Heat equation for an arbitrary doubly-connected region in \(R^ 2\) with mixed boundary conditions
type: Item
instance of: scholarly article
title: Heat equation for an arbitrary doubly-connected region in \(R^ 2\) with mixed boundary conditions
zbMATH Open document ID: 0704.35133
DOI: 10.1007/BF00945010
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1989-01-01T00:00:00Z
review text: Let \(\Omega\) be an arbitrary doubly-connected region in \(R^ 2\) surrounded internally by a simply connected bounded domain \(\Omega_ 1\) with a smooth boundary \(\partial \Omega_ 1\) and externally by a simply connected bounded domain \(\Omega_ 2\) with a smooth boundary \(\partial \Omega_ 2\). Suppose that the eigenvalues \(0<\lambda_ 1\leq \lambda_ 2\leq...\leq \lambda_ j\leq...\to \infty\) as \(j\to \infty\) are given exactly for the equation \((\Delta +\lambda)u=0\) in \(\Omega\) together with one of the following mixed boundary conditions \(\partial u/\partial n_ 1=0\) on \(\partial \Omega_ 1\), \(u=0\) on \(\partial \Omega_ 2\), \(u=0\) on \(\partial \Omega_ 1\), \(\partial u/\partial n_ 2=0\) on \(\partial \Omega_ 2,\) where \(\partial /\partial n_ 1\) and \(\partial /\partial n_ 2\) denote differentiations along the inward pointing normals to \(\partial \Omega_ 1\) and \(\partial \Omega_ 2\) respectively, while \(\Delta\) is the Laplace operator in \(R^ 2\). Determine the geometry of \(\Omega\) from the expansion of \(\theta (t)=\sum^{\infty}_{j=1}\exp (-t\lambda_ j)\) as \(t\to 0\).
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 4155079
zbMATH Keywords: doubly-connected region
zbMATH Keywords: mixed boundary conditions
zbMATH Keywords: Laplace operator
author: Elsayed M. E. Zayed
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4155079