On the spectrum of the Neumann problem for Laplace equation in a domain with a narrow slit

TL;DR

This paper analyzes how the eigenvalues of the Laplace equation with Neumann boundary conditions are affected by a narrow slit in the domain, using matched asymptotic expansions to derive accurate asymptotic formulas.

Contribution

It develops a complete asymptotic expansion for the eigenvalues in a domain with a narrow slit, highlighting the necessity of inner expansions over regular perturbation theory.

Findings

Regular perturbation theory fails to give correct eigenvalue asymptotics.

Inner asymptotic expansion is essential for accurate results.

Asymptotic expansion converges to the eigenvalue of the limiting problem.

Abstract

The Neumann problem in two-dimensional domain with a narrow slit is studied. The width of the slit is a small parameter. The complete asymptotic expansion for the eigenvalue of the perturbed problem converging to a simple eigenvalue of the limiting problem is constructed by means of the method of the matched asymptotic expansions. It is shown that the regular perturbation theory can formally be applied in a natural way up to terms of order $\epsilon^2$. However, the result obtained in that way is false. The correct result can be obtained only by means of inner asymptotic expansion.