Asymptotics for Spectral Problems with Rapidly Alternating Boundary Conditions on a Strainer Winkler Foundation

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We assume that the surface Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varSigma $\end{document} is traction-free out of small regions Tε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T^{\varepsilon }$\end{document}, where we impose Winkler-Robin boundary conditions. This condition links stresses and displacements by means of a symmetric and positive definite matrix-function M(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M(x)$\end{document} and a reaction parameter β(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta (\varepsilon )$\end{document} that can be very large when ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon \to 0$\end{document}. The size of the regions Tε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T^{\varepsilon }$\end{document} is O(rε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(r_{\varepsilon })$\end{document}, where rε≪ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{\varepsilon }\ll \varepsilon $\end{document}, and they are placed at a distance ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon $\end{document} between them. We provide all the possible spectral homogenized problems depending on the relations between ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon $\end{document}, rε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{\varepsilon }$\end{document} and β(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta (\varepsilon )$\end{document}, while we address the convergence, as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon \to 0$\end{document}, of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varSigma $\end{document}. New capacity matrices are introduced to define these strange terms.


Introduction
In this paper, we address the asymptotic behavior of a spectral problem associated with the vibrations of a deformable elastic solid Ω ⊂ R 3+ = {x : x 3 > 0} whose boundary ∂Ω has a part clamped to an absolutely rigid profile Γ Ω and the other part Σ ⊂ {x : x 3 = 0} in contact with a strainer Winkler foundation which can be modeled by a series of small springs periodically placed along Σ , the reaction regions T ε . On these small regions, the boundary conditions are of Winkler-Robin type, also so-called of spring type, while outside, they are traction-free. The small regions T ε have diameter O(r ε ) and are at a distance ε between them, where ε measures the period of the structure. Here ε and r ε are two small parameters r ε ε 1; see Fig. 1.
The elastic coefficients of the small springs are defined through the so-called Robin reaction matrix, which we denote by β(ε)M(x). Matrix M(x) depends on the point where the reaction regions T ε are placed, while the parameter β(ε), which is referred to as the reaction parameter, can range from very small to very large. Each T ε is assumed to be a domain of the plane R 2 homothetic to a fixed domain T , with a Lipschitz boundary. Analyzing the different relations between the three parameters of the problem, ε, r ε and β(ε), it is crucial to detect several behaviors of vibrations of the structure. We study the asymptotic behavior of the eigenvalues and eigenfunctions, when ε → 0; this also involves asymptotics for solutions of associated stationary problems.
The stationary problem, for an isotropic homogeneous media, and a surface Σ which is stuck to the plane along the regions T ε , has been studied in [21] and [6], where they provide a critical size of the stuck regions O(ε 2 ) (cf. (1) with r 0 > 0), which is somehow classical in the literature of applied mathematics. For this size, the asymptotic behavior of the solution is intermediate between the extreme cases. Namely, for r ε ε 2 the stuck regions are large enough and the body behaves as if the whole Σ is stuck to the plane, for r ε ε 2 the stuck regions are very small and the surface behaves as if it were traction-free, while for r ε = O(ε 2 ) a Winkler-Robin boundary condition is asymptotically imposed as an intermediate condition between Dirichlet and Neumann. It contains the so-called strange term, and links stresses and displacements, the elastic coefficients of this spring being given by a constant matrix: the so-called capacity matrix (cf. (25) and (26) for W l,x ≡ W l ).
Here, we deal with a different problem, and obtain the above-mentioned homogenized problems only for a particular relation between the parameters, in the case of the isotropic media (cf. Remark 2). As a matter of fact, in addition to the critical size, it appears a critical relation for parameters (cf. (3) with β * > 0) which also provides asymptotic behavior of solutions different from extreme cases. Now, several kinds of elastic capacity matrices arise, which are obtained from the microstructure of the problem and depend on the macroscopic variable. This dependence is due to both, the nonhomogeneous media filling Ω and the nonconstant Robin matrix M. A formal study of the problem, based on asymptotic expansions, has been addressed in [14] describing convergence as an open problem that we broach here. For the sake of completeness, we provide all the homogenized problems depending on the relations between ε, r ε and β(ε) (cf. Sect. 3).
Notice that other different boundary homogenization problems in linear elasticity have been studied in the literature. Let us mention [29] and [16], which treat stationary homogenization problems for the elasticity system in a perforated media along a plane, the size r ε of the perforations in the plane being r ε = O(ε). Also, [5] considers a cylindrical body, the regions where the displacements vanish being thin bands which are rolled around the body. For the case of a certain non periodical distribution of the regions T ε , for extreme cases, let us mention [30]. For a strongly oscillating boundary, see [26]. Other papers investigating homogenization problems for the elasticity operator, with the same geometrical configuration here considered, are [17] (for r ε ε) and [13] (for r ε = O(ε)). Both deal with spectral problems with alternating boundary conditions of Steklov type and, consequently, they strongly differ from our problem. Also the results are very different.
All these works belong to a large class of boundary homogenization problems for several operators, which have been studied for a long time: in this respect, we refer to [14] for an extensive annotated bibliography on vector and scalar problems. Below, we mention just some of the pioneering works in the literature, either because of the geometry or the key words here used. See [23], [36] and [9] for critical sizes and strange terms in scalar problems. See [24], [20] and [8] for different "sieve" scalar models. For the Stokes fluid problem in a perforated domain along a plane, we mention the works [2] and [12] where, also, a so-called Stokes capacity matrix appears on the transmission condition on Σ when r ε = O(ε 2 ); the large parameter β(ε) appears in [12] related to the adsorption process. See [35] for various effects on the perforated walls in fluid models. See [10] for critical parameters in a fluid homogenization problem. We mention [15] and [33] in connection with the homogenization of spectral problems, for the Laplacian, with large parameters on the boundary conditions; the technique cannot be extended to the vectorial case here considered. [14] and the present work represent the first spectral boundary homogenization models with large parameters in elasticity theory; [14] contains the formal procedure which differs completely from the technique here used for justifications.
Let us introduce parameters r 0 , β 0 and β * which play an important role in the description of the homogenized problems. They are defined through three limits: and lim ε→0 β(ε)r 2 In the case where r 0 > 0 we deal with the classical critical size of the reaction regions T ε mentioned above.
(2) provides a relation between sizes of reaction regions and the reaction parameter which is important in determining the local problems. The case where β * > 0 is referred to as the critical relation between parameters (critical reaction, in short). It occurs when the total area of the reaction regions O(ε −2 r 2 ε ) multiplied by the reaction parameter β(ε) is of order 1.
The most critical situation happens when r 0 > 0 and β 0 > 0 which also amounts to r 0 > 0 and β * > 0, cf. the intersecting lines in Fig. 2. In this case, the strange term has a character completely different from that obtained in the literature. It contains a so-called extended capacity matrix C e (x), cf. (20), which depends on the Robin matrix M(x) in a non trivial way. It also contains the parameter β 0 .
The rest of the critical relations between parameters, for which a spring type boundary condition intermediate between Dirichlet and Neumann is obtained, deal with r 0 > 0 and β 0 = +∞ or β * > 0 and r 0 = +∞. The first one, r 0 > 0 and β 0 = +∞ (also β * = +∞), asymptotically amounts to regions T ε stuck to the plane because of the large reaction parameter and, consequently, the spring boundary condition ignores M(x). It contains a new capacity matrix C(x), which depends on the macroscopic variable x but only due to the nonhomogeneous media filling Ω. The second relation β * > 0 and r 0 = +∞, always keeping r ε = O(β(ε) −1/2 ε), provides an averaged spring type condition on Σ where the Robin reaction matrix is M(x) multiplied by the average constant β * |T |, cf. (27). Let us refer to [28] for other extended capacity matrices in very different problems.
For the sake of brevity, throughout the paper, we address the convergence in the two cases where the strange terms arise, namely r 0 > 0 and β 0 > 0 or β 0 = +∞ (cf. Remark 1). In both cases, the local problems providing microscopic information are elasticity problems posed in R 3+ , cf. Fig. 3, with the macroscopic variable appearing as a parameter, the corresponding media being homogeneous, but anisotropic, while a nonhomogeneous Winkler-Robin boundary condition (a Dirichlet one, respectively) appears on the unit reaction region T . These problems appear for the first time in homogenization theory and, hence, their correct setting in suitable Hilbert spaces and the smoothness properties of solutions in the abovementioned parameter are new in the literature (cf. Sects. 4 and 7.1). Note that constructing test functions to pass to the limit in the variational formulation of homogenization problems relies on the solutions of these problems (cf. Sects. 5-6 and 7.2-7.3).
On the other hand, it should be emphasized that the usual techniques for scalar problems (cf. [3], [24], [20] and [33]) based on results of convergence of measures on manifolds and comparison of measures do not work for the elasticity system under consideration, and therefore, we use a technique based on projections over spaces of finite elements (cf. [23], [32] and [22] in this connection).
Finally, the structure of the paper is as follows. Section 2 contains the setting of the spectral homogenization problem. The corresponding stationary problem and some preliminary results are collected in Sect. 2.1. Section 3 presents the list of homogenized problems both stationary and spectral problems. It also describes the corresponding stationary local problems which allow us to define the strange terms. Throughout Sects. 4-7, we show the convergence. Further specifying, for r 0 > 0 and β 0 > 0, Sect. 4 contains the setting of the parametric family of local problems in the suitable Hilbert spaces, as well as certain smoothness properties of solutions in the macroscopic variable (the parameter). Section 5 deals with the construction of test functions, and Sect. 6 addresses the convergence of solutions and spectra. Section 7 contains proofs for r 0 > 0 and β 0 = +∞.

The Setting of the Problem
Let Ω be a bounded domain of R 3 situated in the upper half-space R 3+ = {x ∈ R 3 : x 3 > 0}, with a Lipschitz boundary ∂Ω. Let Σ be the part of the boundary in contact with the plane {x 3 = 0} which is assumed to be non-empty and let Γ Ω be the rest of the boundary of Ω: ∂Ω = Γ Ω ∪ Σ . Let T denote a bounded domain of the plane {x 3 = 0} with a Lipschitz boundary. Without any restriction we can assume that both Σ and T contain the origin of coordinates while |Σ| and |T | stand for their surface measures.
Let ε be a small parameter, ε 1, and r ε be another parameter such that r ε ε. For k = (k 1 , k 2 ) ∈ Z 2 , we denote by x ε k the point of the plane {x 3 = 0} with coordinates x ε k = (k 1 ε, k 2 ε, 0), and by T ε x k the homothetic domain of T of ratio r ε after translation to the point x ε k : If there is no ambiguity, we shall write x k instead of x ε k , and T ε instead of T ε x k . In this way, for a fixed ε, we have constructed a grid of squares in the plane {x 3 = 0} whose vertices are inside the regions T ε (cf. Fig. 1 Finally, if no confusion arises, T ε implies the union of all the T ε contained in Σ : In what follows x = (x 1 , x 2 , x 3 ) denotes the usual cartesian coordinates, while byx = (x 1 , x 2 ) we refer to the two first components of x ∈ R 3 . Also, we use the summation convention over repeated indexes.

Some Background
Let us denote by V the space obtained by completion of {v ∈ (C 1 (Ω)) 3 : v = 0 on Γ Ω } in the norm generated by the scalar product, an elastic pseudo-energy bilinear form For fixed ε > 0, the weak formulation of problem (8) reads: find λ ε ∈ R, u ε ∈ V, u ε = 0, satisfying On account of (5) and (7), the left hand side of (11) defines a bilinear, symmetric continuous and coercive form on V ⊂ (L 2 (Ω)) 3 . Consequently, (11) has the discrete spectrum: where we have adopted the convention of repeated eigenvalues according to their multiplicities. The corresponding vector eigenfunctions form a basis in V and (L 2 (Ω)) 3 , and we assume that they are subject to the orthonormalization condition u n,ε , u m,ε (L 2 (Ω)) 3 = δ n,m .
Based on the minimax principle we obtain the uniform bound: where C and C n are constants independent of ε. For the sake of completeness, we outline this proof, cf. [14] for further details. The left hand side of (14) is obtained using the Poincaré and Korn inequalities (cf., e.g., [31] and [37]), (5) and (7). Indeed, we have For the right hand side, we write where the minimum has been taken over the set of all the subspaces E n of V of dimension n. For the last inequality, we have taken the particular space E * n generated by the eigenvec- (cf. also (29), (32)-(33)). Therefore, (14) holds true.
In this paper, we address the asymptotic behavior of (λ ε , u ε ) as ε → 0, depending on the different values of r 0 , β 0 and β * in (1), (2) and (3) respectively. The proof of the convergence (cf. Theorems 4 and 8) is based on a general result on spectral perturbation theory (cf. Section III.1 of [31] and Section III.9.1 in [3]). In order to be self-contained, we introduce below a simplified version of such a result, cf. Lemma 1.
On account of this result, (13) and (14), showing the convergence for the eigenpairs of (11) amounts to showing the convergence of solutions of associated stationary problems. Hence, it proves useful to introduce here the stationary homogenization problem: Find where 3 represent given forces acting on the body. Because of the Korn and Poincaré inequalities, (5) and (7), the unique solution of (16) satisfies with C a constant independent of ε, cf. (10). Therefore, for each sequence of {u ε } ε>0 we can extract a subsequence, still denoted by ε, such that for some u 0 ∈ V ⊂ {v ∈ (H 1 (Ω)) 3 : v = 0 on Γ Ω } (cf., e.g., [4], [31] and [37]). As usual in homogenization, we aim to identify u 0 with the solution of a homogenized problem. In Sect. 3, we provide the list of possible stationary homogenized problems depending on the relations between the parameters ε, r ε and β(ε).
The following result links convergence of stationary and spectral problems; we refer to Lemma 1.6 in Section III.1 of [31] for the proof.
We assume that the following properties are satisfied:

respectively) be the corresponding eigenvectors which are assumed to form an orthonormal basis in H.
Then, for each fixed k, μ ε k → μ 0 k , as ε → 0. In addition, for each sequence, still denoted by ε, we can extract a subsequence ε → 0 such that where w * k is an eigenvector of A 0 corresponding to μ 0 k and the set {w * i } ∞ i=1 forms an orthogonal basis of H.

The Homogenized Problems
In this section, for the sake of completeness, we state all the stationary homogenized problems. They can be obtained as in [14], using the technique of matched asymptotic expansions, with minor modifications. We also state the local problems that allow us to describe the strange terms in the boundary conditions. P1). In the most critical situation where β 0 > 0 and r 0 > 0, the homogenized problem reads: where, forx ∈ Σ , the matrix C e = (C e ij ) i,j =1,2,3 is defined as i = 1, 2, 3. Above, and in what follows, y = (y 1 , y 2 , y 3 ) are auxiliary variables in R 3 (cf. (23)), and lower indexes x or y in the components of the stress and strain tensors mean the variable for derivation. The upper indexx is a parameter which refers to the elastic homogeneous media with constant elastic coefficients a ij kl (x). Namely, Also, e l stands for the unit vector in the y l -direction, while l = 1, 2, 3. Macroscopic and local variables, as usual, are related by According to Proposition 2, using the Korn and Poincaré inequalities we deduce that there exists a unique solution of (19) in the space V. P2). For the critical size r 0 > 0, when β 0 = +∞, the homogenized problem reads: Find where the matrix C = (C ij ) i,j =1,2,3 is defined as σx ij,y and e l in (25) and (26) are defined as in the previous item, cf. (22). As in item P 1), problem (24) has a unique solution (cf. Proposition 6). P3). For the critical relation where β * > 0 with r 0 = +∞, the homogenized problem reads: Fig. 3 The domains of setting for homogenized and local problems P4). For the extreme cases where β * = 0 or r 0 = 0, the homogenized problem is the mixed boundary value problem: P5). For the extreme cases where r 0 = +∞ and, β 0 > 0, or β 0 = +∞, or β 0 = 0 and β * = +∞, the homogenized problem is the Dirichlet problem: The existence and uniqueness of solution of (29) and (28) are classical while that of (27) holds as that of (16). In Sects. 6 and 7, we show the convergence of the solutions and the corresponding spectra in the two critical cases P 1) and P 2) (cf. Remark 2 for the rest). Hence, for convenience, we introduce here the associated spectral problems when r 0 > 0 and β 0 > 0, and when r 0 > 0 and β 0 = +∞. We recall the different definition of the elastic capacity matrices C e = (C e ij ) i,j =1,2,3 and C = (C ij ) i,j =1,2,3 appearing in (30) and (31). They depend on the macroscopic variable. However, this dependence for C e ij is due to both, the nonhomogeneous media and the nonconstant Robin matrix M (cf. (20) and (21)), while that for C ij ignores M (cf. (25) and (26)).
The variational formulation of (30) and (31) reads: where B = C e when dealing with (30) and B = C when dealing with (31). We denote the discrete spectrum by where we have adopted the convention of repeated index. Also, we can choose the corresponding vector eigenfunctions {u n,0 } ∞ n=1 to form an orthonormal basis in (L 2 (Ω)) 3 . From the definitions of C e (x) and C(x), cf. (20) and (25), it is self-evident that the discreteness of the spectrum of problems (30) and (31) is linked to the setting of problems (21) and (26), as well as to the properties of their respective solutions. All this is addressed in Sects. 4 and 7.1.
Note that (32) is also the spectral problem associated with (28) when r 0 = 0, and that associated with (27) when we replace matrix r 0 B by the averaged Robin reaction matrix β * |T |M, β * > 0 in (3). The spectral Dirichlet problem associated with (29) is (15).

The Parametric Family of Local Problems
In this section, we describe certain properties of the solutions of the parametric family of local problems (21) which are necessary for the correct setting of the homogenized problem (19), cf. (20). They are also necessary to obtain appropriate estimates for test functions, cf. Sect. 5.
Let (D(R 3+ )) 3 be the space of functions which are the restrictions to R 3+ of the elements of (D(R 3 )) 3 . Consider the space V, completion of (D(R 3+ )) 3 with respect to the norm Due to Korn's inequality in bounded Lipschitz domains of R 3+ whose boundary contains T , the continuous embedding V ⊂ (H 1 loc (R 3+ )) 3 holds. For each fixed l = 1, 2, 3 andx ∈ Σ , problem (21) has the variational formulation: Find Indeed, it is simple to verify that (35) is the weak formulation of (21) 1 -(21) 3 , while the condition at infinity (21) 4 for the solution of (35) is obtained as a consequence of the following theorem which provides the precise convergence rate at infinity.
Theorem 1 For eachx ∈ Σ and l = 1, 2, 3, problem (35) has a unique solution W l,M,x ∈ V and it can be represented in terms of the Green matrix-function Gx(y) as follows: Here, and Gx = (Gx ij ) i,j =1,2,3 is a symmetric tensor which depends on the elastic constants of the media a ij kp (x) and admits the representation where Φx is a symmetric matrix whose elements are smooth functions on the unit semisphere in R 3+ , S 2 + = {y ∈ R 3+ : |y| = 1} ω. In addition, Φx depends continuously on the parameterx ∈ Σ , in such a way that for i, j = 1, 2, 3, where ∇ ω is the gradient-operator on the sphere, and C a certain constant.
Proof In order to show the existence and uniqueness of solution of (35), we denote by ax(., .) the bilinear, symmetric, continuous and coercive form on V: On account of (5) and (7), cf. (22), ax(., .) defines a norm in V equivalent to · V . Also, we consider the linear continuous functional on V Then, the Riesz representation theorem ensures that there exists a unique function W l,M,x ∈ V satisfying ax(W l,M,x , V ) = F l,x (V ) ∀V ∈ V. This is nothing but (35). The representation (36) for the solution of (21) can be derived as that for Bussinesq-Cerutti tensor in the case of an isotropic media, but without explicit computations of the components of the Green matrix-function (37): see, e.g., [19] for the Bussinesq-Cerutti tensor; see also [1] and [18] for the Mindlin tensor and other related tensors. Since the halfspace is a cone with generator the semi-sphere, for anisotropic media, the representation (36) is supported by general results in [25]. Indeed, explicit formulas for the Green matrixfunction and accompanying tensors are known in the case of isotropy and, for their existence and main properties in anisotropic media, we refer to Sects. 2 and 5 of [25] where more general boundary value problems for elliptic systems in conical domains are considered.
To conclude on the structure (37) of the matrix function Gx and its continuous dependence on the parameterx as well as the representation formula (36), it suffices to mention some basic facts: first, the columns of the amplitude part in (37) are eigenfunctions of a pencil A(Λ) of differential operators on the unit sphere and its equator corresponding to the eigenvalue Λ = −1 ∈ C. Second, according to the usual Green's formula for the elasticity equations in R 3+ , the adjoint pencil for A(Λ) is nothing but A(1 −Λ) so that A(−1) * = A(0). Third, the eigenspace of A(0) consists of constant vector functions because any solution |y| Λ Ψ (ω) with Λ = 0 is a translational rigid motion. Finally, the eigenvalue Λ = 0 and, the eigenvalue Λ = −1, are algebraically simple and have geometrical multiplicity 3 the number of translations in R 3 . The above-listed information provides all desired properties of the Green matrix-function on the basis of the theory of non-selfadjoint operators [11], see also the monographs [27] and [18].
To show that the function with components defined by the right-hand side of (36) belongs to V, we follow the technique based on a density argument in Theorem 4.1 of [21], with minor modifications, taking into account that σ l,x denotes the normal component of the stress tensor, corresponding to W l,M,x ∈ V, on the plane {y 3 = 0}, which has a compact support on T . Consequently, (36) is the solution of (35), and this concludes with the proof of the theorem.  7) and (34), we derive Now, we take into account the continuity of a ij kl and M ij , and the inequalities where also C is a constant independent ofx (cf. identity (35) with V = W l,M,x ). In this way, we can choose δ η such that  (21) and (39), we derive that Cx is uniformly bounded forx ∈ Σ, and the proposition is proved.

Proposition 2 For each fixedx ∈ Σ , the matrix C e (x) defined by (20) is symmetric and positive definite and depends continuously onx ∈ Σ .
Proof For each fixed l, we multiply the elasticity equations in (21) In this way, the symmetry of C e comes from that of M and (5) 1 while the positivity is due to (5) 2 and (7). Indeed, it is simple to verify that ∃γ > 0, such that ∀ᾱ ∈ R 3 ,ᾱ = 0, Finally, from Proposition 1 and (41), it follows that the elements C e ij , for i, j = 1, 2, 3, are continuous functions in Σ. Hence, the proposition is proved.

Test Functions for Critical Size and Critical Reaction
In this section, based on the solution of (21), we introduce some functions which prove to be essential to define the test functions for obtaining the convergence of the solution of (16) towards that of the homogenized problem (19), when r 0 > 0 and β 0 > 0.
is the half-ball of radius r centered at the point x k , and C ε,+ x k the half-annulus (cf. Fig. 4) For l = 1, 2, 3, and k ∈ J ε , we construct the functions W l,k,ε (x), W l,k,ε (x) and W l,ε (x) using the solutions W l,M, x k of the local problems (21), as follows: The last one is extended by e l in Ω \ k∈J ε B + x k , r ε + ε 4 . Finally, we set Below, C denotes a positive constant independent of ε and x k , with k ∈ J ε . Also, Ω 1 denotes any Lipschitz domain, Ω 1 ⊆ Ω with Σ 1 := ∂Ω 1 ∩ Σ = ∅.

Proposition 3
There is a constant C such that, for x ∈ C ε,+ x k , and ε sufficiently small, the inequalities and are satisfied. In addition, for l, p = 1, 2, 3, and Ω 1 ⊆ Ω, we have Proof Estimate (45) is a consequence of the definition (42), while estimates (46) are a consequence of (40). From (45) and (46), estimates (47) are also satisfied. In order to show (48), we evaluate where we have employed (47), (23), (39), (4) and r 0 > 0 in (1). Now, we show that the convergence in (48) holds in the topology of (L 2 (Ω)) 3 by applying the Poincaré inequality on each half-ball and the Korn inequality in Ω. Indeed, using (51), we readily obtain Therefore, also (48) is proved. In order to verify (49), we proceed in a similar way as in (51): where o(ε) stands for a function bounded by Cε, and o(1) is any infinitesimal as ε → 0. For these formulas, we have considered (47), (23), (39), (4), the continuity of a ij kp (x) on x ∈ Ω, r 0 > 0 in (1), the fact that the sum of all the terms in which d( x k , ∂Σ 1 ) ≤ r ε + ε 4 is also o(ε), and the inequality which is based on estimates (40). Now, taking into account Proposition 1, the last chain of equalities leads us to (49).
Let us obtain (50). In each integral on T ε x k we introduce the change (23) and obtain β(ε) where we have used (3), (2) and (1), (4), (39), the continuity of M and Proposition 1. Due to the same argument, the last integral converges towards the right hand side of (50) and, therefore, the proposition is proved.

Convergence for Critical Size and Critical Reaction
Throughout the section we set r 0 > 0 and β 0 > 0 in (1) and (2). In Sects. 6.1 and 6.2, we show the convergence of the solutions of the stationary problems, cf. (16) and (19). In Sect. 6.3, we derive the convergence of the eigenvalues of (8) towards those of (30) with conservation of multiplicity. The main results are stated in Theorems 3 and 4.

The Convergence for Solutions of Stationary Problems
Since the solution u ε of (16) already converges towards some u 0 in the weak topology of (H 1 (Ω)) 3 , cf. (18), in this section, we identify u 0 with the solution of (19).
Theorem 2 For any u 0 ∈ V which is the weak limit in (H 1 (Ω)) 3 of a subsequence of u ε , still denoted by ε, cf. (18), we construct a sequence u ε ∈ V such that u ε → u 0 weakly in (H 1 (Ω) and, for any φ ∈ (C 1 (Ω)) 3 , with φ = 0 on Γ Ω , the following convergences occur: and So that, accepting (60) and (61), we write and the limit as ε → 0 gives that u 0 satisfies By a density argument, we get that u 0 is the unique solution of (19), and we have proved the following result.

The Auxiliary Functions: Proof of Theorem 2
In this section, we prove Theorem 2. We follow an idea in [32] based on projections over spaces of finite elements; see also [22] for a scalar problem. We divide the proof in several steps. To make the reading easier, we prove (61) first, under the basis of the existence of u ε satisfying (58), (60) and (59).

The Construction of u ε Satisfying (58)-(60)
Since σ ij,x ( W l,ε ) takes values different from zero only in a neighborhood of Σ , and u ε = 0 on Γ Ω , there is no loss of generality for the proof to assume that the domain Ω is a polyhedron and the boundary Γ Ω can be written as a finite union of plane faces. For each fixed h > 0, we create a regular triangulation { hq } M h q=1 of the domain Ω composed of tetrahedrons of diameter h (see, e.g., [7] and [34]) Let Π h u denote the projection of the element u ∈ H 1 (Ω), with u = 0 on Γ Ω , on the subspace Y h of the continuous functions over Ω which are affine functions on each tetrahedron hq and take the value 0 on Γ Ω . As it is well known, for any u ∈ H 1 (Ω), with u = 0 on Γ Ω , We divide the rest of the proof into four steps. First step: a first approach to the construction of u ε satisfying (58). For u ε = (u ε 1 , u ε 2 , u ε 3 ) the solution of (16), for u 0 the limit in (18), and for l = 1, 2, 3, let u εh l and u 0h l denote the projections on Y h of u ε l and u 0 l respectively. We set On each hq we introduce the polynomial whose coefficients z l,r (ε, h q ), α l (ε, h q ), z l,r (h q ), α l (h q ) are real numbers, and r = 1, 2, 3. On account of (18), for any fixed h > 0, these coefficients satisfy Second step: a first approach to the convergence (60).

The Spectral Convergence
In this section, based on Lemma 1, we derive the convergence for the eigenpairs of (11), when r 0 > 0 and β 0 > 0 in (1) and (2).
Consequently, the convergence of the eigenvalues and the corresponding eigenfunctions in the statement of the theorem holds from Lemma 1.

The Other Critical Case
In this section, we address the convergence of solutions of the stationary problem (16) and the spectral problem (11), as ε → 0, when r 0 > 0 and β 0 = +∞ in (1) and (2). The main results are Theorems 6 and 8.
We follow the scheme in Sects. 4-6 with the suitable modifications. Section 7.1 presents properties of the solutions of thex-dependent family of local problems (26). The convergence for the stationary problem is in Sect. 7.2, while the spectral convergence is in Sect. 7.3. Now, the stationary homogenized problem reads (24), where the matrix C(x) = (C ij (x)) i,j =1,2,3 is defined by (25) with W l,x the solution of (26). The spectral homogenized problem is (32) with B = C, cf. (31).

Abstract Framework for the Stationary Local Problem (26)
Below, we derive the properties ofx-dependent solutions W l,x and those of matrix C(x).
Let (D 1 (R 3+ )) 3 denote the space of functions in (D(R 3+ )) 3 which vanish in a neighborhood of T . Let V and V 1 be the spaces obtained by completion of (D(R 3+ )) 3 and (D 1 (R 3+ )) 3 , respectively, with respect to the norm Due to Korn's inequality in bounded Lipschitz domains, the continuous embedding V 1 ⊂ (H 1 loc (R 3+ )) 3 holds, and the elements of V 1 have null traces on T . For each l = 1, 2, 3, we take a function Ψ l ∈ (D(R 3+ )) 3 , Ψ l = e l in a neighborhood of T .
Then, the variational formulation of (26) 1 -(26) 3 reads: Find W l,x ∈ Ψ l + V 1 satisfying Problem (75) has a unique solution which is independent of Ψ l (see, e.g., Sect. 4 in [21]). The condition at infinity (26) 4 is a consequence of Theorem 5; see [21], [6] and [14] for an isotropic media. Also, σ i3 (W l,x ) y 3 =0 is a distribution having compact support contained in T and belongs to H −1/2 (T ). Thus, applying the Green formula, we write R 3+ σx pj,y (W l,x )e pj,y (V ) dy = σx pj,y n j (W l,x ), The following propositions deal with results analogous to Propositions 1 and 2 about the continuous dependence of W l,x onx ∈ Σ as well as other related functions. In their statements and proofs C denotes a positive constant independent ofx. Proposition 5 For l = 1, 2, 3, the solution W l,x of (75) depends continuously onx ∈ Σ in the topology of V. In addition, for l, p, i, j = 1, 2, 3, the functions R 3+ e ij,y (W l,x )e ij,y (W p,x )dy and σx p3,y (W l,x ), e i p H −1/2 (T )×H 1/2 (T ) depend continuously onx ∈ Σ , and σx p3,y (W l,x ) Proof Let us show that for each η > 0, theres is δ η > 0 such that ifx, x ∈ Σ satisfy |x − x | < δ η , then W l,x − W l, x V ≤ η.
First, we obtain bounds for the norm of W l,x in V which are independent ofx. To this end, we consider (75) taking V ≡ V l,x := W l,x − Ψ l ∈ V 1 , and we obtain R 3+ σx ij,y (V l,x )e ij,y (V l,x ) dy = − R 3+ σx ij,y (Ψ l )e ij,y (V l,x ) dy.
Applying the Cauchy-Buniakovsky-Schwarz inequality, (22) and the continuity of the elastic coefficients provides the uniform bound for V l,x V . Consequently, cf. (77), we obtain Next, we take V = W l,x − W l, x in (75), and similarly, in the formulation (75) for W l, x , we take V = W l,x − W l, x . By subtracting the second identity from the first one, we obtain: Finally, the continuity on Σ of the term σx p3,y (W l,x ), e i p H −1/2 (T )×H 1/2 (T ) is a consequence of (77), cf. (5) and (22).
As for the estimate (78), we use the trace embedding theorem for the space with a fixed R 0 such that T ⊂ B(0, R 0 ). Indeed, because of (26) 1 and (79), we write σx p3,y (W l,x ) which concludes with the proof of the proposition.

Proposition 6
For each fixedx ∈ Σ , C(x) defined by (25) is a symmetric and positive definite matrix. In addition, its coefficients depend continuously onx ∈ Σ .
Proof Considering (77) the symmetry and positivity of C are due to (5): see the reasoning in Proposition 2. In addition, from (77) and Proposition 5, C ij , for i, j = 1, 2, 3, are continuous functions on Σ, and the proposition is proved.
Applying the same arguments as in (55)-(57), the passage to the limit in (83), gives Accepting that the limit in (84) is given by r 0 Σ C ij u 0 i φ j dx (cf. Theorem 7 below), u 0 satisfies Ω σ ij,x (u 0 )e ij,x (φ) dx + r 0 By a density argument, we conclude that u 0 is the unique solution of (24). Therefore, we have proved the following result.
Finally, we obtain the limit in the right hand side of (84) as a consequence of the following theorem.
Proof We follow the steps in Sect. 6.2 with suitable modifications which we outline below. As a matter of fact, some integrals on T transform into dual products in H −1/2 (T )×H 1/2 (T ) and the corresponding proof must be changed. First, note that the construction of u ε satisfying (85) and (86) repeats the proof in Sect. 6.2.2: indeed, it suffices to take into account that all the integrals over T ε (T respect.) containing u ε ( W l,ε respect.) vanish, as well as the definition (77) of C. Now, we show (87) as follows. We repeat the proof in Sect. 6.2.1 to obtain I ε = − Ω σ ij,x (u ε − u ε )e ij,x ( W l,ε )φ l dx = r ε ij,y (W l, x k )e ij,y (d ε φ l ϕ ε )dy + o(1).

The Spectral Convergence
In this section, we show the convergence of the eigenpairs of (11), when r 0 > 0 and β 0 = +∞.
Proof We follow the scheme of the proof of Theorem 4 with minor modifications. We mainly apply Lemma 1 using the result and proof of Theorem 6 instead of Theorem 3.

Remark 1
In connection with the convergence of solutions in the rest of the cases stated in Sect. 3, we observe that when r 0 = 0 the convergence (48) takes place in (H 1 (Ω)) 3 , and the proof of convergence simplifies providing that u 0 in (18) is the solution of (28). When r 0 = +∞, a very different technique should be applied to show convergence towards the solution of (27): cf. e.g., [15] in the case of scalar problem in porous media.
Remark 2 It should be emphasized that our technique allows us to apply and extend the results in [21] and [6], the regions T ε being stuck to the plane, to the case where the media is heterogeneous and anisotropic. The technique can also be applied to other boundary homogenization problems, both scalar and vector, in heterogeneous media.