Dual semigroups and second order linear elliptic boundary value problems
type: Item
instance of: scholarly article
title: Dual semigroups and second order linear elliptic boundary value problems
zbMATH Open document ID: 0535.35017
DOI: 10.1007/BF02774019
author: Herbert Amann
published in: Israel Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: It is shown that second order elliptic boundary value problems on a bounded domain \(\Omega\) of \({\mathbb{R}}^ n\) generate analytic semigroups in \(L_ 1(\Omega)\). The proof is based on Phillip's theory of dual semigroups, which allows to consider \(L_ 1(\Omega)\) as a ''reflexive'' space with ''dual'' \(C({\bar \Omega})\). There are also given several sharp growth estimates for the corresponding semigroups in \(L_ p\), \(1\leq p<\infty\), as well as necessary and sufficient conditions that the semigroups to be contraction semigroups in every \(L_ p\), \(1\leq p<\infty\). Moreover the questions of positive resolvents (''maximum principle'') and the existence of eigenvalues with positive eigenfunctions are studied in great generality.
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 3849686
zbMATH Keywords: dual semigroups
zbMATH Keywords: analytic semigroups
zbMATH Keywords: growth estimates
zbMATH Keywords: positive resolvents
zbMATH Keywords: positive eigenfunctions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02774019
OpenAlex ID: W2027045131
description: scientific article; zbMATH DE number 3849686