Splitting a multiple eigenvalue in the boundary value problem for a membrane clamped on a small part of the boundary
type: Item
instance of: scholarly article
title: Splitting a multiple eigenvalue in the boundary value problem for a membrane clamped on a small part of the boundary
zbMATH Open document ID: 0827.35082
DOI: 10.1007/BF00971218
published in: Siberian Mathematical Journal
publication date: 1994-10-05T00:00:00Z
review text: Let \(\lambda_0\) be an \(m\)-multiple eigenvalue of the Neumann problem for the Laplace operator in a bounded connected domain \(\Omega\) of \(\mathbb{R}^2\) with boundary of class \(C^\infty\) and \(\varphi_i\) \((i = 1, \ldots, m)\) the corresponding eigenfunctions; then there exists \(\varepsilon_m > 0\) such that for \(0 < \varepsilon < \varepsilon_m\) there exist \(m\) simple eigenvalues \(\lambda_{\varepsilon,i}\) of the problem \((\Delta + \lambda_\varepsilon) \varphi = 0\) in \(\Omega\), \(\partial_n \varphi = 0\) on \(\partial \Omega \backslash \overline {\omega_\varepsilon}\), \(\varphi = 0\) on \(\omega_\varepsilon\), where \(\omega_\varepsilon\) is an open connected subset of \(\partial \Omega\) with length \(\varepsilon\), and \(\lambda_{\varepsilon,i} \to \lambda_0\) as \(\varepsilon \to 0\). Moreover the author shows the form of asymptotic expansions of the eigenvalues \(\lambda_{\varepsilon,i}\) and of the corresponding eigenfunctions if \(\sum^m_{i = 1} |\varphi_i (0) |\neq 0\) and if \(\sum^m_{i = 1} |\varphi_i (0) |= 0\).
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35P10
zbMATH DE Number: 641339
zbMATH Keywords: multiple eigenvalue
zbMATH Keywords: Neumann problem for the Laplace operator
zbMATH Keywords: simple eigenvalues
zbMATH Keywords: asymptotic expansion of the eigenvalues
author: R. R. Gadyl'shin
reviewed by: Gianfranco Bottaro
MaRDI profile type: Publication
cites work: Asymptotic Variational Formulae for Eigenvalues
cites work: Singular Hadamard's variation of domains and eigenvalues of the Laplacian. II
description: scientific article; zbMATH DE number 641339