Asymptotic Analysis of Boundary-Value Problems for the Laplace Operator with Frequently Alternating Type of Boundary Conditions

Abstract

This work, which can be considered as a small monograph, is devoted to the study of twoand three-dimensional boundary-value problems for eigenvalues of the Laplace operator with frequently alternating type of boundary conditions. The main goal is to construct asymptotic expansions of the eigenvalues and eigenfunctions of the considered problems. Asymptotic expansions are constructed on the basis of original combinations of asymptotic analysis methods: the method of matching asymptotic expansions, the boundary layer method and the multi-scale method. We perform the analysis of the coefficients of the formally constructed asymptotic series. For strictly periodic and locally periodic alternation of the boundary conditions, the described approach allows one to construct complete asymptotic expansions of the eigenvalues and eigenfunctions. In the case of nonperiodic alternation and the averaged third boundary condition, sufficiently weak conditions on the alternation structure are obtained, under which it is possible to construct the first corrections in the asymptotics for the eigenvalues and eigenfunctions. These conditions include in consideration a wide class of different cases of nonperiodic alternation. With further, very serious weakening of the conditions on the structure of alternation, it is possible to obtain two-sided estimates for the rate of convergence of the eigenvalues of the perturbed problem. It is shown that these estimates are unimprovable in order. For the corresponding eigenfunctions, we also obtain unimprovable in order estimates for the rate of convergence.