A shape of eigenfunction of the Laplacian under singular variation of domains
type: Item
instance of: scholarly article
title: A shape of eigenfunction of the Laplacian under singular variation of domains
zbMATH Open document ID: 0551.35065
author: Shin Ozawa
published in: Proceedings of the Japan Academy. Series A
publication date: 1983-01-01T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(R^ 3\) with smooth boundary \(\partial \Omega\), and let \(\Omega_{\epsilon}\) be obtained by removing from \(\Omega\) a ball of radius \(\epsilon\) and centre a point w in \(\Omega\). The paper considers the eigenfunctions \(\phi_ j\) and \(\phi_{j,\epsilon}\) in the Dirichlet problems for the Laplacian in \(\Omega\) and \(\Omega_{\epsilon}\). The situation where \(\epsilon\) \(\to 0\) is considered. It is stated that, if \(\phi_ j\) corresponds to a simple eigenvalue, the normal derivative of \(\phi_{j,\epsilon}\) on the boundary of the ball satisfies \(\partial \phi_{j,\epsilon}/\partial n=- \phi_ j(w)\epsilon^{-1}+O(\epsilon^{-1/3}).\) It is also stated that the details of the proof will appear elsewhere.
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 3877656
zbMATH Keywords: eigenfunctions
zbMATH Keywords: Dirichlet problems
zbMATH Keywords: Laplacian
MaRDI profile type: Publication
full work available at URL: pjaa.59.315
OpenAlex ID: W2052577101
cites work: Q3963664
cites work: ASYMPTOTICS OF SOLUTIONS OF THE DIRICHLET PROBLEM IN A DOMAIN WITH A TRUNCATED THIN TUBE
cites work: Singular variation of domains and eigenvalues of the Laplacian
cites work: Q3312627
cites work: Q3327038
cites work: On an elaboration of M. Kac's theorem concerning eigenvalues of -\(\Delta\) in a region with randomly distributed small obstacles
DOI: 10.3792/PJAA.59.315
description: scientific article; zbMATH DE number 3877656