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The paper deals with homogenization of a spectral problem for a second order self-adjoint elliptic operator stated in a thin cylinder with homogeneous Neumann boundary condition on the lateral boundary and Dirichlet condition on the bases of the cylinder. We assume that the operator coefficients and the spectral density function are locally periodic in the axial direction of the cylinder, and that the spectral density function changes sign. We show that the behavior of the spectrum depends essentially on whether the average of the density function is zero or not. In both cases we construct the effective 1-dimensional spectral problem and prove the convergence of spectra.

We consider the homogenization of a singularly perturbed selfadjoint fourth order elliptic equation with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order order effective operators with constant coefficients, depending on the large parameters.