báo cáo hóa học:" Research Article Two-Scale Convergence of Stekloff Eigenvalue Problems in Perforated Domains" pot

Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 853717, 15 pages doi:10.1155/2010/853717 Research Article Two-Scale Convergence of Stekloff Eigenvalue Problems in Perforated Domains Hermann Douanla Department of Mathematical Sciences, Chalmers University of Technology, 41296 Gothenbur g, Sweden Correspondence should be addressed to Hermann Douanla, douanla@chalmers.se Received 31 July 2010; Accepted 11 November 2010 Academic Editor: Gary Lieberman Copyright q 2010 Hermann Douanla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By means of t he two-scale convergence method, we investigate the asymptotic behavior of eigenvalues and eigenfunctions of Stekloff eigenvalue problems in perforated domains. We prove a concise and precise homogenization result including convergence of gradients of eigenfunctions which improves the understanding of the asymptotic behavior of eigenfunctions. It is also justified that the natural local problem is not an eigenvalue problem. 1. Introduction We are interested in the spectral asymptotics as ε → 0 of the linear elliptic eigenvalue problem − N  i,j1 ∂ ∂x i  a ij  x, x ε  ∂u ε ∂x j   0, in Ω ε , N  i,j1 a ij  x, x ε  ∂u ε ∂x j ν i  λ ε u ε , on ∂T ε , u ε  0, on ∂Ω, ε  S ε | u ε | 2 dσ ε  x   1, 1.1 where Ω is a bounded open set in N x the numerical space of variables x x 1 , ,x N , with integer N ≥ 2 with Lipschitz boundary ∂Ω, a ij ∈CΩ; L ∞  N y  1 ≤ i, j ≤ N,with 2 Boundary Value Problems the symmetry condition a ji  a ij , the period icity hypothesis: for each x ∈ Ω and for every k ∈ N one h as a ij x, yka ij x, y almost everywhere in y ∈ N y , and finally the ellipticity condition: there exists α>0suchthatforanyx ∈ Ω Re N  i,j1 a ij  x, y  ξ j ξ i ≥ α | ξ | 2 1.2 for all ξ ∈ N and for almost all y ∈ N y ,where|ξ| 2  |ξ 1 | 2  ··· |ξ N | 2 . The set Ω ε ε>0 is a domain perforated as follows. Let T ⊂ Y 0, 1 N be a compact subset in N y with smooth boundary ∂T≡ S and nonempty interior. For ε>0, we define t ε   k ∈ N : ε  k  T  ⊂ Ω  , T ε   k∈t ε ε  k  T  , Ω ε Ω\ T ε . 1.3 In this setup, T is the reference hole, whereas εk T is a h ole of size ε and T ε is the collection of the holes of the perforated domain Ω ε . The family T ε is made up with a finite number of holes since Ω is bounded. Finally, ν ν i  denotes the outer unit normal vector to ∂T ε ≡ S ε  with respect to Ω ε . The asymptotics of eigenvalue problems has been widely explored. Homogenization of eigenvalue problems in a fixed domain goes back to Kesavan 1, 2.Inperforateddomains it was first considered by Rauch 3 and Rauch and Taylor 4, but the first homogenization results on this topic pertains to Vanninathan 5, where he considered eigenvalue problems for the laplace operator a ij  δ ij Kronecker symbol in perforated domains, and combined asymptotic expansion with Tartar’s energy method to prove homogenization results. Concerning homogenization of eigenvalue problems in perforated domains, we also mention the work of Conca et al. 6, Douanla and Svanstedt 7, Kaizu 8, Ozawa and Roppongi 9, Roppongi 10, and Pastukhova 11 and the references therein. In this paper, we deal with the spectral asymptotics of Stekloff eigenvalue problems for an elliptic linear differential operator of order two in divergence form with variable coefficients depending on the macroscopic variable and one microscopic variable. We obtain a very accurate, precise, and concise homogenization result Theorem 3.7 by using the two-scale convergence method 12–16 introduced by Nguetseng 15 and further developed by Allaire 12. A convergence result for gradients of eigenfunctions is provided, which improves the understanding of the asymptotic behavior of eigenfunctions. We also justify that the natural local problem is not an eigenvalue problem. Unless otherwise specified, vector spaces throughout are considered over the complex field, , and scalar functions are assumed to take complex values. Let us recall some basic notations. Let Y 0, 1 N ,andletF N  be a given function space. We denote by F per Y the space of functions in F loc  N  that are Y -periodic a nd by F # Y the space of those functions u ∈ F per Y with  Y uydy  0. Finally, the letter E denotes throughout a family of strictly positive real numbers 0 <ε≤ 1 admitting 0 as accumulation point. The numerical space N and its open sets are provided with the Lebesgue measure denoted by dx  dx 1 ···dx N . Boundary Value Problems 3 The rest of the paper is organized as follows. In Section 2,werecallsomeresultsabout the two-scale convergence method, and the homogenization process is consider in Section 3. 2. Two-Scale Convergence on Periodic Surfaces We first recall the definition and the main compactness theorems of the two-scale convergence method. Let Ω be an open bounded set in N x integer N ≥ 2 and Y 0, 1 N ,theunitcube. Definition 2.1. Asequenceu ε  ε∈E ⊂ L 2 Ω is said to two-scale c onverge in L 2 Ω to someu 0 ∈ L 2 Ω × Y if, as E  ε → 0,  Ω u ε  x  φ  x, x ε  dx −→  Ω×Y u 0  x, y  φ  x, y  dx dy 2.1 for all φ ∈ L 2 Ω; C per Y. Notation 1. We express this by writing u ε 2s −−→ u 0 in L 2 Ω. The following theorem is the backbone of the two-scale convergence method. Theorem 2.2. Let u ε  ε∈E be a bounded sequence in L 2 Ω. Then, a subsequence E  can be extracted from E such that, as E   ε → 0, the sequence u ε  ε∈E  two-scale converges in L 2 Ω to some u 0 ∈ L 2 Ω × Y. Here follows the cornerstone of two-scale convergence. Theorem 2.3. Let u ε  ε∈E be a bounded sequence in H 1 Ω. Then, a subsequence E  can be extracted from E such that, as E   ε → 0, u ε −→ u 0 , in H 1  Ω  -weak, u ε −→ u 0 , in L 2  Ω  , ∂u ε ∂x j 2s −−→ ∂u 0 ∂x j  ∂u 1 ∂y j , in L 2  Ω   1 ≤ j ≤ N  , 2.2 where u 0 ∈ H 1 Ω and u 1 ∈ L 2 Ω; H 1 # Y. In the sequel, we denote by dσyy ∈ Y, dσ ε xx ∈ Ω,ε∈ E, the surface measures on S and S ε , respectively. The surface measure of S is denoted by |S|. The space of squared integrable functions, with respect to the previous measures on S and S ε are denoted by L 2 S and L 2 S ε , respectively. Since the volume of S ε grows proportionally to 1/ε as ε → 0, we endow L 2 S ε  with the scaled scalar product 17  u, v  L 2 S ε   ε  S ε u  x  v  x  dσ ε  x   u, v ∈ L 2  S ε   . 2.3 Definition 2.1 then generalizes as. 4 Boundary Value Problems Definition 2.4. Asequenceu ε  ε∈E ⊂ L 2 S ε  is said to two-scale converge to some u 0 ∈ L 2 Ω × S if as follows. E  ε → 0, ε  S ε u ε  x  φ  x, x ε  dσ ε  x  −→  Ω×S u 0  x, y  φ  x, y  dx dσ  y  2.4 for all φ ∈C Ω; C per Y. The following result paves the way of the general version of Theorem 2.2. Lemma 2.5. Let φ ∈C Ω; C per Y. Then, we have ε  S ε     φ  x, x ε      2 dσ ε  x  ≤ Cφ 2 ∞ 2.5 for some constant C independent of ε and, as E  ε → 0, ε  S ε     φ  x, x ε      2 dσ ε  x  −→  Ω×S   φ  x, y    2 dx dσ  y  . 2.6 Proof. The first part is left to the reader. Let ϕ ∈C Ω and ψ ∈C per Y.Wehave ε  S ε     ϕ  x  ψ  x ε      2 dσ ε  x   ε  k∈t ε  εkS     ϕ  x  ψ  x ε      2 dσ ε  x  . 2.7 Using the second mean value theorem, for any k ∈ t ε ,wehave  εkS     ϕ  x  ψ  x ε      2 dσ ε  x     ϕ  x k    2  εkS     ψ  x ε      2 dσ ε  x  2.8 for some x k ∈ εk  S ⊂ εk  Y .Hence, ε  S ε     ϕ  x  ψ  x ε      2 dσ ε  x   ε  k∈t ε  εkS     ϕ  x  ψ  x ε      2 dσ ε  x   ε  k∈t ε    ϕ  x k     2  εkS     ψ  x ε      2 dσ ε  x   ε  k∈t ε    ϕ  x k     2 ε N−1  kS   ψ  y    2 dσ  y     S   ψ  y    2 dσ  y    k∈t ε ε N    ϕ  x k     2 . 2.9 Boundary Value Problems 5 But, as E  ε → 0,  k∈t ε ε N    ϕ  x k     2 −→  Ω   ϕ  x    2 dx, 2.10 and the proof is completed due to the density of C Ω ⊗C per Y in CΩ; C per Y. Remark 2.6. Even if often used see, e.g., 13, 17, this is the first time Lemma 2.5 is rigorously proved. It is worth noticing that because of a trace issue one cannot replace therein the space C Ω; C per Y by L 2 Ω; C per Y. Theorem 2.2 generalizes as follows. Theorem 2.7. Let u ε  ε∈E be a sequence in L 2 S ε  such that ε  S ε | u ε  x | 2 dσ ε  x  ≤ C, 2.11 where C is a positive constant independent of ε. There exists a subsequence E  of E such that u ε  ε∈E  two-scale converges to some u 0 ∈ L 2 Ω; L 2 S in the sense of Definition 2.4. Proof. Put F ε φε  S ε u ε xφx, x/εdσ ε x for φ ∈CΩ; C per Y.Wehave   F ε  φ    ≤ C  ε  S ε     φ  x, x ε      2 dσ ε  x   1/2 ≤ Cφ ∞ , 2.12 which allows us to view F ε as a continuous linear form on CΩ; C per Y. Hence, there exists a bounded sequence of measures μ ε  ε∈E such that F ε φμ ε ,φ. Due to the separability of C Ω; C per Y there exists a subsequence E  of E such that in the weak ∗ topology of the dual of C Ω; C per Y we have μ ε → μ 0 as E   ε → 0. A limit passage E   ε → 0 in 2.12 yields    μ 0 ,φ    ≤ C   Ω×S   φ  x, y    2 dx dσ  y   1/2 . 2.13 But μ 0 is a continuous form on L 2 Ω; L 2 S by density of CΩ; C per Y in the later space, and there exists u 0 ∈ L 2 Ω; L 2 S such that  μ 0 ,φ    Ω×S u 0  x, y  φ  x, y  dx dσ  y  2.14 for all φ ∈C Ω; C per Y, which completes the proof. In the case when u ε  ε∈E is the sequence of traces on S ε of functions in H 1 Ω, a link can be established between its usual and surface two-scale limits. The following proposition whose proof’s outlines can be found in 13 clarifies this. 6 Boundary Value Problems Proposition 2.8. Let u ε  ε∈E ⊂ H 1 Ω be such that u ε  L 2 Ω  εDu ε  L 2 Ω N ≤ C, 2.15 where C is a positive constant independent of ε and D denotes the usual gradient. The sequence of traces of u ε  ε∈E on S ε satisfies ε  S ε | u ε  x | 2 dσ ε  x  ≤ C  ε ∈ E  , 2.16 and up to a subsequence E  of E, it two-scale converges in the sense of Definition 2.4 to some u 0 ∈ L 2 Ω; L 2 S which is nothing but the trace on S of the usual two-scale limit, a function in L 2 Ω; H 1 # Y. More precisely, as E   ε → 0, ε  S ε u ε  x  φ  x, x ε  dσ ε  x  −→  Ω×S u 0  x, y  φ  x, y  dx dσ  y  ,  Ω u ε  x  φ  x, x ε  dx dy −→  Ω×Y u 0  x, y  φ  x, y  dx dy 2.17 for all φ ∈C Ω; C per Y. 3. Homogenization Procedure We make use of the notations introduced earlier in Section 1. Before we proceed we need a few details. 3.1. Preliminaries We introduce the characteristic function χ G of G  N y \ Θ3.1 with Θ  k∈ N  k  T  . 3.2 It follows from the closeness of T that Θ is closed in N y so that G is an open subset of N y . Next, let ε ∈ E be arbitrarily fixed, and define V ε   u ∈ H 1  Ω ε  : u  0on∂Ω  . 3.3 We equip V ε with the H 1 Ω ε -norm which makes it a Hilbert space. We recall the following classical result 18. Boundary Value Problems 7 Proposition 3.1. For each ε ∈ E there exists a n operator P ε of V ε into H 1 0 Ω with the following properties: i P ε sends continuously and linearly V ε into H 1 0 Ω; iiP ε v| Ω ε  v for all v ∈ V ε ; iii DP ε v L 2 Ω N ≤ cDv L 2 Ω ε  N for all v ∈ V ε ,wherec is a constant independent of ε and D denotes the usual gradient operator. It is also a well-known fact that, under the hypotheses mentioned earlier in the introduction, the spectral problem 1.1 has an increasing sequence of eigenvalues {λ k ε } ∞ k1 , 0 <λ 1 ε ≤ λ 2 ε ≤ λ 3 ε ≤···≤ λ n ε , λ n ε −→ ∞, as n −→ ∞. 3.4 It is to be noted that if the coefficients a ε ij are real valued then the first eigenvalue λ ε 1 is isolated. Moreover, to each eigenvalue, λ k ε is attached to an eigenvector u k ε ∈ V ε and {u k ε } ∞ k1 is an orthonormal basis in L 2 S ε . In the sequel, the couple λ k ε ,u k ε  will be referred to as eigencouple without further ado. We finally recall the Courant-Fisher minimax principle which gives a useful as will be seen later characterization of the eigenvalues to problem 1.1. To this end, we introduce the Rayleigh quotient defined, for each v ∈ V ε \{0},by R ε  v    Ω ε  A ε Dv, Dv  dx  S ε | v | 2 dσ ε  x  , 3.5 where A ε is the N 2 -square matrix a ε ij  1≤i,j≤N and D denotes the usual gradient. Denoting by E k k ≥ 0 the collection of all subspaces of dimension k of V ε , the minimax principle is stated as follows: for any k ≥ 1, the k  th eigenvalue to 1.1 is given by λ k ε  min W∈E k  max v∈W\{0} R ε  v    max W∈E k−1  min v∈W ⊥ \ { 0 } R ε  v   . 3.6 In particular, the first eigenvalue satisfies λ 1 ε  min v∈V ε \ { 0 } R ε  v  , 3.7 and every minimum in 3.6 is an eigenvector associated with λ 1 ε . Now, let Q ε Ω\ εΘ.Thisisanopensetin N ,andΩ ε \ Q ε is the intersection of Ω with the collection of the holes crossing the boundary ∂Ω. We have the following result which implies, as will be seen later, that the holes crossing the boundary ∂Ω are of no effects as regards the homogenization process since they are in arbitrary narrow stripe along the boundary. Lemma 3.2 see 19. Let K ⊂ Ω be a compact set independent of ε.Thereissomeε 0 > 0 such that Ω ε \ Q ε ⊂ Ω \ K for any 0 <ε≤ ε 0 . 8 Boundary Value Problems Next, we introduce the space 1 0  H 1 0  Ω  × L 2  Ω; H 1 #  Y   . 3.8 Endowed with th e following norm v 1 0    D x v 0  D y v 1   L 2 Ω×Y  v   v 0 ,v 1  ∈ 1 0  , 3.9 1 0 is a Hilbert space admitting F ∞ 0  DΩ × DΩ ⊗C ∞ # Y as a dense subspace. This being so, for u, v ∈ 1 0 × 1 0 ,let a Ω  u, v   N  i,j1  Ω×Y ∗ a ij  x, y   ∂u 0 ∂x j  ∂u 1 ∂y j  ∂v 0 ∂x j  ∂v 1 ∂y j  dx dy. 3.10 This defines a hermitian, continuous sesquilinear form on 1 0 × 1 0 . We will need the following results. Lemma 3.3. Fix Φψ 0 ,ψ 1  ∈ F ∞ 0 ,anddefineΦ ε : Ω → (ε>0)by Φ ε  x   ψ 0  x   εψ 1  x, x ε   x ∈ Ω  . 3.11 If u ε  ε∈E ⊂ H 1 0 Ω is such that ∂u ε ∂x i 2s −−→ ∂u 0 ∂x i  ∂u 1 ∂y i , in L 2  Ω  1 ≤ i ≤ N  3.12 as E  ε → 0,whereu u 0 ,u 1  ∈ 1 0 ,then a ε  u ε , Φ ε  −→ a Ω  u, Φ  3.13 as E  ε → 0,where a ε  u ε , Φ ε   N  i,j1  Ω ε a ε ij ∂u ε ∂x j ∂Φ ε ∂x i dx. 3.14 Proof. For ε>0, Φ ε ∈DΩ and all the functions Φ ε ε>0 have their supports contained in a fixed compact set K ⊂ Ω. Thanks to Lemma 3.3,thereissomeε 0 > 0suchthat Φ ε  0, in Ω ε \ Q ε  E  ε ≤ ε 0  . 3.15 Boundary Value Problems 9 Using the decomposition Ω ε  Q ε ∪ Ω ε \ Q ε  and the equality Q ε Ω∩ εG,wegetfor E  ε ≤ ε 0 a ε  u ε , Φ ε   N  i,j1  Ω ε a ij  x, x ε  ∂u ε ∂x j ∂Φ ε ∂x i dx  N  i,j1  Q ε a ij  x, x ε  ∂u ε ∂x j ∂Φ ε ∂x i dx  N  i,j1  Ω∩εG a ij  x, x ε  ∂u ε ∂x j ∂Φ ε ∂x i dx  N  i,j1  Ω a ij  x, x ε  χ εG  x  ∂u ε ∂x j ∂Φ ε ∂x i dx  N  i,j1  Ω a ij  x, x ε  χ G  x ε  ∂u ε ∂x j ∂Φ ε ∂x i dx. 3.16 Bear in mind that, as E  ε → 0, we have see, e.g., 19, Lemma 2.4 N  i,j1 ∂u ε ∂x j ∂Φ ε ∂x i 2s −−→ N  i,j1  ∂u 0 ∂x j  ∂u 1 ∂y j  ∂ψ 0 ∂x j  ∂ψ 1 ∂y j  , in L 2  Ω  . 3.17 We also recall that a ij x, yχ G y ∈CΩ; L 2 per Y1 ≤ i, j ≤ N and that Property 2.1 in Definition 2.1 still holds f or f in C Ω; L 2 per Y instead of L 2 Ω; C per Y whenever the two- scale convergence therein is ensured see, e.g., 14,Theorem15.Thus,asE  ε → 0, a ε  u ε , Φ ε   N  i,j1  Ω a ij  x, x ε  χ G  x ε  ∂u ε ∂x j ∂Φ ε ∂x i dx −→ N  i,j1  Ω×Y a ij  x, y  χ G  y   ∂u 0 ∂x j  ∂u 1 ∂y j  ∂ψ 0 ∂x j  ∂ψ 1 ∂y j  dx dy  N  i,j1  Ω×Y ∗ a ij  x, y   ∂u 0 ∂x j  ∂u 1 ∂y j  ∂ψ 0 ∂x j  ∂ψ 1 ∂y j  dx dy  a Ω  u, Φ  , 3.18 which completes the proof. 10 Boundary Value Problems We now construct and point out the main properties of the so-called homogenized coefficients. Let 1 ≤ j ≤ N,andfixx ∈ Ω.Put a  x; u, v   N  i,j1  Y ∗ a ij  x, y  ∂u ∂y j ∂v ∂y i dy, l j  x, v   N  k1  Y ∗ a kj  x, y  ∂v ∂y k dy 3.19 for u, v ∈ H 1 # Y. Equipped with the seminorm N  u   D y u L 2 Y ∗  N  u ∈ H 1 #  Y   , 3.20 H 1 # Y is a pre-Hilbert space that is nonseparate and noncomplete. Let H 1 # Y ∗  be its separated completion with respect to the seminorm N· and i the canonical mapping of H 1 # Y into H 1 # Y ∗ .Werecallthat i H 1 # Y ∗  is a Hilbert space; ii i is linear; iii iH 1 # Y is dense in H 1 # Y ∗ ; iv iu H 1 # Y ∗   Nu for every u in H 1 # Y; v if F is a Banach space and l a continuous linear mapping of H 1 # Y into F, then there exists a unique continuous linear mapping L : H 1 # Y ∗  → F such that l  L ◦ i. Proposition 3.4. Let j  1, ,N,andfixx in Ω. The noncoercive local variational problem u ∈ H 1 #  Y  ,a  x; u, v   l j  x, v  , ∀v ∈ H 1 #  Y  3.21 admits at least one solution. Moreover, if χ j x and θ j x are two solutions, D y χ j  x   D y θ j  x  a.e. in Y ∗ . 3.22 Proof. Proceeding as in the proof of 19, Lemma 2.5, we can prove that there exists a unique hermitian, coercive, continuous sesquilinear form Ax; ·, · on H 1 # Y ∗  × H 1 # Y ∗  such that Ax; iu, iv  ax; u, v for all u, v ∈ H 1 # Y.Basedonv above, we consider the antilinear form l j x, · on H 1 # Y ∗  such that l j x, iu  l j x, u for any u ∈ H 1 # Y. Then, χ j x ∈ H 1 # Y satisfies 3.21 if and only if iχ j x satisfies i  χ j  x   ∈ H 1 #  Y ∗  ,A  x; i  χ j  x   ,V   l j  x, V  , ∀V ∈ H 1 #  Y ∗  . 3.23 But iχ j x is uniquely determined by 3.23see, e.g., 20, page 216.Wededucethat3.21 admits at least one solution, and if χ j x and θ j x are two solutions, then iχ j x  iθ j x, [...]... Equations and Related Topics, vol 446, pp Lecture Notes in Math.370–379, Springer, Berlin, Germany, 1975 4 J Rauch and M Taylor, “Potential and scattering theory on wildly perturbed domains,” Journal of Functional Analysis, vol 18, pp 27–59, 1975 5 M Vanninathan, “Homogenization of eigenvalue problems in perforated domains,” Proceedings of the Indian Academy of Sciences Mathematical Sciences, vol 90, no 3,... periodic setting,” Communications in Mathematical Analysis, vol 11, no 1, pp 61–93, 2011 8 S Kaizu, “Homogenization of eigenvalue problems for the Laplace operator with nonlinear terms in domains in many tiny holes,” Nonlinear Analysis: Theory, Methods & Applications, vol 28, no 2, pp 377–391, 1997 9 S Ozawa and S Roppongi, “Singular variation of domain and spectra of the Laplacian with small Robin conditional... and M Vanninathan, “Existences and location of eigenvalues for fluid-solid structures,” Publications del departemento de matematicas y ciencias de la computation informes tecnicos, Facultad de ciencias fisicas y matematicas, Universitad de Chile, Informe Interno, No MA88-8-352 7 H Douanla and N Svanstedt, “Reiterated homogenization of linear eigenvalue problems in multiscale perforated domains beyond... Allaire, A Damlamian, and U Hornung, Two-scale convergence on periodic surfaces and applications,” in Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media (May 1995), A Bourgeat et al., Ed., pp 15–25, World Scientific, Singapore, 1996 14 D Lukkassen, G Nguetseng, and P Wall, Two-scale convergence, ” International Journal of Pure and Applied Mathematics, vol... 403–429, 1992 10 S Roppongi, “Asymptotics of eigenvalues of the Laplacian with small spherical Robin boundary,” Osaka Journal of Mathematics, vol 30, no 4, pp 783–811, 1993 11 S E Pastukhova, “On the error of averaging for the Steklov problem in a punctured domain,” Differential Equations, vol 31, no 6, pp 975–986, 1995 12 G Allaire, “Homogenization and two-scale convergence, ” SIAM Journal on Mathematical... dx 0 3.45 Thanks to the arbitrariness of ψ0 and the weak derivative formula, we conclude that λk , uk 0 0 is the k th eigencouple to 3.30 and the whole sequence 1/ε λk ε∈E converges ε Finally, by using 3.28 and a similar line of reasoning as in the proof of Lemma 2.5, we arrive at lim ε E ε→0 Sε P ε uk P ε ulε dσε x ε |S| Ω uk ul0 dx 0 3.46 The normalization condition in 3.30 follows thereby, and moreover... Value Problems Regarding ii , pick any χj x solution to the cell problem 3.21 , and put z x ∂uk /∂xj x χj x 0 By multiplying both sides of 3.21 by − ∂uk /∂xj x and then summing over 1 ≤ j ≤ 0 N, we see that z x satisfies 3.31 Hence, i z x i uk x by uniqueness of the solution 1 ∗ to the coercive variational problem in H# Y corresponding to the noncoercive variational problem 3.31 see the proof of Proposition... admits a sequence of eigencouples with similar properties to those of problem 1.1 However, this is also proved by our homogenization process Now, fix k ≥ 1 There exists a constant 0 < c1 < ∞ independent of ε such that 0 < λk ≤ c1 μk , ε ε 3.33 where μk ε min W∈Ek max v∈W\{0} Ωε Sε |Dv|2 dx |uε |2 dσε x , 3.34 Ek still being the collection of subspaces of dimension k of Vε But it is proved in 5, Proposition... orthogonal 0 k basis in L2 Ω References 1 S Kesavan, “Homogenization of elliptic eigenvalue problems I,” Applied Mathematics and Optimization, vol 5, no 2, pp 153–167, 1979 2 S Kesavan, “Homogenization of elliptic eigenvalue problems II,” Applied Mathematics and Optimization, vol 5, no 3, pp 197–216, 1979 Boundary Value Problems 15 3 J Rauch, “The mathematical theory of crushed ice,” in Partial Differential... Proposition 12.1 that 0 < μk < c2 ε, c2 being a constant independent of ε Hence the ε sequence 1/ε λk ε∈E is bounded in ε Clearly, for fixed E ε > 0, uk lies in Vε and ε N i,j 1 Ωε aε ij ∂uk ∂v ε dx ∂xj ∂xi 1 k λ ε ε ε Sε uk v dσε x ε 3.35 Boundary Value Problems 13 1, and chose v uk in 3.35 The for any v ∈ Vε Bear in mind that ε Sε |uk |2 dσε x ε ε k boundedness of the sequence 1/ε λε ε∈E and the ellipticity . Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 853717, 15 pages doi:10.1155/2010/853717 Research Article Two-Scale Convergence of Stekloff Eigenvalue Problems in. perturbed domains,” Journal of Functional Analysis, vol. 18, pp. 27–59, 1975. 5 M. Vanninathan, “Homogenization of eigenvalue problems in perforated domains,” Proceedings of the Indian Academy of Sciences cited. By means of t he two-scale convergence method, we investigate the asymptotic behavior of eigenvalues and eigenfunctions of Stekloff eigenvalue problems in perforated domains. We prove a