An Asymptotic Formula for Eigenvalues of the Neumann Laplacian in Domains with a Small Star-shaped Hole

TL;DR

This paper derives an asymptotic formula for how eigenvalues of the Neumann Laplacian in a 2D domain with a small star-shaped hole approach those of the unperforated domain, including convergence rates.

Contribution

It provides the first asymptotic expansion for eigenvalues considering both the size and shape of a small star-shaped hole in a domain.

Findings

Eigenvalues converge to those of the unperturbed domain as the hole shrinks.

An explicit asymptotic expansion for simple eigenvalues is established.

The rate of convergence depends on the size and shape of the hole.

Abstract

This article investigates a spectral problem of the Laplace operator in a two-dimensional bounded domain perforated by a small arbitrary star-shaped hole and on the smooth boundary of which the Neumann boundary condition is imposed. It is proved that the eigenvalues of this problem converge to the eigenvalues of the Laplacian defined on the unperturbed domain as the size of the hole approaches zero. Furthermore, our main theorem provides the rate of convergence by showing an asymptotic expansion for all simple eigenvalues with respect to the size and shape of the hole.