Perturbation formula of eigenvalues in singularly perturbed domain
type: Item
instance of: scholarly article
title: Perturbation formula of eigenvalues in singularly perturbed domain
zbMATH Open document ID: 0785.35069
author: Shuichi Jimbo
published in: Journal of the Mathematical Society of Japan
publication date: 1994-04-27T00:00:00Z
review text: The author deals with partial degeneration of a domain (the Dumbbell shaped domain) and an asymptotic behavior of a certain class of eigenvalues of the Laplacian with Neumann boundary conditions. For small \(\zeta>0\), the domain \(\Omega(\zeta)\) is expressed as follows \(\Omega(\zeta)=D_ 1\cup D_ 2\cup Q(\zeta)\subset\mathbb{R}^ n\) where \(Q(\zeta)\) is a thin cylindrical region which approaches a line segment \(L\) as \(\zeta\to 0\). The set of the eigenvalues \(\{\mu(\zeta)\}^ \infty_{m=1}\) of Laplacian (with Neumann boundary conditions) is expressed as follows, \[ \{\mu_ k(\zeta)\}^ \infty_{k=1}=\{\omega_ k(\zeta)\}^ \infty_{k=1}\cup\{\lambda_ k (\zeta)\}^ \infty_{k=1},\;\lim_{\zeta\to 0}\omega_ k(\zeta)=\omega_ k,\;\lim_{\zeta\to 0}\lambda_ k(\zeta)=\lambda_ k,\;(k\geq 1), \] where \(\omega_ k\) is the \(k\)th eigenvalue of \(-\Delta\) in \(D_ 1\cup D_ 2\) (Neumann boundary conditions) and \(\lambda_ k\) is the \(k\)th eigenvalue of \(-d^ 2/dz^ 2\) in \(L\) with Dirichlet boundary conditions. The purpose of this paper is to give some elaborate characterization of \(\omega_ k(\zeta)\) when \(\zeta\to 0\). Precisely speaking the value of limit \(\lim_{\zeta\to 0}{\omega_ k(\zeta)-\omega_ k\over\zeta^{n- 1}}\) is given for any \(k\geq 1\). For the study of the asymptotic behavior of the eigenfunctions of the above eigenvalue problem [see the author, J. Differ. Equations 77, No. 2, 322-350 (1989; Zbl 0703.35138)].
reviewed by: Shuichi Jimbo
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 469066
zbMATH Keywords: Neumann boundary conditions
zbMATH Keywords: Dumbbell shaped domain
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: eigenvalues
zbMATH Keywords: Laplacian
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04520339
OpenAlex ID: W2068582328
DOI: 10.2969/JMSJ/04520339
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description: scientific article; zbMATH DE number 469066