Approximation durch Lösungen partieller Differentialgleichungen
type: Item
instance of: scholarly article
title: Approximation durch Lösungen partieller Differentialgleichungen
zbMATH Open document ID: 0581.35015
DOI: 10.4171/ZAA/154
author: J. Wanka
author: Alfred Göpfert
author: Gert Wanka
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1985-01-01T00:00:00Z
review text: Let \(\Omega \subset {\mathbb{R}}^ m\) (m\(\geq 2)\) be a bounded domain, \[ Lu(x)=\sum^{m}_{i,k=1}a_{ik}(x)\frac{\partial^ 2u}{\partial x_ i\partial x_ k}+\sum^{m}_{i=1}b_ i(x)\frac{\partial u}{\partial x_ i}+c(x)u(x);\quad x=(x_ 1,...,x_ m) \] is an elliptic differential operator, \(a_{ik}\in C^ 2({\bar \Omega})\) \(b_ i\in C^ 1({\bar \Omega})\), \(c\in C({\bar \Omega})\). \(\Gamma \in C^ 1\) is a closed (m-1)-dimensional surface in the interior of \({\bar \Omega}\) and \(\Omega_ i\) the interior part of \(\Gamma\) in \(\Omega\) \((\partial \Omega_ i=\Gamma)\). Suppose, the Dirichlet problem \(Lu=0\) in \(\Omega\), \(u|_{\partial \Omega}=0\) has n (\(\geq 0)\) linearly independent eigensolutions \(u^{(1)},...,u^{(n)}\). Suppose further that the problem \(Lu=0\) in \(\Omega_ i\) and \(u|_{\Gamma}=0\) has p eigensolutions, that of the adjoint problem are denoted by \(\tilde v^{(1)},...,\tilde v^{(p)}\) (which are chosen orthonormal in \(L_ 2(\Omega_ i))\). Consider the following linear and bounded functionals on C(\(\Gamma)\): \[ l_ k(h)=\int_{\Gamma}h(x)(d\tilde v^{(k)}(x)/dn_ x) d\sigma (x)\quad k=1,2,...,p \] where \(n_ x\) is the conormal at x to \(\Gamma\), \(\sigma\) denotes the Lebesgue surface- measure, \[ a(x)=[\sum^{m}_{i=1}(\sum^{m}_{k=1}a_{ik}(x)\cos (n,x_ k))^ 2]^{1/2}. \] The following notation is introduced: \(H(\Gamma)=\cap^{p}_{k=1}\ker l_ k\) if \(p>0\); \(H(\Gamma)=C(\Gamma)\) if \(p=0\) and \(M(\Gamma)=\{u|_{\Gamma}:\) \(L(u)=0\) in \(\Omega\), \(u|_{\partial \Omega}=f\}\), where \(f\in C(\partial \Omega)\) and \(f=0\) on \(\partial \Omega \setminus V\), V is an open subdomain of \(\partial \Omega\). The main result of the paper states that the restrictions to \(\Gamma\) of the solutions of the Dirichlet problem in M(\(\Gamma)\) are dense in the space H(\(\Gamma)\).
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 3929525
zbMATH Keywords: approximation by solutions
zbMATH Keywords: dense subset
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: eigensolutions
reviewed by: Istvan Fenyö
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3929525