On Friedrichs inequality and Rellich's theorem
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).
scientific article; zbMATH DE number 4155204