Localization of eigenfunctions in a thin cylinder with a locally periodic oscillating boundary

Abstract

We study a Dirichlet spectral problem for a second-order elliptic operator with locally periodic coefficients in a thin cylinder. The lateral boundary of the cylinder is assumed to be locally periodic. When the thickness of the cylinder ε tends to zero, the eigenvalues are of order ε−2 and described in terms of the first eigenvalue μ(x1) of an auxiliary spectral cell problem parametrized by x1, while the eigenfunctions localize with rate ε.