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

摘要

We consider singular perturbed eigenvalue problem for Laplace operator in atwo-dimensional domain. In the boundary we select a set depending on acharacter small parameter and consisting of a great number of small disjointparts. On this set the Dirichlet boundary condition is imposed while on therest part of the boundary we impose the Neumann condition. For the case ofhomogenized Neumann or Robin boundary value problem we obtain highly weakrestrictions for distribution and lengths of boundary Dirichlet parts of theboundary under those we manage to get the leading terms of asymptoticsexpansions for perturbed eigenelements. We provide explicit formulae for theseterms. Under weaker assumptions we estimate the degrees of convergence forperturbed eigenvalues.