Eigenvalues of the Laplacian for the third boundary value problem: An extension to higher dimensions
type: Item
instance of: scholarly article
title: Eigenvalues of the Laplacian for the third boundary value problem: An extension to higher dimensions
zbMATH Open document ID: 0652.35093
DOI: 10.1016/0022-247X(88)90387-3
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: Let \(\Omega:=\{(r,\theta,\phi):\) \(a\leq r\leq b\), \(0\leq \theta \leq \pi\), \(0\leq \phi \leq 2\pi \}\) be a spherical shell in \({\mathbb{R}}^ 3\). Suppose the spectrum of eigenvalues \(0<\lambda_ 1\leq \lambda_ 2\leq...\leq \lambda_ j\leq...\to \infty\) is known exactly for the impedance problem \[ (\Delta +\lambda)u=0\quad in\quad \Omega,\quad | (\partial u/\partial r+\gamma_ 1u)_{r=a}=0,\quad | \partial u/\partial r+\gamma_ 2u|_{r=b}=0, \] where \(\gamma_ 1\) and \(\gamma_ 2\) are positive constants. The basic problem is that of determining the geometry of the spherical shell \(\Omega\) and the impedances \(\gamma_ 0\) and \(\gamma_ 1\). This problem is attacked by a careful analysis of the asymptotic behaviour of the trace function \(\theta (f):=\sum^{\infty}_{j=1}e^{-\lambda_ jt}\) for small positive t.
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 4063690
zbMATH Keywords: spectrum of eigenvalues
zbMATH Keywords: impedance problem
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: trace function
author: Elsayed M. E. Zayed
reviewed by: G. A. Philippin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4063690