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.

We consider boundary value problems for the Laplace operator in a domain with boundary conditions of rapidly varying type: the Dirichlet homogeneous condition and the third (Fourier) boundary condition or a Steklov type condition. We construct the limit (homogenized) problem and prove that solutions, eigenvalues, and eigenfunctions of the original problem converge respectively to solutions, eigenvalues, and eigenfunctions of the limit problem. Bibliography: 47 titles. Illustrations: 2 figures.

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In the paper, we study the Steklov-type problem for the system of elasticity with rapidly changing boundary condition depending on a small parameter . We construct the homogenized (limit) problem and estimate the rate of convergence of the eigenvalues and eigenfunctions of the initial problem