Singularly perturbed spectral problems in a thin cylinder with Fourier conditions on its bases

The paper deals with the bottom of the spectrum of a singularly perturbed second order elliptic operator defined in a thin cylinder and having locally periodic coefficients in the longitudinal direction. We impose a homogeneous Neumann boundary condition on the lateral surface of the cylinder and a generic homogeneous Fourier condition at its bases. We then show that the asymptotic behavior of the principal eigenpair can be characterized in terms of the limit one-dimensional problem for the effective Hamilton–Jacobi equation with the effective boundary conditions. In order to construct boundary layer correctors we study a Steklov type spectral problem in a semi-infinite cylinder (these results are of independent interest). Under a structure assumption on the effective problem leading to localization (in certain sense) of eigenfunctions inside the cylinder we prove a two-term asymptotic formula for the first and higher order eigenvalues.