Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole

Abstract

Let ${\mathbb{A}}(\epsilon)$ be the annular domain obtained by removing from a bounded open domain ${\mathbb{I}}^{o}$ of ℝⁿ a small cavity of size ϵ>0. Then we assume that for some natural index l, $\lambda_{l}[{\mathbb{I}}^{o}]>0$ is a simple Neumann eigenvalue of −Δ in ${\mathbb{I}}^{o}$, and we show that there exists a real valued real analytic function $\hat{\lambda }_{l}(\cdot,\cdot)$ defined in an open neighborhood of (0,0) in ℝ² such that the lth Neumann eigenvalue $\lambda_{l}[{\mathbb{A}}(\epsilon)]$ of −Δ in ${\mathbb{A}}(\epsilon)$ equals $\hat{\lambda}_{l}(\epsilon,\kappa_{n}\epsilon\log\epsilon)$ and such that $\hat{\lambda}_{l}(0,0)= \lambda_{l}[{\mathbb{I}}^{o}]$. Here κ _n =1 if n is even and κ _n =0 if n is odd. Thus in particular, we show that if n is even $\lambda_{l}[{\mathbb {A}}(\epsilon)]$ can be expanded into a convergent double series of powers of ϵ and ϵlogϵ and that if n is odd $\lambda_{l}[{\mathbb{A}}(\epsilon)]$ can be expanded into a convergent series of powers of ϵ. Then related statements have been proved for corresponding eigenfunctions.