Asymptotic expansion of eigenvalues of the Neumann problem in a domain with a thin bridge
type: Item
instance of: scholarly article
title: Asymptotic expansion of eigenvalues of the Neumann problem in a domain with a thin bridge
zbMATH Open document ID: 0808.35083
DOI: 10.1007/BF00971127
author: O. R. Polyakova
author: Sergei A. Nazarov
published in: Siberian Mathematical Journal
publication date: 1993-03-28T00:00:00Z
review text: This article deals with the spectral problem \[ \begin{aligned} \Delta u(\varepsilon, x)+ \lambda(\varepsilon) u(\varepsilon,x) &= 0\qquad (x\in\Omega_ \varepsilon),\\ \partial_ n u(\varepsilon, x) &=0\qquad (x\in\partial\Omega),\\ (\partial u/ \partial\nu^ \pm) (\varepsilon, x) &=0\qquad (x\in G_ \varepsilon^ \pm), \end{aligned} \] where \(\Omega\) is a domain in \(\mathbb{R}^ 3\) with smooth boundary \(\partial\Omega\), \(G\) is a smooth two-dimensional submanifold of \(G\) with boundary \(\partial G\), \(\Omega_ \varepsilon= \Omega\setminus \omega_ \varepsilon\), \(\omega_ \varepsilon\) is a thin set over \(G\) of `thickness' \(\varepsilon\) in \(\Omega\), \(G_ \varepsilon^ \pm\) are `shores' of \(G_ \varepsilon\). The main goal of this article is to study the asymptotic behavior for eigenvalues \(\lambda(\varepsilon)\) of this problem. To this end the limit problems are determined and then asymptotics of three types \[ \lambda(\varepsilon)\sim \lambda_ 0+ \varepsilon\lambda_ 1, \quad \lambda(\varepsilon)\sim \lambda_ 0+ (\varepsilon \ln\varepsilon) \lambda_ 1, \quad \lambda(\varepsilon)\sim \lambda_ 0+ \varepsilon^{1/2} \lambda_ 1, \] where \(\lambda_ 0\) is an eigenvalue of the limit problem are presented.
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 130832
zbMATH Keywords: Laplace equation
zbMATH Keywords: limit problems
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00971127
OpenAlex ID: W1970246041
description: scientific article; zbMATH DE number 130832