Homogenization of spectral problems in bounded domains with doubly high contrasts

TL;DR

This paper develops a homogenization theory for spectral problems in bounded domains with high contrasts in stiffness and density, deriving two-scale asymptotics and explicit eigenvalue expansions with error bounds.

Contribution

It introduces a novel homogenization approach for spectral problems with doubly high contrasts, including explicit asymptotic eigenvalue formulas and error estimates.

Findings

Derived two-scale limit equations for eigenvalues and eigenfunctions.

Constructed asymptotic expansions with explicit first two terms.

Proved an error bound of order ^{5/4} for the eigenvalue approximation.

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 \epsilon^{5/4} proved.