Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients

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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 for example §9 of the monograph [11], 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 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 [2,8].
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 ( ), ε > 0, (1.1) and the resolvent of the operator representing the "homogenisation limit" −div A hom ∇u , u ∈ D hom ⊂ L 2 ( ). (1.2) Here is an open connected subset of R d , the matrix function A is [0, 1) dperiodic, 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 for example [8], one should in principle expect the better rate of convergence of order O(ε) suggested by the formal asymptotic analysis (assuming that the domain is sufficiently regular). The work [3] contains the related result for problems in the whole space ( = R d ), via a combination of spectral theoretic machinery based on the Bloch fibre decomposition of periodic PDE and asymptotic analysis. Earlier works [4,13] 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 [10,16] use different techniques to show an improved rate of convergence of order O(ε| log ε| σ ), σ > 0, for problems in bounded domains: [16] for scalar equations and [10] for systems. Finally, the papers [5,12] prove the "expected" order O(ε) convergence for such problems: [5] by using the method of periodic unfolding (and only for scalar problems), and [12] by combining the earlier results of [3] with some elements of the approach of [16] (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 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 [14].
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 leading-order contribution to the behaviour of the resolvent as ε → 0. Bearing this in mind, we analyse the asymptotic behaviour of the operator fibres 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 quasi- 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 [14]. 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 [14] and the relation of its result to our convergence statement.

Problem Setup
In what follows we study the problem In the above equation We assume that A 0 ≥ ν I , ν > 0 and that A 1 ≥ ν I on a Lipschitz 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 (2.1). 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 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) 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 H 1 loc (R d ) functions that are Q-periodic and by H −1 # (Q) its dual. For a normed space X and its dual X * , we write f, v for the action of f ∈ X * on v ∈ X . 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 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 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 : and the scaling transform T ε : of the inverse of T ε given by and the inverse of G 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 for example [2].

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 b hom ε,θ (c, u), (d, v) := A hom θ · θ cd 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 , 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 C × L 2 (Q 0 ) equipped with the inner product (c, u), (d, v) 0 = I(c, u), I(d, v) L 2 (Q) . 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 Sections 1 and 8, the expression εθ in (4.1) 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 (3.1) of the form Substituting (5.1) into (3.1) and comparing the coefficients in front of ε −2 on both sides of the resulting equation we find or, equivalently, which is a space naturally isometric to H 0 via the mapping I defined above: Further, comparing the coefficients in front of ε −1 and using (5.3) yields we note that, up to an arbitrary additive constant, one has The concrete choice of the constant added to the right-hand side of (5.6) 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 The formula (5.6) and the solvability condition for (5.7) 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 with the problem (5.9) such that (c in the inner region of θ.
Proof. For each θ as in the lemma, consider the pairs and By setting (d, ϕ) = (c ε , v ε ) in (5.11) and noting that v ε ∈ H 1 0 (Q 0 ) we arrive at the a priori bound for some constant C.

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": 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 (6.1) is given below by a variant of the Lax-Milgram lemma. The related ideas are inspired by the work [10], where Poncaré-type inequalities similar to (6.2), for the case κ = 0, were shown to be sufficient for homogenisation (in the strong two-scale resolvent sense) of "partially degenerate problems" with periodic rapidly oscillating coefficients. For the normresolvent asymptotics, however, a κdependent version of the inequality is required, which we prove by following the line of the argument of [10].
Then, for all values of κ: In particular, if w is a solution to (6.1) then P V (κ) ⊥ w is the unique part in V (κ) ⊥ of any solution to (6.1).
Proof. (i) The inequality (6.3) holds if there exists a constant C > 0 such that for We shall now verify this for two distinct cases.
Notice that such an extension exists for connected Q 1 (cf. [8, Section 3.1]). Defining where the first inequality is a variant of the standard Poincaré inequality.
Denoting the map as above, we find that which proves the result. Here we have used the 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 (6.1) and let ϕ ∈ V (κ). Then, using the symmetry of A 1 and (6.2), which yields G, ϕ = 0 for all ϕ ∈ V (κ). Conversely, suppose G, ϕ = 0 for all ϕ ∈ V (κ), and seek w ∈ H 1 κ (Q) that satisfies (6.1). By (6.4), the identity Now by the Lax-Milgram lemma, there exists a unique solution w ∈ V (κ) ⊥ to the problem and hence to (6.1). (iii) If w satisfies (6.1) and v ∈ V (κ), then A 1 ∇v = 0 and hence w + v also satisfies (6.1). Assuming further that w 1 and w 2 both satisfy (6.1), notice that v = w 1 − w 2 is a solution of (6.1) with G = 0. Finally, setting ϕ = v in (6.5) yields implying that (A 1 ) 1/2 ∇v = 0 and hence A 1 ∇v = 0, that is 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. holds.

Elliptic Estimates
In our proof of Theorem 4.1 we use the following two statements.
is the solution to (5.9) with F ∈ L 2 (Q), and let u (1) θ ∈ H 1 # (Q) be the solution (5.6) to the unit-cell problem (5.4). Then the following estimates hold with some C > 0 : θ ) in (5.9), and dropping the scripts "(0)" and "θ " for convenience, yields and (6.8) follows by the Cauchy-Schwarz inequality. Setting Using the estimate along with the positivity of A hom and the bound (6.8), we infer (6.7). The estimate (6.9) is now a direct consequence of (6.7) and (6.6).

Lemma 6.3. For each
with F ∈ L 2 (Q). We denote by u (1) ε,θ a solution to the unit-cell problem ε,θ = 0. (6.11) Then the following estimates hold with some C > 0: 14) Proof. Taking the unique solution w ε,θ ∈ V ⊥ to the problem we find by Corollary 6.1 that it is clear that (6.11) holds. By the properties of boundedness and ellipticity of A 1 we find that In particular, the estimate holds.
Proof. The functions u (0) and u (1) are chosen so that F θ satisfies the solvability condition for the equation (5.7), thus the existence of a solution u (2) is guaranteed by Lemma 6.1. Denoting by R θ the unique part in V ⊥ of any such solution, that is letting R θ ∈ V ⊥ be such that we find, by choosing ϕ = R θ in (6.17) and using the assumptions on A 1 , that where A 1/2 1 is the square root of the matrix 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 some constant C > 0 independent of ε, θ.

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.

The Outer Expansion and Principal Term for B hom
ε,θ in the Outer Region |θ | ≥ ε −1/2 For fixed κ = 0 we shall study the asymptotics of the following problem: find Let us consider an asymptotic expansion for the solution to the above problem of the form Substituting (8.2) in (8.1) and comparing the coefficients in front of ε −2 on both sides of the resulting equation yields κ ∈ H 1 0 (Q 0 ), see (6.2). Further, comparing the coefficients in front of ε 0 yields The existence of a solution to (8.
for some constant C independent of κ. Comparing the terms with ε 2n , for n ≥ 1, yields The existence of a solution to (8.7) is guaranteed by requiring that P V (κ) w (n) κ satisfies the identity for some constant C. Therefore, by Lemma 6.1, (8.9) and (8.7), there exists a constant C > 0 independent of κ such that In particular, by recalling (8.6) we find that Now constructing the function we have the following result.
Theorem 8.1. Let w ε,κ be the solution to (8.1). Then for any positive integer N there exists a constant C N > 0 independent of κ and ε such that In particular, in to (8.1) and equating powers of ε yields Using (8.10) and the standard ellipticity estimates results in an estimate similar to (8.11) for the difference w ε,κ −U κ and once again employing (8.10) yields the required estimate (8.11).
Denote by [g] the multiplication operator for a given function g, and denote by B 0 the operator associated with the problem (8.4) such that w (0) where P 0 is the orthogonal projection of L 2 (Q) onto H 1 0 (Q 0 ). Theorem 8.1 implies that that B 0 is ε-close to B hom ε,θ in the region |θ | ≥ ε −1/2 , in the following sense.
holds 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 principal 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. Furthermore, Lemma 5.1 states that in the region |θ | ≤ 1 the function u (0) is the principal 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 is suitable for order O(ε) estimates. This leads to an interpretation of B hom ε,θ as a 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 into the boundary layer one can only achieve O(ε 2/3 )-estimates. I −1 P f dθ There exists a constant C = C(α) independent of ε such that 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 −1 .
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 θ (Section 5) in the new "inner" region |θ | ≤ ε β−1 and for the functions w ε,κ (the present section above) in the new "outer" region |κ| ≥ ε 1−γ (equivalently, |θ | ≥ ε −γ ) for β, γ > 0 such that β + γ ≤ 1. A direct construction of this kind yields operator-norm estimates for (A ε + I ) −1 of order O ε β(N +1) + ε 2γ (N +1) , which is optimised to O ε 2(N +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 ≤ |θ | ≤ ε β−1 , when β + γ < 1, one can then construct a θ -uniform asymptotic expansion, whose N -th truncation [that is replacing "∞" with "N " in (5.1) and (8.2)] yields an error of order O ε σ (N +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 for example [7], especially in the context spatial representation of solutions to PDE. Our approach exploits similar ideas in the "dual" formulation, with respect to the quasimomenta (θ, 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.

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 Setting v = 0 in (9.1) with an arbitrary d ∈ C yields Further, setting d = 0 in (9.1) with an arbitrary v ∈ H 1 0 (Q 0 ) yields 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 (Q), defined by the sesquilinear form on the maximal possible domain D(A 0 ), or λ / ∈ S 0 and 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 by 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 (9.2) implies Q u = 0, while in the latter case c ∈ C is arbitrary and by substituting (9.3) into (9.2) one gets (9.4) The expression obtained by setting εθ = 0 in the right-hand side of (9.4), appeared in the work [14], 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 [15] 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 as ε → 0 to the union of the set S 0 and the set

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: Q 0 = ∅
This is the case when V consists of constant functions on Q. The inequality (6.3) trivially holds for κ = 0, and for κ = 0 it takes the form of the usual Poincaré inequality for functions with zero mean over Q. 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 where the matrix A hom is given by (4.2). Indeed, in this example the operator family B hom ε,θ does not depend on ε and for each specific value of ε represents θ -components of the direct fibre decomposition of the operator A hom treated as an operator with ε-periodic coefficients, that is 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
This was considered in the work by Zhikov [15], where the spectrum of doubleporosity problems in R d was analysed, following an earlier work [14] concerning double-porosity models in bounded domains.
The paper [15] contains a proof of the strong two-scale convergence of the sequence of solutions u = u ε to the problems (2.1) to the solution (v 1 , v 0 ) ∈ H dp : , y), of the problem where the form a dp , with D(a dp ) = H dp , is given by The author of [15] 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 [14] 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 (10.1) 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 (10.1) can also be written in the form < C( f )ε, (10.2) 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 (10.2), 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 (4.3) therefore rectifies this drawback and captures the operator-norm resolvent asymptotic behaviour of the sequence A ε .
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