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

TL;DR

This paper investigates the asymptotic behavior of eigenvalues and eigenfunctions of the Laplacian in a 2D domain with a boundary featuring numerous small Dirichlet parts interchanging with Neumann conditions, providing explicit asymptotic formulas.

Contribution

It introduces new asymptotic formulas for perturbed eigenvalues of the Laplacian with complex boundary conditions involving frequent nonperiodic boundary condition interchange.

Findings

Derived explicit asymptotic formulas for eigenvalues.

Established weak restrictions on boundary partition distribution.

Estimated convergence rates of perturbed eigenvalues.

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.