On a problem with nonperiodic frequent alternation of boundary condition imposed on fast oscillating sets

TL;DR

This paper investigates a singular perturbed eigenvalue problem with rapidly and nonperiodically alternating boundary conditions on a cylinder, providing estimates for eigenvalue convergence as the perturbation parameter varies.

Contribution

It introduces a general framework for analyzing eigenvalue problems with nonperiodic, fast oscillating boundary conditions, extending previous periodic case results.

Findings

Derived two-sided estimates for eigenvalue convergence.

Established bounds that depend on the oscillation characteristics.

Extended understanding of boundary perturbations in eigenvalue problems.

Abstract

We consider singular perturbed eigenvalue problem for Laplace operator in a cylinder with frequent and nonperiodic alternation of boundary conditions imposed on narrow strips lying in the lateral surface. The width of strips depends on a small parameter in a arbitrary way and may oscillate fast, moreover, the nature of oscillation is arbitrary, too. We obtain two-sided estimates for degree of convergences of the perturbed eigenvalues.