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