Asymptotics and Estimates for Eigenelements of Laplacian with Frequent Nonperiodic Interchange of Boundary Conditions

Abstract

We consider singular perturbed eigenvalue problem for Laplace operator in a two-dimensional domain. In the boundary we select a set depending on a character small parameter and consisting of a great number of small disjoint parts. On this set the Dirichlet boundary condition is imposed while on the rest part of the boundary we impose the Neumann condition. For the case of homogenized Neumann or Robin boundary value problem we obtain highly weak restrictions for distribution and lengths of boundary Dirichlet parts of the boundary under those we manage to get the leading terms of asymptotics expansions for perturbed eigenelements. We provide explicit formulae for these terms. Under weaker assumptions we estimate the degrees of convergence for perturbed eigenvalues.