Homogenization of spectral problems in bounded domains with doubly high contrasts

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September 2008, 3(3): 413-436. doi: 10.3934/nhm.2008.3.413

Homogenization of spectral problems in bounded domains with doubly high contrasts

Natalia O. Babych ^1, , Ilia V. Kamotski ^1, and Valery P. Smyshlyaev ^1,

1.	Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom, United Kingdom, United Kingdom

Received April 2008 Published June 2008

Abstract
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Homogenization of a spectral problem in a bounded domain with a high contrast in both stiffness and density is considered. For a special critical scaling, two-scale asymptotic expansions for eigenvalues and eigenfunctions are constructed. Two-scale limit equations are derived and relate to certain non-standard self-adjoint operators. In particular they explicitly display the first two terms in the asymptotic expansion for the eigenvalues, with a surprising bound for the error of order $\varepsilon^{5/4}$ proved.

Keywords: high-contrasts, Homogenization, eigenvalue asymptotics, periodic media.

Mathematics Subject Classification: Primary: 35B27; Secondary: 34.

Citation: Natalia O. Babych, Ilia V. Kamotski, Valery P. Smyshlyaev. Homogenization of spectral problems in bounded domains with doubly high contrasts. Networks & Heterogeneous Media, 2008, 3 (3) : 413-436. doi: 10.3934/nhm.2008.3.413

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