Singular Perturbations of Curved Boundaries in Three Dimensions. The Spectrum of the Neumann Laplacian

Abstract

We calculate the main asymptotic terms for eigenvalues, both simple and multiple, and eigenfunctions of the Neumann Laplacian in a three-dimensional domain $\Omega (h)$ perturbed by a small (with diameter $O(h)$) Lipschitz cavern $\overline{{\omega }_h}$ in a smooth boundary $\partial \Omega =\partial \Omega (0)$. The case of the hole $\overline{{\omega }_h}$ inside the domain but very close to the boundary $\partial \Omega$ is under consideration as well. It is proven that the main correction term in the asymptotics of eigenvalues does not depend on the curvature of $\partial \Omega$ while terms in the asymptotics of eigenfunctions do. The influence of the shape of the cavern to the eigenvalue asymptotics relies mainly upon a certain matrix integral characteristics like the tensor of virtual masses. Asymptotically exact estimates of the remainders are derived in weighted norms.