On Friedrichs inequality and Rellich's theorem
type: Item
instance of: scholarly article
title: On Friedrichs inequality and Rellich's theorem
zbMATH Open document ID: 0704.46022
DOI: 10.1016/0022-247X(90)90417-E
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: Given an open set \(\Omega \subseteq {\mathbb{R}}^ n\), denote by K(\(\Omega\)) the number \[ K(\Omega)=\sup \{\| \phi \|:\;\phi \in C_ 0^{\infty}(\Omega),\quad \| \nabla \phi \| \leq 1\}, \] where \(\| \cdot \|\) is the \(L_ 2\)-norm. (Note that \(K(\Omega)<\infty\) iff \(\Omega\) satisfies the Poincaré inequality.) The author proves that \(K(\{x:\;x\in \Omega,\quad \| x\| >r\})\to 0\), as \(r\to \infty\), iff the imbedding \(H^ 1_ 0(\Omega)\hookrightarrow L_ 2(\Omega)\) is compact. A similar characterization for the measure of noncompactness of this imbedding was given by \textit{C. J. Amick} in J. London Math. Soc. II. Ser. 18, 81-93 (1978; Zbl 0391.46029).
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46B99
zbMATH DE Number: 4155204
zbMATH Keywords: Friedrichs inequality
zbMATH Keywords: Rellich's theorem
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: imbedding
zbMATH Keywords: measure of noncompactness
author: Fernando Galaz Fontes
reviewed by: Jürgen Appell
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4155204