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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

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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