Homogenization of the spectral Dirichlet problem for a system of differential equations with rapidly oscillating coefficients and changing sign density

Abstract

We study the asymptotic behavior of the spectrum of the Dirichlet problem for a formally selfadjoint elliptic system of differential equations with rapidly oscillating coefficients and changing sign density ρ. Since the factor ρ at the spectral parameter changes sign, the problem possesses two – positive and negative – infinitely large sequences of eigenvalues. Their asymptotic structure essentially depends on whether the mean $$ \overline \rho $$ over the periodicity cell vanishes. In particular, in the case $$ \overline \rho = 0 $$ , the homogenized problem becomes a quadratic pencil. Bibliography: 20 titles.