Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. II

We construct an order-sharp theory for a double-porosity model in the full linear elasticity setup. Crucially, we uncover time and frequency dispersive properties of highly oscillatory elastic composites.


Introduction
"Quantitative" asymptotic methods in the analysis of parameter-dependent families of PDEs (in particular, multiscale asymptotic analysis), see e.g. [41,3,21,43,23], have emerged recently as a natural improvement of the classical semi-formal ad-hoc asymptotic approach. Their key feature is pursuit of an estimate on the difference between the "exact" (usually inaccessible) solution to the wave propagation problem and its asymptotic approximation in terms of some non-dimensional parameters, e.g. the ratio ε between the wavelength and the period of material oscillations. This idea has brought forth the physically relevant possibility to account for degenerate problems (such as the "flipped double-porosity" setup [10]), which are both technically more challenging and generally not amenable to the said classical approach. On the physical side of the issue, the principal application of the related analysis is to the development of a rigorous understanding of metamaterials [7]. It can be argued, see e.g. the discussion in [12], that generically metamaterial behaviour is due to the "corrector" terms in the relevant asymptotic expansion [10]. Indeed, if one assumed that the family admits a "limiting" operator in a strong enough topology, then the latter must inherit the defining properties of the family, which would not permit the non-trivial effects expected of a metamaterial. This calls for a new outlook on the asymptotic analysis of operator families that would lead to quantitatively tight asymptotic expansions capturing fully those features of the solution that are of value for specific physical problems at hand.
In connection with this goal, operator theory has emerged as a source of powerful tools, due to its general functional-analytic framework for bounding the norm of the solution in terms of the norm of the problem data, in other words estimating the resolvent. Indeed, a typical consequence of the related bound is the guarantee that the resolvent of the operator associated with the approximation is close, in a quantitatively controlled fashion, to the resolvent of the original, "inaccessible", operator. The availability of these operator-theoretic tools is complemented by the interest in using resonant elements within composite media for controlling wave motion and by recent advances in the mathematical analysis of boundary-value problems for differential equations. Indeed, for problems involving natural length-scale ratios such as the parameter ε mentioned above -another example being heterogeneous "thin" structures (rods, plates, shells, and their combinations), where a similar non-dimensional parameter may represent the ratio of the structure thickness to the typical length of variation of material properties [15,6,16] -the relevant geometric information can be translated into a parametrised family of "transmission" or "boundary value" problems for partial differential equations (PDE), for which an analysis of the "asymptotically leading" part of the solution behaviour can be carried out. For linear setups, the analytical access to this leading-order term is afforded by a spectral decomposition, either for the problem itself or for its appropriate reformulation. In particular, in [9,10,11] we used the link (known in operator theory as the "Krein formula") between the resolvent and the Dirichlet-to-Neumann (DtN) operator, or the "M-matrix", to obtain sharp operatornorm convergence estimates in the "high-contrast" setup for a scalar elliptic PDE describing a composite containing "resonating" components, which admit rapidly oscillating waves -the related setup has often been referred to in the literature as "double-porosity model" [2,1,42]. An "asymptotic diagonalisation" of the DtN map allowed us to quantify the associated resonant effect and formulate an approximating model of a "non-classical" type. Prior to the work [8], all techniques for the analysis of resonant media relied on weaker notions of convergence (e.g. the so-called two-scale convergence [1]), which do not allow one to achieve the same degree of control of the solution, i.e. an order-explicit estimate of its L 2 norm by the similar norm of the problem data. The paper [8] introduced the idea of developing power-series asymptotic approximations (in terms of powers of ε) that are uniform with respect to the wave-length 2π{χ in a spatial integral representation of L 2 functions (such as the Bloch-Floquet-Gelfand representation, for which χ is often referred to as "quasimomentum"). These were then used to obtain operator-norm estimates for scalar problems with high-contrast, thereby providing a quantitative version of the result of [42]. The same idea of using power-series expansion for the decomposition by wave-length was applied in [15,16] in the context of simultaneous homogenisation and dimension reduction for elastic plates and rods, where power series in terms of the quasimomentum χ were constructed first and then converted in the order-sharp estimates in terms of the original parameter ε.
In the present paper we obtain order-sharp norm-resolvent estimates for systems of PDEs with critical contrast, by adopting expansions with respect to the quasimomentum χ cast in the operator-theoretic framework of [10]. In order to reach a key milestone on the way to the asymptotic analysis of resonant media, namely the so-called Krein formula for the resolvent of the operator in question, we adopt the approach of [34] via what we call "Ryzhov triple", which provides ingredients for defining the "M-operator", i.e. the Dirichlet-to-Neumann map on the interface between the resonant components and the "host" medium, as a function of the spectral parameter. As the vector setup can not be addressed as a simple generalisation of the scalar case, we develop a new method for constructing the Ryzhov triple, based on the existing theory of boundary-value problems for elliptic PDE. In effect, this sheds a new light on the Ryzhov triple and the corresponding Krein formula as natural tools for quantitative studies of boundary-value problems, Notably, the utility of these tools extends well beyond our analysis of critical contrast as inspired by the homogenisation perspective of [1] and [42]. In particular, the new framework allows one to re-formulate a number of existing problems in material characterisation and wave propagation in a way that will provide a general methodology for asymptotic approximations with order-sharp error control, for example the problems of scattering of waves on inhomogeneous obstacle, see [12] and references therein. The second key aspect of the present work is a demonstration that the mentioned Krein formula is amenable to asymptotic analysis with respect to the quasimomentum χ yielding uniform order-sharp error estimates. Overall, the two mentioned analytic features constitute the present, second part of our work on quantitative descriptions of time dispersion in media with resonant inclusions. As a by-product of our analysis, we recover the homogenization result on perforated domain, see Remark 7.6. For an additional comparison of the present work with [10], see Remark 6.12. Although here we analyse the elasticity system, it can be seen that the same analysis can be applied to any elliptic system of the form [3]. It is also worth mentioning that the techniques used here can tackle the situation of stiff inclusion in the soft matrix, as was done in [10] in the scalar setting.
Next we outline the structure of the paper. In Section 2, we describe the setup and state the main results (see Theorem 2.2 and Theorem 2.3). We also show how to decompose the operator using Gelfand transform and reformulate the resolvent problem as a family of transmission problems. In Section 3 we introduce the abstract framework of boundary triples and we show how to put a transmission problem into this framework.
In Section 4 we construct the asymptotics that we use for the proof of the main results. Section 5 is devoted to the proof of Theorem 2.2, Section 6 is devoted to the proof of Theorem 2.3 (a) and Section 7 is devoted to the proof of Theorem 2.3 (b) as well as additional result on dispersion relation. Finally, in the Appendix we prove some auxiliary claims needed in the paper.

Notation
In this section we introduce the notation which we will use throughout the paper.
For a vector a P R k , we denote by a j , j " 1, . . . , k its components. Similarly, the entries of a matrix A P R kˆk , are referred to as A i j , i, j " 1, . . . , k, and sym A denotes the symmetric part of a matrix A. The vectors of the standard orthonormal basis in R k are denoted by e i , i " 1, . . . , k. Furthermore, for a, b P R k , we denote by a b b P R kˆk , the matrix with entries a i b j , and set a d b :" sympa b bq. The inner product of matrices A, B is denoted by A : B :" TrpA J Bq, and R kˆk skew (C kˆk skew ) stands for the space of antisymmetric matrices with real (complex) entries.
For an operator A (or a bilinear form a) the domain of A (respectively a) is denoted by DpAq (respectively Dpaq). We use the notation A for the closure of a closable operator A, and denote by σpAq the spectrum of an operator A. For normed vector spaces X, Y, we denote by BpX, Yq the space of bounded linear operators from X to Y. Furthermore, when indicating a function space X in the notation for a norm }¨} X , we omit the physical domain on which functions in X are defined whenever it is clear from the context. For example, we often write }¨} L 2 , }¨} H 1 instead of }¨} L 2 pΩ;R k q , }¨} H 1 pΩ;R k q , k P N.
For A, B Ă R k by distpA, Bq we denote the distance between the sets A and B. For f P L 1 pAq, f denotes f :" ş A f and ½ A denotes the characteristic function of A. Finally, δ i j denotes the Kronecker delta.

The operator of linear elasticity
Consider the "reference cell" Y :" r0, 1q 3 Ă R 3 , and let Y soft Ă Y be a connected open set with smooth boundary Γ (possessing, e.g., C 1,1 regularity) such that the closure of Y soft is a subset of the interior of Y and Y stiff " YzY soft . For a fixed period of material oscillations ε ą 0, we would like to study the behaviour of a composite elastic medium with components whose properties are in high contrast to one another. We refer to the component materials as "soft" and "stiff" accordingly. We this goal in mind, we view R 3 as being composed of two complementary subsets, the stiff part Ω ε stiff ("matrix") and the soft complement Ω ε soft ("inclusions"): Ω ε stiff :" R 3 zΩ ε soft , Ω ε soft :" We are interested in the approximation properties, when ε Ñ 0, of the operator family pA ε q εą0 , where, for every ε ą 0, the operators A ε are defined as the unique self-adjoint unbounded operators on L 2 pR 3 ; R 3 q corresponding to the differential expressions´div`A ε px{εq sym ∇˘on domain DpA ε q Ă H 1 pR 3 , R 3 q. These operators are defined by the bilinear forms a ε pu, vq :" A ε px{εq sym ∇u : sym ∇v, u, v P H 1 pR 3 ; R 3 q.

The tensors of material coefficients
A ε pyq " " A stiff pyq, y P Y stiff , ε 2 A soft pyq, y P Y soft are defined on the unit cell Y and extended to R 3 via periodicity. We make the following assumptions about the tensor-valued functions A stiffpsoftq .
We are interested in the approximation properties of the resolvent operator associated with A ε , for small ε ą 0. Namely, we would like to obtain asymptotics of the resolvents pA ε´z Iq´1 , as ε Ñ 0, in the L 2 Ñ L 2 operator norm. The spectral parameter z P C is assumed uniformly separated from the spectrum of the operator A ε : more precisely, we fix σ ą 0 and consider a compact set K σ Ă tz P C : distpz, Rq ě σu . We introduce the following space of functions supported only on the stiff (respectively, soft) component: Furthermore, we denote by P stiff ε the orthogonal projector from L 2 pR 3 ; R 3 q onto L stiff ε . In order to state the main theorem, we first define the following operators, which make up crucial components of the leadingorder term of the resolvent asymptotics.
Definition 2.1 (Macroscopic operator). Consider the tensor A macro P R 3ˆ3ˆ3ˆ3 defined by (2) We define the macroscopic operator (the operator of perforated domain) A macro as the self-adjoint unbounded operator on L 2 pR 3 ; R 3 q corresponding to the differential expressioń div pA macro sym ∇¨q , with domain DpA macro q Ă H 1 pR 3 , R 3 q, defined by the bilinear form a macro pu, vq :" A macro sym ∇u : sym ∇v, u, v P H 1 pR 3 ; R 3 q.
We will require the following lemma, the proof of which is standard.
Proof. The proof is based on a standard extension argument, see e.g. [6,Proposition 3.4].
In addition to the properties highlighted in Lemma 2.1, the leading-order term in the resolvent asymptotics retains information on the microstructure, via the spectrum of the "Bloch operator" A Bloch associated with the bilinear form a Bloch pu, vq :" as a nonnegative self-adjoint operator on L 2 pY soft ; R 3 q. Furthermore, we define the matrix-valued "Zhikov function" B by where pη k , ϕ k q, k P N are the eigenpairs of the associated Bloch operator A Bloch on L 2 pY soft ; R 3 q and z is an arbitrary complex number different from all η k , k P N (an increasing sequence converging to infinity). The above defined operators can be naturally extended to complex-valued functions. The appearance of the matrix-valued Zhikov function in the approximation of the operator in high-contrast is standard, even in qualitative analysis (see, e.g. [44]).

Main results
We will now state the main results of the paper which deal with the approximation of the resolvent of the operator. We will state ε 2 and ε approximation result. The benefits of ε approximation is that it provides simpler operator that, when restricted to the stiff component, is second order differential operator. On the other hand, the approximation with ε 2 precision provides the operator which is (even when restricted to the stiff component) pseudodifferential. The operators A app ε and A eff ε and the coresponding projections Θ app ε , Θ eff ε are defined in Section 5.3 and Section 6 respectively. The proof of Theorem 2.2 is also given in Section 5.3 and the proof of claim (a) in Theorem 2.3 is given in Section 6, while claim (b) is proved in Section 7.
where the constant C ą 0 depends only on σ and diampK σ q. The operator A app ε is a self-adjoint operator defined on the subspace Θ app ε L 2 pR 3 ; C 3 q in what follows, and Θ app ε is a projection.
where the constant C ą 0 depends only on σ and diampK σ q . The operator A eff ε is a self-adjoint operator defined (on the subspace Θ app ε L 2 pR 3 ; C 3 q) in the continuation of the text and Θ eff ε is a projection.
where A macro is a differential operator of linear elasticity with constant coefficients defined by (3).
For an additional comment about approximating operators see Remark 7.7 and Remark 6.13 below. It is well known that the spectrum of A ε has band-gap structure (see [42]). Theorem 2.2 and Theorem 2.3 enable us to estimate the gaps in the spectrum of A ε on any compact interval with the gaps in the spectrum of A app ε , i.e. A eff ε . Corollary 2.4. For every M ą 0, one has dist`σpA ε q X r´M, Ms, σpA app ε q X r´M, Ms˘ď CpM`1q 2 ε 2 , dist`σpA ε q X r´M, Ms, σpA eff ε q X r´M, Ms˘ď CpM`1q 2 ε.
where C ą 0 is a fixed constant independent of M Proof. The proof can be obtained by taking z " i in Theorem 2.2 and Theorem 2.3 (a). It is well known that for the self-adjoint bounded linear operators A and B on a Hilbert space X we have dist pσpAq, σpBqq ď }A´B} XÑX , see e.g. [22]. We have The claim follows from the fact that for arbitrary λ, µ P R we have: |λ´µ| " |λ´i||µ´i|ˇˇˇˇ1 λ´i´1 λ´iˇˇˇˇď p|λ|`1qp|µ|`1qˇˇˇˇ1 λ´i´1 λ´iˇˇˇˇ.
We also can extend the claim of Theorem 2.2 and 2.3 outside K σ , provided that we stay away from the spectrum in a bounded region. We can also aposteriori estimate the dependence of constant C on the distance of z from the spectrum of the operator A ε and the modulus of z.
Corollary 2.5. The claims of Theorem 2.2 can be extended to all z R σpA ε q Y σpA app ε q and the constant C " Cpzq depends on z in the following way where C is independent of z. The analogous claim is valid for Theorem 2.3 (a).
Proof. For the unbounded operators A and B on the Banach space X, the orthogonal projection P that commutes with the operator B and z 1 , z 2 R σpAq Y σpBq it is easy to check the identity Notice also that by the functional calculus for a self-adjoint A we have We take again z 1 " i in Theorem 5.10 and A " A ε and B " A app ε and P " Θ app . Notice that as a consequence of Theorem 2.2 we have pI´Θ app ε q pA ε´i Iq´1 L 2 pR 3 ;C 3 qÑL 2 pR 3 ;C 3 q ď Cε 2 .
Remark 2.6. By combining Corollary 2.4 and 2.5 one can make the constant C " Cpzq in Theorem 2.2 and 2.3 dependent only on |z| and distpz, σpA ε qq, i.e. only on |z| and distpz, σpA app{eff ε qq. By the analysis of the proof of Theorem 2.3 (b) presented in Section 7 the same can be concluded for Theorem 2.3 (b). Also, the fixed constant C ą 0 in Corollary 2.4 and 2.5 that corresponds to z " i in Theorem 2.2 and 2.3 can be easily estimated from the proofs of these theorems.2

.4 Gelfand transform and periodic decomposition
The purpose of this chapter is to decompose the original differential operator into a family of differential operators with compact resolvents that act on functions defined on the unit cell Y. This is carried out in a standard way by the Gelfand transform.
Denote by L 2 # pY; C 3 q and H 1 # pY; C 3 q the spaces of Y-periodic functions in L 2 pR 3 ; C 3 q, H 1 pR 3 ; C 3 q, respectively. The Gelfand transform G is defined on L 2 pR 3 ; C 3 q by the formula where Y 1 :" r´π, πq 3 . We denote with x the variable in R 3 , while we use the notation y for the variable in Y. The Gelfand transform is a unitary operator: in the sense that xu, vy L 2 pR 3 ;C 3 q " xGu, Gvy L 2 pY 1 ;L 2 # pY;C 3 qq for all u, v P L 2 pR 3 ; C 3 q. A function can be reconstructed from its Gelfand transform as follows: For a systematic overview of the properties of Gelfand transform, the reader is encouraged to consult [4].
In order to deal with the setting of highly oscillating material coefficients, we consider the following scaled version of Gelfand transform. For a fixed ε ą 0 and all u P L 2 pR 3 ; C 3 q, we set pG ε uqpy, χq :"´ε 2π¯3 The scaled Gelfand transform G ε transforms functions into Y-periodic functions, namely: pG ε uqpy`1, χq " pG ε uqpy, χq @y P Y, χ P Y 1 .
Note that the scaled Gelfand transform G ε is a composition of classical Gelfand transform G and the unitary scaling operator S ε : L 2 pR 3 ; C 3 q Ñ L 2 pR 3 ; C 3 q defined by S ε upxq :" ε 3{2 upεxq, u P L 2 pR 3 ; C 3 q.
It follows that G ε is also unitary, i.e.
xu, vy L 2 pR 3 ;C 3 q " xG ε u, G ε vy L 2 pY 1 ;L 2 The original function is recovered from its Gelfand transform by the formula Also, by noting that for the scaled Gelfand transform of a derivative one has where the for each χ P Y 1 the operator X χ acting on L 2 # pY; C 3 q is defined by One should note that C 1 |χ|||u|| L 2 pY;C 3 q ď ||X χ u|| L 2 pY;C 3ˆ3 q ď C 2 |χ|||u|| L 2 pY;C 3 q @u P L 2 # pY; C 3 q. With the Gelfand transformation at hand, one can show that For each χ P Y 1 and ε ą 0, we introduce the self-adjoint operator A χ,ε :" psym ∇`iX χ q˚A ε psym ∇`iX χ q : DpA χ,ε q Ă H 1 # pY; C 3 q Ñ L 2 pY; C 3 q associated with the form a χ,ε . Here we use the notation p¨q˚for the formal adjoint of the operator. Applying the scaled Gelfand transform to the resolvent yields which is an example of the classical von Neumann "fibre decomposition" formula. Due to the compactness of the embedding H 1 # pY; C 3 q ãÑ L 2 pY; C 3 q, the resolvents`ε´2A χ,ε´z I˘´1 are, in fact, compact. We interpret (9) as follows: by applying the Gelfand transform to the problem, we have decomposed the resolvent operator pA ε´z Iq´1 into a continuum family of resolvent operators`ε´2A χ,ε´z I˘´1 indexed by χ P Y 1 . In contrast to the original resolvent operator, this family consists of compact operators, which have discrete spectra.

The fibers of the resolvent problem
For each ε ą 0, the resolvent problem χ-fibre (χ P Y 1 ) of the operator A ε consists in finding, for a fixed z P ρpA χ,ε q and every f P L 2 # pY, C 3 q, the solution u P DpA χ,ε q to the equation`ε´2A χ,ε´z I˘u " f . This is equivalent to the following problem.
The subscripts "`" and "´" refer to the (spatial) limits of from inside Y stiff and Y soft , respectively, while n`, n´denote the outward pointing unit normals to Γ from Y stiff and Y soft , respectively.

Operator theoretic approach -Ryzhov triples
The purpose of this section is to introduce the abstract framework for the problem (10). In Section 3.1 we develop the abstract framework, while in Section 3.2 and Section 3.3 we show how transmission problem can be seen as part of this abstract framework, proving also the main properties of the operators appearing. First we prepare the necessities needed in order to work under the paradigm of Ryzhov triples, a suited version of boundary triples for the analysis of boundary value problems.

Abstract notion of Ryzhov triples
The concept of Ryzhov triple was introduced in [33,34]. The main results of this section are Theorem 3.8 which expresses the solution of abstract Robin boundary value problem and Theorem 3.9.
Definition 3.1. Let H be a separable Hilbert space and E an auxiliary Hilbert space. Let: • A 0 be a self-adjoint operator on H with 0 P ρpA 0 q, • Π : E Ñ H be a bounded operator such that DpA 0 q X RpΠq " t0u, kerpΠq " t0u.
• Λ be a self-adjoint operator on E with domain DpΛq Ă E.
We then refer to the triple pA 0 , Π, Λq as a Ryzhov triple on pH, Eq.
Our intention is to study boundary value problems, and we introduce the following objects.
Notice that, by definition, we have Λg " Γ 1 Πg @g P DpΛq, In the above definition, the operators Γ 0 and Γ 1 typically play the rôles of the trace of a function and of its co-normal derivative on the boundary, respectively. The following result is then well expected and can be found in [34].
Definition 3.3. Suppose z P ρpA 0 q. Define the operator S pzq mapping g P E to the solution u P DpAq " DpΓ 0 q of the spectral boundary value problem " Au " zu, The operator-valued function Mpzq defined on DpΛq by Mpzq :" Γ 1 S pzq, is called the Weyl M-function of the Ryzhov triple pA 0 , Π, Λq.
In [34] the following formulae for the operators S pzq and Mpzq were proven (z P ρpA 0 q): Note also that`I´z and therefore S pzq " Π`zpA 0´z Iq´1Π.
The next proposition collects some properties of the M-function.

Proposition 3.2. (Properties of the Weyl M-function)
• The following representation holds: • Mpzq is an analytic operator-valued function with values in the set of closed operators on E with (z-independent) domain DpΛq.
• For u P kerpA´zIq X DpA 0 q 9 ΠDpΛq ( , the following formula holds: Remark 3.3. In the case when the operators Γ 0 and Γ 1 the operators of the trace of a function and of the trace of its co-normal derivative, the formula (18) clearly reveals the M-function Mpzq to be the Dirichletto-Neumann map associated with the resolvent problem.R emark 3.4. Due to the fact that DpMpzqq " DpΛq independently on z P C, one can define the operators on DpΛq. Note that by (17) and the fact that Λ is self-adjoint, one has Mpzq˚" Mpzq, and therefore ℑMpzq " 1 2i`M pzq´Mpzq˚˘" ℑz pS pzqq˚S pzq, ℑMpzq " ℑpMpzq´Λq " ℑpMpzq´Mp0qq. (19) Remark 3.5. It is clear from (17) and (15) that This formula will prove to be one of the key elements in deriving the asymptotics, as ε Ñ 0, of the resolvents ε´2A χ,ε´z I˘´1 of the operators A χ,ε introduced in Section 2.4.9 For a given A 0 , we define A 00 to be the restriction of A 0 to the set DpA 00 q :" kerpΓ 0 q X kerpΓ 1 q. We refer to A 00 as the minimal operator.
Remark 3.6. It was shown in [34] that DpA 00 q does not actually depend on the choice of the operator Γ 1 (or Λ) but instead can be characterised as the subspace of DpA 0 q consisting of those elements u for which A 0 u is orthogonal to the range of Π.Ő ne can characterize all densely defined closed extensions of A 00 contained in A to be the operators A β 0 ,β 1 associated with the spectral boundary value problem Au´zu " f subject to an abstract Robin-type condition by varying over the choice of the operators β 0 , β 1 on E. The rigorous definition of the extension operators A β 0 ,β 1 is postponed to Theorem 3.9. Note that A 0 is then the self-adjoint extension of A 00 corresponding to the choice β 0 " I, β 1 " 0. However, in order to clarify the meaning of (21), it is necessary to make additional assumptions on the operators β 0 , β 1 , e.g., as follows.
Assumption 3.1. The operators β 0 , β 1 are linear operators in E such that β 0 is defined on the domain Dpβ 0 q Ą DpΛq and β 1 is defined everywhere on E and bounded. The operator β 0`β1 Λ, defined on DpΛq is closable in E.
Under the above assumption, the operator β 0`β1 Mpzq is closable, and we denote its closure by B . The condition (21) is shown to be well posed on a certain Hilbert space associated with the closure of β 0`β1 Λ in E.
Definition 3.4. Let H be a separable Hilbert space, E an auxiliary Hilbert space, and pA 0 , Π, Λq a Ryzhov triple on pH, Eq. Suppose also that β 0 , β 1 are linear operators on E satisfying Assumption 3.1. Consider the space equipped with the norm The following lemma is also proved in [34].
Lemma 3.7. The space pH β 0 ,β 1 , ¨ β 0 ,β 1 q is a Hilbert space. The operator β 0 Γ 0`β1 Γ 1 : We are now in a position to assign a meaning to the abstract spectral Robin-type boundary value problem and establish its well-posedness.
Theorem 3.8. Suppose that z P ρpA 0 q is such that the operator β 0`β1 Mpzq is boundedly invertible in E. Then, for given f P H, g P E, the unique solution u P H β 0 ,β 1 to the Robin boundary-value problem is provided by the formula By plugging g " 0, it is seen that (22) defines the resolvent of a closed, densely defined operator in H that extends the minimal operator A 00 . Theorem 3.9. Suppose that z P ρpA 0 q be such that the operator β 0`β1 Mpzq defined on DpBq is boundedly invertible in E. Then the operator R β 0 ,β 1 pzq defined by is the resolvent pA β 0 ,β 1´z Iq´1 of a closed densely defined operator A β 0 ,β 1 in H such that Remark 3.10. Notice that the pointwise nature of Theorem 3.8, with respect to the parameter z, yields the same formula (22) under an even more general assumption that the operator β 1 depends on z. This remark is crucial for the remainder of the paper.1

Operators associated with boundary value problems
In this section we introduce some operators required to reformulate the transmission boundary value problem (10) in the context of the abstract theory of Ryzhov triples. First we define the spaces H :" L 2 # pY; C 3 q, E :" L 2 pΓ; C 3 q, as well as the associated orthogonal projections P soft : H Þ Ñ H soft , P stiff : H Þ Ñ H stiff . Note that the following orthogonal decomposition holds:

Differential operators of linear elasticity
Relative to the decomposition (24), we define self-adjoint operators A stiff 0,χ , A soft 0,χ on the spaces H stiff , H soft , respectively, by the sesquilinear forms Clearly, A stiffpsoftq 0,χ correspond to the differential expressions with zero boundary condition on Γ. The following proposition is easy to prove. Proof. Due to Assumption 2.1, there exist χ-independent constants C 1 , C 2 ą 0 such that Furthermore, due to (121) there exists a χ-independent C ą 0 such that so the forms in question are uniformly coercive.

Lift operators
The way we introduce the lift operators required for the analysis of boundary value problems via Krein's formula is the following. First, we introduce classical lift operators r Π stiffpsoftq χ as the operators mapping g P H 1{2 pΓ; C 3 q to the weak solutions u P H 1 # pY stiffpsoftq ; C 3 q of the boundary value problems u is Y-periodic. (27) As the consequence of Proposition 8.8, the following statement holds. is bounded and satisfies the following bounds for all g P H 1{2 pΓ; C 3 q : where C ą 0 is independent of χ.
Proof. The proof uses the classical approach of rewriting the problem (27) with zero boundary condition and non-zero right-hand side and using Proposition 8.3 and Proposition 8.8.
We would like to introduce operators Π stiffpsoftq χ as the extensions of r Π stiffpsoftq χ to E " L 2 pΓ; C 3 q. We do so by defining their adjoints first. To this end, we consider the operatorś Here B stiffpsoftq ν is the trace of the co-normal derivative: where n stiffpsoftq is outward unit normal on Γ from Y stiffpsoftq . Furthermore, we denote by Π with respect to the pair of inner products x¨,¨y H stiffpsoftq and x¨,¨y E , namely , and Proof. Due to results on elliptic regularity, under Assumption 2.1 the operators pA stiffpsoftq 0,χ q´1 are bounded from H stiffpsoftq to H 2 pΩ stiffpsoftq ; C 3 q by Lemma 8.14. Thus, using the trace theorem, we infer that Ξ stiffpsoftq χ is bounded as an operator with values in H 1{2 pΓ; C 3 q. Due to the compactness of the embedding H 1{2 pΓ; C 3 q ãÑ E, it follows that Ξ stiffpsoftq χ is compact. Next, one can easily verify that for g P H 1{2 pΓ; Proceeding to the proof of (30), note that holds in the sense of distributions. In other words, for all v P C 8 c pY stiffpsoftq ; which can be seen through approximation. Under the condition Π stiffpsoftq χ g P DpA from which it follows immediately that Π stiffpsoftq χ g " 0. This proves the second property in (30). To prove the first property in (30), choose an arbitrary g P H 1 pΓ; C 3 q and let u P H 2 pY stiffpsoftq ; The existence of such u is guaranteed by the Lemma 8.16. Now, denoting f , and, due to the density of H 1 pΓ; C 3 q in E, we conclude that

Dirichlet-to-Neumann maps
Here we define the operators Λ stiffpsoftq χ and prove their self-adjointness. The main result is Theorem 3.14. Due to the elliptic regularity, under Assumption 2.1 one can define the operators where g P H 3{2 pΓ; C 3 q and u is the unique solution to (27). Notice that as a consequence of Lemma 8.14 we have where the constant C is independent of χ. We have the following.
Theorem 3.14. The operators r Λ stiffpsoftq χ can be uniquely extended to unbounded nonpositive self-adjoint Proof. The first step of the proof consists in obtaining a larger extension of the operator r Λ stiffpsoftq χ for which the desired operator Λ stiffpsoftq χ is simply a restriction onto H 1 pΓ; C 3 q. The second step is to show that this restriction is, in fact, self-adjoint. For this, it suffices to show that the restriction is symmetric, and that the resolvent set of the restriction contains at least one real number, for example 1.
We begin by taking g P H 3{2 pΓ; C 3 q, h P H 1{2 pΓ; C 3 q and considering the solutions u P H 2 pY stiffpsoftq ; It follows that r Λ stiffpsoftq χ g defines an element of H´1 {2 pΓ; C 3 q. Due to (28), the bound holds, with C independent of χ P Y 1 . In particular, we can define a unique bounded extension Since`H 1 pΓ; C 3 q, L 2 pΓ; C 3 q˘is an interpolation pair with respect to the pairs`H 3{2 pΓ; is bounded as an operator from H 1 pΓ; C 3 q to Proceeding to the second step, we prove that Λ stiffpsoftq χ is self-adjoint as an unbounded operator on is non-positive and symmetric. Indeed, as a consequence of (32) holding for g P H 3{2 pΓ; In order to prove self-adjointness of Λ stiffpsoftq χ , it suffices to show that ρpΛ stiffpsoftq χ q X R ‰ H. We claim that p´Λ stiffpsoftq χ`I q´1 : L 2 pΓ; C 3 q Ñ H 1 pΓ; C 3 q is bounded. To see this, first assume that h P H 1{2 pΓ; C 3 q and seek the solution g P H 3{2 pΓ; C 3 q to the problem´Λ stiffpsoftq χ g`g " h. This is equivalent to seeking u P H 2 pY stiffpsoftq ; in the weak sense and setting g :" u| Γ . Invoking Lemma 8.13 for the existence and uniqueness of such a solution, we thus obtain an operator p´Λ stiffpsoftq χ`I q´1 : H 1{2 pΓ; C 3 q Ñ H 3{2 pΓ; C 3 q. Second, applying the Korn's inequality (see Proposition 8.8) to the weak form of (34), namely where a stiffpsoftq χ is defined by (25) and Dpa stiffpsoftq χ q :" H 1 # pY stiffpsoftq ; C 3 q, where we set v " u, we obtain the apriori estimate u H 1 pY stiffpsoftq ;C 3 q ď C h H´1 {2 pΓ;C 3 q . Therefore, p´Λ stiffpsoftq χ`I q´1 can be extended to a bounded operator from H´1 {2 pΓ; C 3 q to H 1{2 pΓ; C 3 q. Using, once again, an interpolation argument, we conclude that p´Λ stiffpsoftq χ`I q´1 is bounded from L 2 pΓ; C 3 q to H 1 pΓ; C 3 q and is hence compact as an operator from L 2 pΓ; C 3 q to itself. Thus, unity is in the resolvent set of Λ stiffpsoftq χ . Remark 3.15. Notice that as a consequence of interpolation theorem and (31) and (33) where the constant C does not depend on χ.Ű

sing (11), we define operators
Actually, a more general approach to defining the operators´Λ stiffpsoftq χ (and thus Λ stiffpsoftq χ ) can be developed under weaker assumptions on the regularity of the domain and the operator coefficients by considering sesquilinear forms λ We have the following proposition. are symmetric, non-negative, closed and densely defined in E, respectively. Furthermore, there exists a χ-independent constant C ą 0 such that Proof. For g P H 1{2 pΓ; C 3 q, due to the pointwise coercivity of A stiff , one has Furthermore, due to the estimate (126) and the trace theorem, we obtain Kato's representation theorem yields the existence of (non-positive) self-adjoint operators Λ stiffpsoftq χ This approach to defining Dirichlet-to-Neumann maps allows us to relax the regularity assumptions (on the boundary and the operator coefficients). However, as a result, we lose the information on the domains of these maps. In particular, we no longer have the equality DpΛ stiff χ q " DpΛ soft χ q " H 1 pΓ; C 3 q, as in the Theorem 3.14. We will not push this approach further, but we will use Proposition 3.16 for estimating the smallest eigenvalues of the operator Λ stiffpsoftq χ . The eigenvalues and eigenfunctions of Dirichlet-to-Neumann maps are usually referred to as Steklov eigenvalues and eigenfunctions, respectively.

Reformulation of the transmission problem in terms of a Ryzhov triple
The coupled operator A 0,χ,ε is defined by relative to the decomposition (24). Equivalently it can be defined by the form With these definitions at hand, we can define the coupled lift operator as follows: From Theorem 3.13 we have kerpΠ χ q " t0u, DpA 0,χ,ε q X RpΠ χ q " t0u. In order to describe the condition of continuity of normal derivatives from (10), we introduce the operator Λ χ,ε :" ε´2Λ stiff χ and Λ soft χ are self-adjoint on the common domain H 1 pΓ; C 3 q and non-negative. Therefore, the operator Λ χ,ε is self-adjoint on DpΛ χ,ε q :" H 1 pΓ; C 3 q. This fact very probably belongs to the domain of folklore, still we include the proof of it as kindly shared with the authors by Dr. V. Sloushch, to whom we express our deep gratitude. Proof. Without loss of generality, assume that A and B are positive definite. By the closed graph theorem, AB´1 and BA´1 are bounded in H. Pick a κ such that }κBA´1} ď 1. One clearly has DpAq Ď DpκBq and }κBu} ď }Au} for u P DpAq.
By the Heinz inequality (see, e.g., [4, Chapter 10, Section 4.2, Theorem. 3]), Bq´1 is bounded and defined on the whole H, and therefore closed. Therefore, A`B is self-adjoint on DpAq, as required.
Note that, while the operators Λ stiff χ and Λ soft χ are used to calculate the normal derivative on the boundary Γ, the map Λ χ,ε instead yields the jump on the boundary Γ between the normal derivative from the soft component and scaled normal derivative from the stiff component. We can also introduce the transmission boundary operators pA χ,ε , Γ 0,χ , Γ 1,χ,ε q associated with (37). Clearly, we have Γ 1,χ,ε " ε´2Γ stiff 1,χ P stiff`Γ soft 1,χ P soft . Along with these triples, we use the Definition 3.3 to define the solution operators as well as M-functions indexed by complex z in the respective resolvent sets: Obviously, one has In the context of introduced boundary triples, the transmission boundary problem (10) can be formulated as finding u P DpA χ,ε q such that # A χ,ε u´zu " f , The corresponding solution operator is given by the "Krein formula" and we know it to be the resolvent of a closed extension pA χ,ε q 0,I of A 0,χ,ε .

Asymptotic properties of the Ryzhov triples -stiff component
The aim of this section is to provide operator asymptotics with respect to the quasimomentum χ P Y 1 for the operators of the Ryzhov triple associated with the stiff component, introduced in the previous section. As we will see, the approximation on the stiff component is the crucial part of the approximation in total (cf. Remark 7.7 below).
We also show that the eigenspace of the Dirichlet-to-Neumann map Λ stiff χ corresponding, for small χ, to Steklov eigenvalues of order |χ| 2 is finite-dimensional. This fact is one of the key ingredients for providing resolvent asymptotics for the transmission problem (10).
As we are about to see, the vector nature of the problem does not allow one to infer any asymptotic expansion for Steklov eigenvalues. However it turns out to be sufficient for our purposes to obtain the asymptotics of the resolvents of Dirichlet-to-Neumann maps, and consequently, the asymptotics of their spectral projections.

Asymptotic properties of the lift operators
We are interested in the asymptotics for the lift operator Π stiff χ : H 1{2 pΓ; C 3 q Ñ H 1 # pY stiff ; C 3 q defined by (27). Naturally, the leading-order term is provided the operator Π stiff 0 : H 1{2 pΓ; C 3 q Ñ H 1 # pY stiff ; C 3 q, which does not depend on χ. Note that for g P H 1{2 pΓ; C 3 q, the function Π stiff The operator Π stiff 0 satisfies the estimate provided by the following lemma.
Lemma 4.1. There exist constants C 1 , C 2 ą 0 such that for every g P H 1{2 pΓ; C 3 q one has Proof. The first inequality is (128). To prove the second one we observe Remark 4.2. Notice here that the operator Π stiff 0 lifts the constant functions on the boundary g " C P C 3 ãÑ E the to constant functions defined by the same constant: The following theorem is crucial for understanding the asymptotics of the Dirichlet-to-Neumann maps. Since its proof follows a standard asymptotic argument, we provide it in Appendix. Theorem 4.3. For each n P N, the operator Π stiff χ admits the asymptotic expansion where the operators r Π χ,k , r Π error χ,n : H 1{2 pΓ; C 3 q Ñ H 1 # pY stiff ; C 3 q, k " 1, . . . , n are bounded, and satisfy for all n P N, where the constant C ą 0 does not depend on χ P Y 1 . Furthermore, the following is valid: where the operator-valued tensors Π k are symmetric: where S n is the permutation group of order n.

Remark 4.4.
Here we make an observation that proves to be crucial in the understanding of the homogenisation properties of the effective operator (see Lemma 4.9). By following the proof of Theorem 4.3 and observing the equation (140), which actually serves as the definition of the operator r Π χ,1 , one concludes that for g " The following lemma yields the estimates on the stiff components which are useful in the spectral analysis in the continuation of the paper. We identify the spaces of constant functions on Y stiff and Γ with the space C 3 .
Proposition 4.5. There exists a constant C ą 0 such that, for all χ P Y 1 zt0u we have: • For every g P H 1{2 pΓ; C 3 q we have: • For every g P H 1{2 pΓ; C 3 q, g K C 3 (in L 2 pΓ; C 3 q inner product), we have: Proof. The estimate (44) is a clear consequence of estimates (122), (123) and the trace theorem. In order to prove (45), it is enough to show that for g P H 1{2 pΓ; C 3 q, g K C 3 we have: To see this, assume that (46) is true. By employing the trace theorem and Korn's inequality (121) we calculate: Furthermore, (46) is an easy consequence of (128) which can be seen by plugging u " Π stiff χ g into (128).

The smallest Steklov eigenvalues
The operator Λ stiff χ on L 2 pΓ; C 3 q has discrete spectrum, due to the compactness of its resolvent established above. The eigenvalues ν χ of Λ stiff χ are equivalently characterised as solutions to either one of the following two problems: Next, we define the Rayleigh quotient associated with Λ stiff χ , namely where λ stiff χ is defined by (36). The eigenvalues pν χ n q nPN of Λ stiff χ can be characterised with the min-max principle´ν The following lemma is crucial for estimating the smallest eigenvalues.
Lemma 4.6. There exist constants C 1 ą C 2 ą 0 such that Proof. The proof of the first and third points is a direct consequence of Proposition 4.5. The second point is verified by a direct computation.
The asymptotics of eigenvalues with respect to |χ| is given in the following lemma.
Lemma 4.7. The three smallest eigenvalues of Λ stiff χ are of order Op|χ| 2 q, and the remaining eigenvalues uniformly separated from zero. Namely, there exist constants c 1 , c 2 ą 0 that do not depend on χ such that We refer to ν χ k , k " 1, 2, 3, as order Op|χ| 2 q eigenvalues and to ν χ k , k ě 4, as order Op1q eigenvalues.
Proof. The proof is a direct consequence of (47) and Lemma 4.7.
Consider the decomposition where p P χ is the orthogonal projection onto the three-dimensional space p E χ ă E spanned by the eigenfunctions associated with order Op|χ| 2 q eigenvalues of Λ stiff χ , and q P χ is the orthogonal projection onto q E χ ă E, the infinite-dimensional space spanned by the eigenfunctions associated with order Op1q eigenvalues of Λ stiff χ , so that p P χ " I´q P χ . Since the decomposition (48) is reducing for Λ stiff χ , we have Both p Λ stiff χ and q Λ stiff χ are self-adjoint operators on p E χ and q E χ , respectively (for the first one it is obvious, while for the second one it can be easily deduced). The operator p Λ stiff χ is clearly bounded since it is essentially a matrix. Notice also that the domain of the second operator is H 1 pΓ; C 3 q. Due to the Lemma 4.7 we have On the other hand, Lemma 4.7 implies that the operator q Λ stiff χ , while unbounded, is uniformly bounded from below, where the estimate does not depend on |χ|, namely where the constant C is independent of χ.

Asymptotics for the operator
The results of this Section are needed in Section 6 and Section 7 for the proof of Theorem 2.3. They are not needed for the proof of Theorem 2.2. The main tool for proving Theorem 2.3 is the approximation of the Dirichlet-to-Neumann map and finding the approximating homogenized operator and its variational definition (see (53)). Let us notice that, in order to do that, we cannot follow the approach of [10], since one cannot do the expansion of eigenfunctions and eigenvalues with respect to (vector valued) quasimomentum χ for systems, see [22,Example 5.12]. Instead we do the asymptotics for the resolvent of the Dirichlet-to-Neumann map and show in Section 6 and 7 that this is enough to prove Theorem 2.3. We also notice that through the variational definition of the approximating homogenized operator (53) we are able to prove its characterization through A macro see Lemma 4.9.
We calculate the estimates on the distance between the resolvents of´|χ|´2Λ stiff χ and´|χ|´2Λ hom χ , where the latter plays the rôle of the effective Dirichlet-to-Neumann map and is defined in the continuation of the text. In Corollary 4.13 we use this to obtain the approximation error with respect to the resolvents of ε 2 scaled operators. The similar approach (in different context) was used in [15,16].
Recall that the lift operator Π stiff in the sense of Theorem 4.3, where the operator r Π χ,1 is defined by (140) and its dependence on the parameter χ is linear. The error term r Π error χ,1 satisfies the bound (see (42)) r Π error We define the effective operator to be a constant matrix Λ hom where c corr P H 1 # pY stiff ; C 3 q is the unique solution with where we denote Ξ χ :" iX χ Π stiff 0`s ym ∇ r Π χ,1 . The definition of homogenised operator takes the form @´Λ hom χ c, d In next two lemmas we give important properties of the operator (matrix) Λ hom χ . Lemma 4.8. There exists a constant ν ą 0 such that ν|χ| 2 |c| 2 ď x´Λ hom χ c, cy ď ν´1|χ| 2 |c| 2 for all c P C 3 .

Proof. Our first observation is the formula
where d corr is defined by (52) from d, so Λ hom χ is a Hermitian matrix. Also, we have the coercivity bound @´Λ hom Lemma 4.9. The matrix Λ hom χ is quadratic in χ, namely Λ hom χ "´|Γ|´1 piX χ q˚A macro iX χ , where A macro is the constant symmetric tensor defined by (2).
Proof. The first step is to conclude that for c P C 3 , the equation (52) can actually be tested against a wider class of functions, namely ż This can be seen by noting that for an arbitrary v P H 1 # pY stiff ; C 3 q one has the decomposition The vanishing of the expression (55) when testing against functions of the type Π stiff 0 h, h P H 1{2 pΓ; C 3 q is stated in (52), while the case of testing with w P H 1 # pY stiff ; C 3 q, w " 0 on Γ is covered with (41) and (43). Introducing the notation u j corr :" Π stiff 0 c j corr`r Π χ,1 c j , j " 1, 2, where c 1 , c 2 P C 3 , and invoking (51) we have ż and also @´Λ hom where we have employed the definition of the macroscopic tensor (2).
Next, we state the theorem on the norm-resolvent estimates.
Proof. The proof of this theorem is constructive in nature. We start from the weak formulation of the resolvent problem for the operator Λ stiff χ . For h P L 2 pΓ; C 3 q, our goal is to expand the solution g of the problem in the form g " g 0`g1`g2`gerr , where the terms satisfy the bounds g 0 " Op1q, g 1 " Op|χ|q, g 2 " Op|χ| 2 q, g err " Op|χ| 3 q, with respect to the H 1{2 pΓ; C 3 q norm, and are calculated from a sequence of boundary value problems obtained from (56), which are introduced in the continuation of the proof. By equating the terms in (56) which are of order Op1q, we obtain the following identity for the leading-order term g 0 : ż Y stiff A stiff sym ∇Π stiff 0 g 0 : sym ∇Π stiff 0 v " 0 @v P H 1{2 pΓ; C 3 q.
From here, it is clear that the term g 0 " c P C 3 . However, at this point, we cannot determine g 0 yet. Furthermore, by combining the terms of order Op|χ|q, we obtain the identity ż which has a unique solution satisfying ş Γ g 1 " 0. Furthermore, comparing the terms of order Op|χ| 2 q, we define g 2 as the solution to the identity ż The uniqueness and existence of solution to this equation can be established with two additional constraints. We require g 2 to have zero mean: ş Γ g 2 " 0, and that the right-hand side vanishes when tested against constants 1 . There is a priori no reason for the second constraint to be valid. However, we still have an additional degree of freedom in the choice of g 0 P C 3 , which can be used here to obtain well posedness of the problem. Setting v " v 0 P C 3 in (58) yields Using (54) and (57), it follows from (59) that or, equivalently,´1 Thus, by defining the leading-order term as the solution to the above resolvent problem, we have ensured the solvability of (58). Next, we prove the estimates for the correctors, and the final error estimate. It is clear from (60) and the coercivity of Λ hom Furthermore, from (57) we have sym ∇Π stiff 0 g 1 L 2 pY;C 3ˆ3 q ď iX χ Π stiff 0 g 0 L 2 pY;C 3ˆ3 q` sym ∇ r Π χ,1 g 0 L 2 pY;C 3ˆ3 q .
By virtue of Theorem 4.3 and Proposition 8.12, we obtain In a similar manner, it can be seen from (58) that Next, we formulate the equation for the error term g err :" g´g 0´g1´g2 , where g is the solution to the full problem (56).
Remark 4.11. The averaging operator S : E Ñ E coincides with the projection operator p P 0 " p P χ | χ"0 .R emark 4.12. The norm-resolvent estimate provided in Theorem 4.10 also yields the estimate where Cpzq depends on the distance of z to the spectrum of |χ|´2Λ  There exists a constant C ą 0, independent of χ P Y 1 , ε ą 0, such that the following norm-resolvent estimate holds: where the homogenised operator Λ hom χ is defined by the formula (51).
Before proceeding to the proof, we first provide some important auxiliary results. There are several crucial remarks which allow us to rewrite the norm resolvent estimates in terms of the parameter ε. Both operators |χ|´2Λ stiff χ and |χ|´2Λ hom χ (where the latter is in fact a multiplication by a constant matrix depending on χ) have exactly 3 eigenvalues of order Op1q, and the set of these eigenvalues can be enclosed with a fixed contour γ Ă C uniformly in |χ| (for small |χ|).
Proof. The proof of this fact stems from the fact that the spectrum of order Op1q is uniformly separated from the rest for small enough |χ|. This is guaranteed by the estimates in Lemma 4.7. The same estimates also provide the fact that this part of the spectrum lies in a fixed interval independent of χ.
Theorem 4.10 has another direct consequence, namely the asymptotics of the spectral projections p P χ and the truncated lift operators Π stiff χ p P χ with respect to the quasimomentum χ.
Corollary 4.16. The operators p P χ and Π stiff χ p P χ satisfy the asymptotics where the constant C ą 0 does not depend on χ P Y 1 .
Proof. By choosing a contour γ as above, and applying the Cauchy formula to the constant function gpzq " 1, we obtain the asymptotics of projectors p P χ defined by (48): as a consequence of Remark 4.12. This proves the estimate To improve it, we write it in the following way: Notice that also p P χ : L 2 pΓ; C 3 q Ñ H 1 pΓ; C 3 q is bounded as a consequence of the fact that D`Λ stiff χ˘" H 1 pΓ; C 3 q and that p P χ is the projection onto the eigenspace corresponding to eigenvalues of order |χ| 2 . The first term in (70) is bounded in L 2 pΓ; C 3 q Ñ H 1 pΓ; C 3 q norm with order |χ| due to (69) and the boundedness of p P χ . The second term in (70) is also bounded in the same norm with the same order, as a consequence of (69) and the fact that p P 0 is a projection onto a fixed finite-dimensional subspace of H 1 pΓ; C 3 q. Combining this with the result of Theorem 4.3, it follows that where Op|χ|q is understood in the sense of the L 2 pΓ; C 3 q Ñ H 1 pY stiff ; C 3 q operator norm. Similarly, where Op|χ|q is understood in the sense of the L 2 pΓ; C 3 q Ñ L 2 pY stiff ; C 3 q operator norm.

Asymptotics on the soft component
Next, we state some asymptotic properties of the boundary operators for the soft component.
Moreover, the eigenvalues of the operator A soft 0,χ are independent of χ, and e iχy`Asoft 0,χ´z I˘´1e´i χy "`A soft 0,0´z I˘´1, where the operator A soft 0,0 is defined by (26) (with χ " 0). Furthermore, let pµ n q n be a sequence of eigenvalues of the operator A soft 0,0 and pϕ n q n the associated sequence of eigenfunctions. Then the corresponding sequence of eigenfunctions of A soft 0,χ is given by pϕ χ n q n " pe´i χy ϕ n q n .
Proof. The proof is based on the fact that H 1 χ pY soft ; C 3 q " H 1 pY soft ; C 3 q. Recall the definition of Π soft 0 : Writing u " e iχy w P H 1 pY soft ; C 3 q, we obtain Π soft 0 g " u ðñ #´d iv`A soft sym ∇`e iχy w˘˘" 0 on Y soft , e iχy w " g on Γ ðñ #´d iv`e iχy A soft psym ∇`iX χ q w˘" 0 on Y soft , w " e´i χy g on Γ ðñ w " Π soft χ e´i χy g ðñ u " e iχy Π soft χ e´i χy g.
Similarly, one has h " e iχy Λ soft χ e´i χy g. Furthermore, there exists a χ-independent constant C " Cpzq ą 0 such that Proof. By recalling the identity (20), we have the representation formula Employing the identities (71) and (72), as well as Remark 3.15 and (66) yields the claim.
Remark 4.19. Notice that, due to the fact that Λ soft 0 | p E 0 " 0, one also has The following corollary is easy to prove.
Corollary 4.20. The operator Π soft χ p P χ satisfies the asymptotics where the constant C ą 0 does not depend on χ P Y 1 .
Proof. The proof is the direct consequence of (71) and (66).

ε 2 -order resolvent asymptotics of the transmission boundary problem
In this section we aim at proving Theorem 2.2. The starting point is the Krein formula (40). The approximation is carried out in two steps. The first step (see Section 5.1, Theorem 5.2) is to use the Schur-Frobenius inversion formula by restricting the traces onto the space p E χ , as well as imposing the equality of projections to the same space of the traces of co-normal derivatives. The second step (Section 5.2, Theorem 5.5) is to approximate the M-function. We recall the formulae (39), (38) and (40).
Recalling (15), we write where Opε 2 q is understood in the sense of the E Þ Ñ H stiff norm. The formula (20) yields We can also obtain the full expansion by using the following geometric series (in the L 2 Ñ L 2 sense): This yields

Steklov truncation
Similarly as in (49), we define the following truncated Dirichlet-to-Neumann maps: so one obviously has p Λ χ,ε " ε´2 p Λ stiff χ`p Λ soft χ , q Λ χ,ε " ε´2 q Λ stiff χ`q Λ soft χ . The first operator sum is well defined due to the fact that these objects are essentially Hermitian matrices. The second sum defines a self-adjoint operator, by the similar argument as in the Theorem 3.17, where the domain is given by Dp q Λ χ,ε q " Dp q Λ stiff χ q " Dp q Λ soft χ q " q P χ`D pΛ χ,ε q˘.
One can define the boundary triples and M-functions associated with the Ryzhov triples introduced above, denoted in the same fashion. Notice that, for example, the domain of the operator p A soft χ coincides with DpA soft 0,χ q 9p Π χ p p E χ q, so the trace operator p Γ 0,χ takes values in p E χ . By recalling that representation formula (17) decomposes the M-function into the sum of a self-adjoint operator (Dirichlet-to-Neumann map) and a bounded operator, we know that its domain actually coincides with the domain of the associated "Λoperator", and thus one has and a similar claim holds for q M χ pzq stiffpsoftq , q M χ,ε pzq, and p M χ,ε pzq. We introduce the notation p H stiffpsoftq χ :" Π stiffpsoftq χ p E χ , together with the notation P p H stiffpsoftq χ for the respective orthogonal projections (with respect to the H inner product). We also define Θ χ : H Ñ H as the orthogonal projection Before stating our approximation result, we need one helpful lemma, whose proof is found in the Appendix. It establishes the equivalence of the H 1 and L 2 norms on p H stiffpsoftq χ uniformly in the quasimomentum χ.
Lemma 5.1. There exists a χ-independent constant C ą 0 such that The following theorem is the first approximation result.
Theorem 5.2. There exists C ą 0 such that for the resolvent of the transmission boundary problem (10) one has `p A χ,ε q 0,I´z I˘´1´`pA χ,ε q q Proof. Notice that, as a consequence of (20) and (39), we can write: where the operator B χ,ε is bounded uniformly in χ, ε. So, the question of boundedness of a certain truncation of M χ,ε actually comes down to the boundedness of associated truncation of Dirichlet-to-Neumann map. But, since p E χ Ă DpΛ χ,ε q " DpΛ stiff χ q " DpΛ soft χ q and p E χ is finite-dimensional, one has and thus the operators p P χ M χ,ε pzq q P χ , q P χ M χ,ε pzq p P χ , are bounded as well, uniformly in χ, ε. We next show that the operator q P χ M χ,ε pzq q P χ is boundedly invertible with a bound depending on ε. This is the point where we stress the importance of Steklov truncations and the bound (50).
where we have shown that A, B, E are bounded (the bound of A depends on ε), and F is boundedly invertible F´1 EÑE ď Cε 2 , where C does not depend on ε. Our next aim is to show that A is boundedly invertible with the bound independent of χ. To this end, notice that (76) implies where Opε 2 q is understood in the sense of the L 2 Ñ L 2 operator norm, uniformly in χ. Using (19), (81), Lemma 5.1, the trace inequality, and the fact that z P K σ , we infer the existence of a χ-independent constant C ą 0 such that, for all f P L 2 pΓ; C 3 q and ε small enough, one hašˇˇ@ By virtue of Corollary 8.2, it now follows that where C ą 0 does not depend on χ. Using the Schur-Frobenius inversion formula, see [39], we have where S :" F´EA´1B. However, since On the other hand, by Schur-Frobenius' formula, we havè Thus, by putting β 0 " q P χ , β 1 " p P χ and recalling the Theorem 3.9 we recognise that the operator-valued function z Þ Ñ pA 0,χ,ε´z Iq´1´S χ,ε pzq´q P χ`p P χ M ε,χ pzq¯´1 p P χ S χ,ε pzqi s the resolvent of the closed extension of A 0,χ,ε associated with the boundary conditioǹ q P χ Γ 0,χ`p P χ Γ 1,χ˘u " 0 in the sense of Theorem 3.9. Thus we have proven the desired result.
Remark 5.3. The resolvent ppA χ,ε q q P χ , p P χ´z Iq´1 is associated with boundary value problem ( f P H) The solution u P H satisfies the following constraints: • On one hand, only the 3-dimensional component which lies in the space p E χ of the traces of their co-normal derivatives (from inside Y soft , Y stiff respectively) coincide on the boundary Γ.
• On the other hand, the traces from both inside Y soft and Y stiff actually belong to the 3-dimensional space p E χ (they clearly coincide, as seen in Remark 3.18).
Thus, by approximating the resolvent ppA χ,ε q 0,I´z Iq´1 associated with the transmission problem (10) by the resolvent ppA χ,ε q q P χ , p P χ´z Iq´1, one relaxes the condition on the continuity of co-normal derivatives and tightens the constraint on the traces, while making an error of order ε 2 .
The standard intuition yields that the homogenisation procedure should replace the solution on the stiff component with the 3-dimensional constant vector. However, in this step, the finite-dimensionality is only imposed on the trace of the solution.5 .2 Refinement of the approximation: truncation of p M stiff χ We introduce the following notation for the truncated solution operator p S χ,ε pzq :" S χ,ε pzq| p E χ . Using the formula (16), we have which is precisely the solution operator associated with the triple pA 0,χ,ε , p Π χ , p Λ χ,ε q in the sense of Definition 3.3. Similar representation formulae are obtained for the operators p S softpstiffq χ pzq (with an obvious definition).
Also, one has which follows directly from the definition.T he operator p Γ 0,χ is left inverse to the operator p Π χ in the sense of the Definition 3.2, so it is clear, due to (83), that it is the left inverse to the operator p S χ,ε pzq as well. Similar holds for the operators p S stiffpsoftq χ pzq. In particular, we have p Γ 0,χ p S χ,ε pzq " p Γ stiffpsoftq 0,χ p S stiffpsoftq χ pzq " I| p E χ . In the Theorem 5.2 we have obtained an approximation of the original resolvent in terms of the resolvent of another operator, where the relative simplification is not immediately evident. However, by doing simple additional approximations the result becomes much clearer. We perform these additional approximations by separately analyzing the block components of the resolvent (see (84)) relative to the decomposition (24). It follows from (81) and (82) that We introduce the following operator-valued function which plays a great rôle in the formulation of the homogenisation results: Definition 5.1. We refer to the operator valued function p Q app χ,ε pzq : p E χ Ñ p E χ given with the following expression: as the transmission function.
We have the following result on resolvent asymptotics in which we significantly simplify the solution on the stiff component in the approximation of the resolvent.
Theorem 5.5. There exists C ą 0, which depends only on σ and diampK σ q, such that for the resolvent of the transmission boundary problem (10) we have: for all χ P Y 1 , where the operator valued function R app χ,ε pzq is defined by the following formula: relative to the decomposition H soft ' H stiff .
Proof. The proof consists of applying the asymptotic formula (86) to the separate blocks of the resolvent ppA χ,ε q 0,I´z Iq´1. We have Next, we use the asymptotic formula (75) for S stiff χ : For calculating the remaining block, we use the fact that P stiff pA 0,χ,ε´z Iq´1 P stiff "`ε´2A stiff 0,χ´z I˘´1 " Opε 2 q in H Ñ H operator norm. Finally, we have which concludes the proof.
Remark 5.6. Notice that we can rewrite (88) in the following way:

The fiberwise approximating operator
It remains to identify the selfadjoint operator which resolvent is in a sense represented with (88). To this end, we consider the Hilbert space H soft ' p H stiff χ and define the following operator on it: The following theorem establishes the link between the resolvent of the operator (91) and the operator valued function (88).
Theorem 5.7. For every χ P Y 1 , the operator A app χ,ε is self-adjoint and its resolvent for all z P ρpA app χ,ε q is given by the formula (88), relative to the decomposition H soft ' p H stiff χ .
Proof. First we show that the operator A app χ,ε is symmetric. For pu, p uq, pv, p vq P DpA app χ,ε q we calculate: By the Green formula (see (13)) and by the self-adjointness of p Next, we fix f P H soft , p f P p H stiff χ . For every z P ρpA app χ,ε q we consider the problem which is equivalent to Using the fact that p u " p Π stiff χ p Γ soft 0,χ u, the problem (93) is equivalent to However, by recalling (12) we have that p Γ stiff Now we define the operators β 0,χ,ε pzq :" ε´2 p Λ stiff χ`z p p Π stiff χ q˚p Π stiff χ , β 1 " I and introduce the transmission function defined by (87) to obtain The operator p Q app χ,ε pzq is boundedly invertible (as can be seen by looking its imaginary part and using Corollary 8.2) and satisfies the assumptions of Theorem 3.8. The solution u is then given by the formula (22), with g "´p p Π stiff χ q˚p f :

Now, one has
Another insight into the operator (91) is obtained by considering its associated sesquilinear form, given in the following lemma.
Lemma 5.8. The sesquilinear form a app χ,ε on HˆH associated with the operator (91) is given by Proof. The proof goes by direct computation.
whereů,v P Dpa soft 0,χ q and g u , g v P p E χ . With this notation at hand, the form a app χ,ε can be written as he following theorem yields the final conclusion of this section, namely, the norm resolvent asymptotics in terms of the simplified operator A app χ,ε .
Theorem 5.10. There exists C ą 0, which depends only on σ and diampK σ q, such that for the resolvent of the transmission boundary problem (10) one has where the operator A app χ,ε is defined by (90), (91), and is an orthogonal projection defined by with respect to L 2 pY stiff ; C 3 q inner product.
Proof. The proof consists of using (89) combined with Theorem 5.7 in order to conclude that R app χ,ε pzq " Θχ`A app χ,ε´z I˘´1 Θ χ .
Proof of Theorem 2.2. The proof is the direct consequence of Theorem 5.10, using the fact that the (scaled) Gelfand transform is an isometry and defining

General outlook on the approach
An alternative way to rewrite formula (88) is the following: The operator valued function R app,soft χ,ε pzq is the solution operator (in the sense of Theorem 3.9) to the problem of finding u P D`A soft χ˘s uch that # p A soft χ u´zu " f , β 0,χ,ε pzq p Γ soft 0,χ u`β 1 p Γ soft 1,χ u " 0.
where transmission operators are given by β 0,χ,ε pzq :" ε´2 p Λ stiff χ`z p p Π stiff χ q˚p Π stiff χ , β 1 " I. Problems like these, where the boundary conditions depend on the spectral parameter are called the boundary value problems with "impedance". One should note that the "impedance" here appears in a linear manner.
Proposition 5.11. The generalised resolvent R soft χ,ε pzq is the solution operator of the "impedance" boundary value problem on H soft that consists in finding u P D`A soft χ˘s uch that where the operators of boundary transmission are given by r β 0,χ,ε pzq :" ε´2M stiff χ pε 2 zq, r β 1 " I.
The "impedance" of the boundary value problem (96) is highly nonlinear, due to the structure of the M-function M stiff χ pzq. On the abstract level, both solution operators R app,soft χ,ε and R soft χ,ε represent the so-called generalised resolvents [29,30,36,37,38]. A generalised resolvent can be equivalently characterised as either an operator of the form PpA´zIq´1| P for a self-adjoint A in a Hilbert space H and an orthogonal projection P, or a solution operator of an abstract spectral boundary value problem where A is a densely defined linear operator on PH, pE, Γ 0 , Γ 1 q is an abstract boundary triple of A, and Bpzq is an analytic in the upper half-plane operator-function with positive imaginary part (i.e., an operator R-function) on E, extended into the region ℑz ă 0 by the identity Bpzq " B˚pzq.
The system (97) can be thus re-cast in the form of the operator equation A z u " zu, where A z is a closed densely defined linear operator on PH with domain DpA z q " tu P DpAq Ă PH : Γ 1 u " BpzqΓ 0 uu.
The operator A z is shown to be maximal dissipative for z P C`and maximal antidissipative for z P C´.
From the point of view of generalised resolvents, one can therefore view the homogenisation procedure we have performed above as obtaining the main order term in the asymptotic expansion of the generalised resolvent R soft χ,ε for every fixed χ P Y 1 as ε Ñ 0. Moreover, we point out that in order to determine the main order term of ppA χ,ε q 0,I´z Iq´1 as ε Ñ 0, or in other words to recover the operator describing the homogenised medium, it is in fact necessary and sufficient to construct an asymptotic expansion of the generalised resolvent described above. This follows from the fact that under a natural and non-restrictive "minimality" condition the operator A giving rise to the generalised resolvent PpA´zIq´1| PH is in fact uniquely determined based on the latter up to a unitary gauge transform Φ such that ΦPH " PH.
This can be viewed in the homogenisation problem at hand as taking the "down, right, up" detour in the commutative diagram ppA χ,ε q 0,I´z Iq´1 Ý ÝÝÝ Ñ A app χ,ε " pA app χ,ε q˚in H Ą H soft : where the double solid line represents the unitary gauge. As far as the asymptotic analysis of the generalised resolvent R soft χ,ε is concerned, the required analysis is essentially reduced to the derivation of the asymptotics of the operator r β 0,χ,ε pzq which governs its impedance boundary conditions. This, due to (95), in turn reduces to a well-understood problem of perturbation theory for the Dirichlet-to-Neumann map pertaining to the stiff component of the medium and thus presents no complications.
Having said that, we point out that however appealing this argument appears, it meets two significant difficulties. Firstly, at present we don't have an explicit way to construct the operator A app χ,ε in (98) for arbitrary impedance boundary conditions parameterised by a generic´R-operator function Bpzq in (97). In the problem at hand, this presents no challenge as the main order term (92) of the boundary operator is in fact linear in z. Generalised resolvents of this form have already appeared in problems of dimension reduction, most notably in the works concerned with the convergence of PDEs defined on "thin" networks to ODEs on limiting metric graphs, see, e.g., [32,24,25,18], and in particular our recent paper [17] where an approach akin to the one utilised in the present work is extended to the context of thin networks. In the area of linear elasticity in particular this analysis is thought to be applicable to the analysis of pentamodes [28], which will be further discussed elsewhere.
Secondly and crucially, once the asymptotics of the family of generalised resolvents is obtained in some desired strong topology, the same type of convergence for the family of resolvents ppA χ,ε q 0,I´z Iq´1 cannot be inferred from the general operator theory. In fact, one can argue that norm-resolvent convergence of R soft χ,ε only yields strong convergence of ppA χ,ε q 0,I´z Iq´1. It is here that the specifics of the problem at hand must play a crucial rôle in the analysis, leading to a result of the type formulated in Theorem 5.5 above.
Despite the deficiencies of the general operator-theoretic outlook based on generalised resolvents explained above, we point out that this way of considering the dimension reduction problem at hand is very natural in that it presents one with a physically motivated understanding of the problem.
As the argument of [19,20], see also references therein, demonstrates, generalised resolvents appear naturally in physical setups where one forcefully removes certain degrees of freedom from consideration in an otherwise conservative setting in view of simplifying the latter. Conversely, the procedure of reconstructing the self-adjoint generator of conservative dynamics must be viewed as adding those "hidden", or concealed, degrees of freedom back in a proper way. In doing so one frequently faces a situation (and in particular, in the setup of linear elasticity discussed in the present paper) where the resulting model is drastically simplified owing to only a certain limited number of concealed degrees of freedom appearing in it in a handily transparent way. The procedure of the diagram (98) can be therefore seen as a non-trivial generalisation of the seminal idea of Lax and Phillips [26], with a dissipative generator expressing the scattering properties of the system being replaced by a more general one, corresponding to an R-function which non-trivially depends on the spectral parameter z.
At the same time, as explained in [12] (see also references therein), the concept of dilating (in the sense of (97)) a generalised resolvent to a resolvent of a self-adjoint generator gives rise to the understanding of homogenisation limits in the setup of double porosity models as essentially operators on soft component of the media with singular surface potentials possessing internal structure, see also [14] where a similar argument applied to high-contrast ODEs has led to a Kronig-Penney-type model. In the problem considered in the present paper, the mentioned singular surface can be shown to be the periodic lattice of the original composite.
Moreover, the argument of [13] can be immediately invoked for the homogenised family (91) to obtain its functional model in an explicit form in certain explicitly constructed Hilbert space of complex-analytic functions, giving rise to a Clark-Alexandrov measure serving as the spectral measure of the family. This latter program will be pursued elsewhere, together with the study of effective scattering problems of the high-contrast composite which can be considered naturally on this basis.

ε-order resolvent asymptotics of the transmission boundary problem
In this section, our aim is to further approximate the resolvent related to the transmission boundary problem and to prove Theorem 2.3, part (a). In doing this, we will worsen the order in ε of the estimate, but will obtain more familiar objects in the asymptotics.
To this end, we state the following two results which are crucial for understanding the perturbation properties of the spaces p H stiff χ with respect to the quasimomentum χ P Y 1 . The proof of the first result is postponed to the Appendix.
The following lemma establishes that the continuity of the orthogonal projections P p H stiff χ as χ approaches zero.
Lemma 6.1. There exists a constant C ą 0 independent on χ P Y 1 such that Our aim is to provide further approximation to the operator hich constitutes the resolvent (89), where the approximation error is of order not worse than ε. A crucial result is the following estimate on the inverse of the transmission function: Lemma 6.2. There exists C ą 0 which does not depend on ε ą 0, z P K σ , χ P Y 1 , such that: Proof. First, note that For u P p E χ we calculate using Lemma 5.1 and trace inequality: where the constant C depends only on K σ . Thus, due to Corollary 8.2 we conclude that where the constant C is independent of χ and z. Furthermore, using Corollary 4.18 and Remark 4.19, we infer that there exists a constant r C, which depends on |z| and σ such that p M soft χ pzq L 2 pΓ;C 3 qÑL 2 pΓ;C 3 q ď r C.
By using Lemma 4.7 and the fact that p Π stiff χ is uniformily bounded we conclude that one can choose a constant D, independent of χ, ε, such that for |χ| ě Dε, one has: where C 2 is independent of χ, ε. For such |χ|, by the application of the Corollary 8.2 we obtain which together with (100) concludes the proof.
Next we introduce a version of the transmission function which will appear in the final homogenisation result.
Definition 6.1. We refer to the operator valued function p Q eff ε,χ pzq : p E 0 Ñ p E 0 given with the following expression: as the effective transmission function. Associated with this we also introduce the following operator-valued function on H where the constant C ą 0 is independent of χ and z. Also, by using the estimate (68), as well as the identities (84), (85), we have `p Π stiff χ˘˚´`p Π stiff 0˘˚ L 2 pY stiff ;C 3 qÑL 2 pΓ;C 3 q ď C|χ|.F or the inverse of the effective transmission function we have an estimate similar to (99).
Proof. The proof follows the steps of the proof of Lemma 6.2 by using Lemma 4.8.
The following lemma provides the estimate on the distance between the two transmission functions.
Lemma 6.5. There exists a constant C ą 0 that does not depend on ε ą 0, z P K σ , χ P Y 1 such that The case of χ " 0 is trivial. It is clear that for all χ P Y 1 zt0u we have where γ is the contour provided by Lemma 4.14. Therefore, by applying the Theorem 4.10 (cf. Remark 4.12), we obtain 1 The claim now follows from (87) and (101) by using Corollary 4.18 and Corollary 4.16.
The following lemma is crucial for obtaining ε-order asymptotics of the resolvent ppA χ,ε q 0,I´z Iq´1 and relating it to an object which incorporates the effective transmission function. Lemma 6.6. There exists a constant C ą 0 which does not depend on ε ą 0, z P K σ , χ P Y 1 , such that Proof. By a direct calculation, we see that Next, we estimate using Lemma 6.2, Lemma 6.4 and Lemma 6.5: Furthermore, by employing Corollary 4.16 and Lemma 6.2, one easily estimates Similarly using again Corollary 4.16 and Lemma 6.4 we estimate: This concludes the proof.
Finally, we are able to sum up these results into the following theorem. Proof. The proof of this fact consists of estimating the four blocks of the matrix or which all the arguments go analogously. Thus, we showcase the estimate only for one of the blocks. By using triangular inequality and Corollary 4.20, (84) and Lemma 6.2 we obtain in the same way as the estimate (102) : where Opεq denotes the term of order ε in the operator norm L 2 Ñ L 2 . Using Lemma 6.6 we complete the proof.
Definition 6.2. We define the effective operator as follows: where .R ecall that p E 0 are constant functions and p Λ soft 0 g " 0, for every constant function. In a similar manner as in the Theorem 5.7 one can show the following theorem, the proof of which we omit. Theorem 6.9. For every χ P Y 1 , the operator A eff χ,ε is self-adjoint and its resolvent is given, for all z P ρpA eff χ,ε q, by the formula (103) relative to the decomposition H soft ' p H stiff 0 . Remark 6.10. The sesquilinear form a eff χ,ε on HˆH associated with the operator (104) is given by Recalling Lemma 4.9, we can see that the same form was obtained in [8] as εapproximation in the case of scalar equation by using different techniques.T he following theorem provides the norm resolvent asymptotics of order ε with the leading order term being the "sandwiched"-resolvent of the effective operator A eff χ,ε . Its proof is the direct consequence of Theorem 6.7, cf. the proof of Theorem 5.10. Theorem 6.11. There exists C ą 0, independent of z P K σ and ε, such that for the resolvent of the transmission boundary problem (10) where the operator A eff χ,ε is defined by (104), and Θ 0 is the orthogonal projection (77).
Next, we make a remark that the projection operator P stiff is simply a multiplication with an indicator function associated with Y stiff , namely P stiff u " ½ Y stiff pyqu, u P H. Similarly, for the operator P stiff ε i.e. the orthogonal projector from L 2 pR 3 ; C 3 q onto L stiff ε , which is defined by (1), we have P stiff ε u " ½ Ω ε stiff pxqu, u P L 2 pR 3 ; C 3 q. Also, for u P H stiff , we have We will need the following lemma.
where the operator A macro is defined by (3).
Proof. To see this, we introduce the operator A χ,macro on H by its associated form: By invoking the properties of the Gelfand transform (8), it is clear that By virtue of (108), it remains to show that ε´2A χ,macro´B pzq˘´1SP p First conclusion is that for z P K σ , u P H, we have ℑ`ε´2 piX χ q˚A macro iX χ´B pzq˘" ℑ`ε´2A χ,macro´B pzq˘"´ℑ`Bpzq˘, so the operators are invertible by taking into account Corollary 8.2 and the estimates (106). In order to show (110), we take f P H and consider the unique solution u P C 3 " p H 0 to the resolvent problem ε´2 piX χ q˚A macro iX χ u´Bpzqu " SP p By multiplying the above equation with arbitrary v P C 3 , and integrating over Y one obtains Now, it is straightforward to check that u is also the solution to the problem: find u P H 1 # pY; C 3 q such that which is unique. The formula (109) now follows from (110).
The following lemma allows us to drop the smoothing operator Ξ ε from the resolvent asymptotics while not making the error of higher order then ε 2 .
Lemma 7.4. Let z P K σ . There exists a constant C ą 0 such that `A macro´B pzq˘´1 pI´Ξ ε q L 2 pR 3 ;C 3 qÑL 2 pR 3 ;C 3 q ď Cε 2 , where A macro is a differential operator of linear elasticity with constant coefficients defined by (3).
Proof. We start with the identity F pΞ ε f qpξq " ½ r´1 2ε , 1 2ε s 3 pξq F p f qpξq, valid for all f P L 2 pR 3 ; C 3 q, where ξ P C 3 is the Fourier variable and F p¨q denotes the Fourier transform (see, e.g. [16,Section 2.5.3]). The estimate (111) follows from the fact that for f P L 2 pR 3 ; C 3 q we have F``A macro´B pzq˘´1 f˘pξq "`piX ξ q˚A macro iX ξ´B pzq˘´1 F p f qpξq.
Finally, we are able to prove Theorem 2.3 (b).

Proof of Theorem 2.3 (b).
The asymptotic estimate (105) immediately yields Invoking (109), we obtain Combining this with (112) and using the fact that Gelfand transform is a unitary operator, we obtain P stiff ε pA ε´z Iq´1 P stiff ε´P stiff ε`Amacro´B pzq˘´1Ξ ε P stiff ε L 2 pR 3 ;C 3 qÑL 2 pR 3 ;C 3 q ď Cε, The last step is to drop the smoothing operator for which we use Lemma 7.4.
Remark 7.5. The operator A macro´B pzq which plays the role of the leading order term in the resolvent asymptotics in Theorem 2.3 (b) is clearly a differential operator of second order with constant coefficients. Remark 7.6. With the arguments presented in this section we are able to recover the result for the perforated domain, obtained in [35], L 2 Ñ L 2 estimate. To see this we have to take P stiff ε " I and Bpzq " z. In order to obtain L 2 Ñ H 1 estimate, we would have to go one step further in the asymptotics of Dirichlet-to-Neumann operator in Section 4.3, see also the results obtained in [15,16].R emark 7.7. Both operators A app ε and A eff ε are pseudodifferential. However, A eff ε is simpler (on the stiff component). If one wants to obtain approximation of the resolvent of the original problem, we cannot further simplify (homogenise) the operator A eff ε . Indeed, on the stiff component the approximation is already fully homogenised (cf. Remark 7.6), while on the soft component it cannot be further simplified. On the other hand for the operator A app ε it seems that by going further at the expansion in Section 4.3 one would be able to obtain simpler approximating operator (simplification would again be done on the stiff component), while keeping the precision ε 2 . It can be shown that the obtained operator would again be pseudodifferential (even on the stiff component). We refrained ourselves from doing this, for the efficiency of the exposition. We emphasize the fact that when dealing with high contrast the limit or approximating operator is always ε-dependent. In the case of the limit operator this is seen through the fact that it is defined on the spaces which have additional (micro) variable (see [42]), representing the oscillations on the soft component. Thus we can say that, in the case of high contrast, the homogenisation is necessarily partial. From the point of view of wave propagation this is expected since on the soft inclusions the wave length (in the regime where we are interested) is proportional to the size of the cell, while on the stiff matrix the wave length is much bigger than the size of the cell.4 By invoking the formulas (15), (14) and (113), the dispersion function can be expressed as Denote now by pη k q kPN Ă R`, pϕ χ k q kPN Ă H soft the eigenvalues and associated eigenfunctions of A soft 0,χ . We write the expansion of the resolvent of A soft 0,χ with respect to its spectral projections: Proof. First, we show that RpAq " H " RpAq. In order to do this, we take x P kerpA˚q. Since x P DpAq, one has 0 " xx, A˚xy " xAx, xy ě C x 2 and therefore x " 0. Therefore, RpAq "`kerpA˚q˘K " t0u K " H. Clearly, A is an injection. Thus A´1 exists. For y P RpAq we put x " A´1y. Thus, one has From this we have the claim.

Auxiliary estimates
Here we state some of the results which we use in the main text. In this part we state various version of Korn's and trace inequalities.
The following proposition is the special situation of the classical trace theorem.
There exists a constant C ą 0 such that for every g P H 1{2 pΓ; C 3 q there is an extension G P H 1 χ pY stiffpsoftq ; C 3 q satisfying G H 1 pY stiffpsoftq ;C 3 q ď C g H 1{2 pΓ;C 3 q . Next is a well known Korn's inequality (see, e.g., [31]).

Proposition 8.4.
Let Ω Ă R 3 be a bounded open set with Lipschitz boundary. There exists a constant C ą 0 such that for every u P H 1 pΩ; C 3 q, we have u H 1 pΩ;C 3 q ď C´ u L 2 pΩ;C 3 q` sym ∇u L 2 pΩ;C 3ˆ3 q¯, where the constant C depends only on the domain Ω.
The following version of Korn's inequality is also well known. We will provide its proof by a contradiction from Proposition 8.4.
where the constant C depends only on Ω and Γ.
Proof. By the inequality (117) we see that it suffices to show that u L 2 pΩ;C 3 q ď C` sym ∇u L 2 pΩ;C 3ˆ3 q` u L 2 pB;C 3 q˘.
Suppose the opposite, namely that we have a sequence pu k q kPN Ă H 1 pΩq 3 , u k L 2 pΩ;C 3 q " 1 such that u k L 2 pΩ;C 3 q ą k` sym ∇u k L 2 pΩ;C 3ˆ3 q` u k L 2 pB;C 3 q˘, k P N.
By (117), the sequence pu k q kPN is bounded in H 1 pΩ; C 3 q and therefore converges weakly in H 1 pΩ; C 3 q and strongly in L 2 pΩ; C 3 q to some u P H 1 pΩ; C 3 q, u L 2 pΩ;C 3 q " 1. By lower semicontinuity of the norm, we have sym ∇u L 2 pΩ;C 3ˆ3 q ď lim kÑ8 sym ∇u k L 2 pΩ;C 3ˆ3 q " 0.
The following version of Korn's inequality is also well known and is a direct consequence of [31, Theorem 2.5].
Proposition 8.6. Let Ω be a bounded open set with Lipschitz boundary. There exists C ą 0 dependent only on Ω such that for every u P H 1 pΩ; C 3 q the estimate u´w H 1 pΩ;C 3 q ď C|| sym ∇u|| L 2 pΩ;C 3ˆ3 q holds with w " Ax`c, where A " The following proposition is simple to prove.
where the constant C depends only on the domain Ω.
Proof. By plugging w " e iχy u, u P H 1 pΩ; C 3 q, in the inequality (118), we obtain from Proposition 8.7: where we have also used (120).
The following proposition gives us necessary estimates for periodic functions with periodicity cell Y.
Proposition 8.9. There exists a constant C ą 0 such that for all u P H 1 # pY; C 3 q, χ P Y 1 zt0u one has Proof. For a function u P H 1 # pY; C 3 q we have the Fourier series decomposition: a k e 2πik¨y , ∇u " ÿ kPZ 3 e 2πik¨y a k b p2πikq , for which we can calculate the norms Furthermore, we have and therefore psym ∇`iX χ qu Combining this with the inequality |a d b| ě |a||b|{ ? 2 , we infer (122)-(124).
The following is the result on the extension with dominated norm of symmetrized gradients: Proposition 8.10. There exists an extension operator from H 1 # pY stiff ; C 3 q to H 1 # pY; C 3 q such that (u Þ Ñ u): for all χ P Y, where the constant C depends only on Y stiff .
Proof. This is a clear consequence of an analogous result for the extension operator applied to quasiperiodic functions. For the construction of the extension operator see [31].
We can now state the following two propositions.
Proposition 8.12. There exists a constant C ą 0 such that for all u P H 1 # pY stiff ; C 3 q, χ P Y 1 one has Proof. The bound (127) is deduced from (124) and Proposition 8.10 similar to how Proposition 8.11 was established. To prove (128), note first that by (127) and the continuity of traces, one has (129) Second, from Proposition 8.8 and the trace inequality, one has The bound (128) now follows from (130) by using (129), (126), and the continuity of traces.

Well posedness and regularity of boundary value problems in elasticity
Here we state some results regarding the properties of the weak solution to the boundary value problem "´d iv pA sym ∇uq " f on Ω, pA sym ∇uq n| BΩ`u " g on BΩ, where Ω Ă R 3 is a bounded domain, f P L 2 pΩ; R 3 q, g P H´1 {2 pBΩ; R 3 q and the tensor of material properties A P L 8 pΩ; R 3ˆ3ˆ3ˆ3 q satisfies the following assumption.
The weak form of this problem is stated in the following definition: Definition 8.1 (Robin boundary problem for elasticity, weak form). For given functions f P L 2 pΩ; R 3 q, g P H´1 {2 pBΩ; R 3 q find u P H 1 pΩ; R 3 q such that ż Ω Apxq sym ∇upxq : sym ∇vpxq`ż BΩ u¨v " ż Ω f¨v`ż BΩ g¨v @v P H 1 pΩ; R 3 q.
Notice that the map v Þ Ñ ş BΩ g¨v is a bounded linear functional on H 1 pΩ; R 3 q. Also, the form .
The following result can be found in, e.g. [27]. Let Ω Ă R 3 be a bounded domain with boundary BΩ of class C 1,1 . Let f P L 2 pΩ; R 3 q, g P H 1{2 pBΩ; R 3 q and, in addition to 8.1, A P C 0,1 pΩ; R 3ˆ3ˆ3ˆ3ˆ3 q.
Then the unique weak solution to the problem (131) belongs to H 2 pΩ; R 3 q and we have u H 2 pΩ;R 3 q ď C´ f 2 L 2 pΩ;R 3 q` g 2 H 1{2 pBΩ;R 3 q¯1
We are also interested in the solution to the Dirichlet type boundary value problem: "´d iv pA sym ∇uq " f on Ω, u " g on BΩ, where f P H´1pΩ; R 3 q, g P H 1{2 pBΩ; R 3 q. It is straight forward to apply theorem of Lax-Milgram to conclude that there exists unique weak solution u P H 1 pΩ; R 3 q. Again the proof of the following result can be found in [27] and is the standard regularity result for Dirichlet boundary problem.
Lemma 8.14 (Regularity of the solution of Dirichlet problem). Let Ω Ă R 3 be a bounded domain with boundary BΩ of class C 1,1 . Let f P L 2 pΩ; R 3 q, g P H 3{2 pBΩ; R 3 q and A P C 0,1 pΩ; R 3ˆ3ˆ3ˆ3ˆ3 q. Then the unique weak solution to the problem (132) belongs to H 2 pΩ; R 3 q and we have u H 2 pΩ;R 3 q ď C´ f 2 L 2 pΩ;R 3 q` g 2 H 3{2 pBΩ;R 3 q¯1
The constant C depends on Ω and C 0,1 norm of A.

Trace extension lemma
A version of the following theorem can be found in [5].
Theorem 8.15. Let Ω Ă R 3 be a bounded domain with boundary BΩ of class C 0,1 . Let g 0 P H 1 pBΩ; R 3 q, g 1 P L 2 pBΩ; R 3 q. Then, there exists u P H 2 pΩ; R 3 q such that " B n u " g 1 on BΩ, u " g 0 on BΩ if and only if ∇ BΩ g 0`g1 b n P H 1{2 pBΩ; R 3 q.
We will prove the following lemma.
Lemma 8. 16. Let Ω Ă R 3 be a bounded domain with boundary BΩ of class C 1,1 and A P C 0,1 pΩ; R 3ˆ3ˆ3ˆ3 q. Let g P H 1{2 pBΩ; R 3 q. Then there exists u P H 2 pΩ; R 3 q such that " A sym ∇u¨ n " g on BΩ, u " 0 on BΩ.
Proof. The first step is to conclude the following: if a function u P H 2 pΩ; R 3 q satisfies u " 0 on Γ then ∇u| BΩ " B n u| BΩ b n.
From this we have (135). With this at hand, the first equation in (134) becomes A pB n u| BΩ d nq¨ n " g. We proceed by introducing the new variable ω :" B n u| BΩ and we consider the problem of finding ω such that A pω d nq¨ n " g.
The second step of the proof is to conclude the following: There exists a unique solution ω P H 1{2 pBΩ; R 3 q to the problem (136). Notice that (136) is an algebraic equation for every x P BΩ. For a fixed point x P BΩ, we introduce the operator L x : R 3 Ñ R 3 with the expression L x ω :" A pω d npxqq¨ npxq. To see that it is symmetric and positive definite we perform the following calculations: xL x ω, wy R 3 " xA pω d npxqq¨ npxq, wy R 3 " A pω d npxqq : pw d npxqq " xL x w, ωy R 3 , where we have used the Assumption 8.1. Thus, the problem L x ωpxq " gpxq has a unique solution for a.e.
x P BΩ. Clearly, from (137) it follows that det pL x q ě α ą 0, uniformly in x P BΩ. By C 0,1 regularity of both the material coefficients and the normal n, we know that both det pL x q and pdet pL x qq´1 are of class C 0,1 . Then, by the application of Cramer's rule, we arrive at the conclusion that the function ωpxq :" pL x q´1 gpxq belongs to H 1{2 pBΩ; R 3 q. Thus, we have reduced the problem (134) to finding u P H 2 pΩ; R 3 q such that " B n u " ω on BΩ, u " 0 on BΩ, where ω is a solution of (136).In order to complete the proof we only need to check the validity of the condition (133) and apply Theorem 8.15, which can be applied since ω b n P H 1{2 pBΩ; R 3 q, due to the C 0,1 pBΩ; Rq regularity of the normal n.
such that the H 1 norm of each term u n is bounded with a bound of order |χ| n , while the error of approximation u error,n :" u´u 0´n ÿ k"1 u k is of order |χ| n`1 . We assume that all terms of the series satisfy zero boundary condition on Γ. The leading order term u 0 corresponds to the harmonic lift in the case χ " 0. With this apriori intention, we plug the expansion (139) into (138) and equate the terms which would be of order Op1q to obtain ż Y stiff A stiff sym ∇u 0 pyq : sym ∇vpyqdy "´ż Y stiff A stiff sym ∇Gpyq : sym ∇vpyqdy, @v P H 1 # pY stiff ; C 3 q, v| Γ " 0.
We recognise here the definition of the zeroth order lift operator Π stiff 0 , namely Π stiff 0 g " u 0`G ": r u. It is clear that u 0 H 1 pY stiff ;C 3 q ď C G H 1 pY stiff ;C 3 q ď C g H 1{2 pΓ;C 3 q .
Next, with this term in hand, we proceed to define u 1 with: ż Clearly, one has u 1 H 1 pY stiff ;C 3 q ď C|χ| g H 1{2 pΓ;C 3 q . Proceeding next with terms of second order, we define next term in the expansion by ż Y stiff A stiff sym ∇u 2 pyq : sym ∇vpyqdy A stiff sym ∇u 1 pyq : iX χ vpyqdy @v P H 1 # pY stiff ; C 3 q, v| Γ " 0.
In this way, we can define the operators r Π χ,n : h Þ Ñ u n , n P N. The error term u error,n :" u´n ÿ k"0 u k , n P N, A stiff iX χ u n pyq : sym ∇vpyqdy´ż Y stiff A stiff sym ∇u n pyq : iX χ vpyqdy @v P H 1 # pY stiff ; C 3 q, v| Γ " 0.
Proof of Lemma 6.1. The proof goes by representing the action of the projections P p H stiff χ in the appropriate orthonormal basis. First, we denote with e 1 , e 2 , e 3 an orthonormal basis of constant functions for p E 0 . Due to (66) we have: where Op|χ|q is understood in the sense of H 1{2 pΓ; C 3 q norm. Here we have used the fact that p P 0 e i " e i . Furthermore, due to (67), one has Π stiff χ p P χ e j " Π stiff 0 p P 0 e j`O p|χ|q " Π stiff 0 e j`O p|χ|q " e j`O p|χ|q, j " 1, 2, 3, where Op|χ|q is understood in the sense of H 1 pY stiff ; C 3 q norm and we interpret Π stiff 0 e i as in Remark 4.2. Notice that " e 1 e 1 L 2 pY stiff ;C 3 q , e 2 e 2 L 2 pY stiff ;C 3 q , e 3 e 3 L 2 pY stiff ;C 3 q * is an orthonormal basis for p H stiff 0 . Now we define an orthonormal basis te χ 1 , e χ 2 , e χ 3 u for p H stiff χ by using the Gramm-Schmidt procedure: (142) From (141) we also infer that Π stiff χ p P χ e i L 2 pY stiff ;C 3 q " e i L 2 pY stiff ;C 3 q`O p|χ|q, i " 1, 2, 3, @ Π stiff χ p P χ e i , Π stiff χ p P χ e j D L 2 pY stiff ;C 3 q " xe i , e j y L 2 pY stiff ;C 3 q`O p|χ|q " Op|χ|q, i ‰ j.