Homogenization of spectral problems in bounded domains with doubly high contrasts

Pages: 413 - 436, Volume 3, Issue 3, September 2008

doi:10.3934/nhm.2008.3.413       Abstract        Full Text (351.9K)       Related Articles

Natalia O. Babych - Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom (email)
Ilia V. Kamotski - Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom (email)
Valery P. Smyshlyaev - Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom (email)

Abstract: 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:  Homogenization, periodic media, high-contrasts, eigenvalue asymptotics
Mathematics Subject Classification:  Primary: 35B27; Secondary: 34E.

Received: April 2008;      Published: June 2008.