Spectral radius properties for layer potentials associated with the elastostatics and hydrostatics equations in nonsmooth domains
type: Item
instance of: scholarly article
title: Spectral radius properties for layer potentials associated with the elastostatics and hydrostatics equations in nonsmooth domains
zbMATH Open document ID: 0932.45004
DOI: 10.1007/BF01259379
author: Irina Mitrea
published in: The Journal of Fourier Analysis and Applications
publication date: 2000-01-03T00:00:00Z
full work available at URL: https://eudml.org/doc/59614
review text: The double layer electrostatics operator corresponding to the pseudostress conormal derivative and the double layer hydrostatics operator corresponding to the stress conormal derivative are studied. Let \(K\) be one of those operators, and let \(\Omega\) be a bounded Lipschitz domain in \(\mathbb R^n\) with the Lipschitz character \(char (\Omega)\). It is proved that there exists a constant \(C_0\) such that if \(char (\Omega) < C_0\) then for any \(\beta\in\mathbb C\), \(|\beta|\geq 1\), \(\beta\neq 1\), the operator \((\beta I - K)\) is invertible on \(L^2(\partial\Omega)\) where \(I\) is the identity operator. Moreover, in the electrostatics case the operator \(I - K\) is Fredholm with index zero on \(L^2(\partial\Omega)\) and \(I - K^{\ast}\) is invertible on \(L^2_0(\partial\Omega)\), the subspace of functions with zero mean value on \(\partial\Omega\); in the hydrostatics case again the operator \(I - K\) is Fredholm with index zero on \(L^2(\partial\Omega)\) but \(I - K^{\ast}\) is invertible now on \(L^2_{\psi}(\partial\Omega)\), the subspace of functions whose weighted mean values, with certain weights, are zero. For \(\Omega\) being a bounded, simply connected Lipschitz polyhedron \(\mathcal P\) in \(\mathbb R^3\), the same conclusions are drawn under certain size restrictions for polyhedron's dihedral angles.
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45C05
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1340864
zbMATH Keywords: layer potentials
zbMATH Keywords: nonsmooth domains
zbMATH Keywords: spectral radius
zbMATH Keywords: double layer electrostatics operator
zbMATH Keywords: double layer hydrostatics operator
zbMATH Keywords: Fredholm operator
reviewed by: M. Shapiro
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1340864