The existence of a solution to (8.60) is guaranteed by Lemma 6.1 if, and only if, w κ satisfies the identity Furthermore, by Lemma 6.1 and (8.60) the unique part of such a solution satisfies the following inequality for some constant C independent of κ. Existence and uniqueness of w (0) κ is implied by the ellipticity of A 0 in Q 0 and standard ellipticity estimates give the following inequality Comparing the powers of ε 2n , for n ≥ 1, yields The existence of a solution to (8.63) is guaranteed by requiring that P V (κ) w (n) κ satisfies the identity for some constant C. Therefore, by Lemma 6.1 there exists a constant C > 0 independent of κ such that In particular, by recalling (8.62) we find that Now constructing the function we have the following result.
Theorem 8.1. Let w ε,κ be the solution to (8.58). Then for any positive integer N there exists a constant C N > 0 independent of κ and ε such that In particular, Proof. Substituting U (N ) ε,κ in to (8.58) and equating powers of ε yields The results follow by employing (8.65) and the standard ellipticity estimates.
Denote by [g] the multiplication operator for a given function g and denote by B 0 to be the operator associated with the problem (8.61) such that w ε,θ in the region |θ| ≥ ε −1/2 , in the following sense.
for all θ ∈ ε −1 Q ′ such that |θ| ≥ ε −1/2 . Corollary 8.1 and Theorem 7.1 imply that in the region |κ| ≥ ε 1/2 the term w (0) κ is the principle term in the approximation to w ε,κ (y) = exp(iκ · y)u ε ε −1 κ (y), y ∈ Q, in the "slow" variable κ. Further, Lemma 5.1 states that in the region |θ| ≤ 1 the function u is the principle term in the approximation to u ε θ in the "fast" variable θ = κ/ε. This leads to the presence of a boundary layer in the Bloch space in the region 1 ≤ |θ| ≤ ε −1/2 , where neither the "outer" operator B 0 nor the "inner" operator B hom 0,θ are suitable for order O(ε) estimates. This leads to an interpretation of B hom ε,θ as being the non-trivial matching of B 0 and B hom 0,θ in the boundary layer necessary to achieve order O(ε) estimates. This interpretation is further supported by the following result, which states that by extending B 0 and B hom 0,θ in to the boundary layer one can only achieve O(ε α ) estimates for any α ∈ (0, 1). independent of ε such that 9 Spectra of the operators B hom ε,θ Using the definition of the form b hom ε,θ , see Section 4, we infer that a pair (c, u) ∈ H 0 is an eigenvector of the operator B hom ε,θ corresponding to an eigenvalue λ if and only if where ϕ * j (y) := ϕ j (y) exp(iεθ · y), y ∈ Q, 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 where the form a dp , with D(a dp ) = H dp , is given by a dp (v 1 , v 0 ), (ϕ 1 , ϕ 0 ) := The author of [14] 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 [13] implies that where v 0 is the Q-periodic extension of the function v 0 = v 0 (x, y) after setting it to zero for y ∈ Q 1 . In the estimate (10.70) the constant C = C(f ) > 0 is independent of ε, but it can not be replaced by C f L 2 (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 C(f ε ) → ∞ as ε → 0.) The estimate (10.70) can also be written in the form < C(f )ε, (10.71) where in the expression (A dp + I) −1 f the function f is treated as an element of L 2 (R d × Q), and the operator S ε : The inequality (10.71), 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) −1 , where λ ∞ is such that β(λ) → ∞ 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 Q.) Our estimate (4.7) therefore rectifies this drawback and captures the operator-norm resolvent asymptotic behaviour of the sequence A ε .