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

DOI10.3233/ASY-2009-0976zbMATH Open1200.35081arXivmath-ph/0208039MaRDI QIDQ4930609

A. M. Il'in, R. R. Gadyl'shin

Publication date: 1 October 2010

Published in: Asymptotic Analysis (Search for Journal in Brave)

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 $epsilo{n}^2$. However, the result obtained in that way is false. The correct result can be obtained only by means of inner asymptotic expansion.

Full work available at URL: https://arxiv.org/abs/math-ph/0208039

zbMATH Keywords

singular perturbationasymptoticseigenvaluesNeumann problem

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Singular perturbations in context of PDEs (35B25) General topics in linear spectral theory for PDEs (35P05) Estimates of eigenvalues in context of PDEs (35P15) Asymptotic expansions of solutions to PDEs (35C20)

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