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