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Compactness of embeddings of function spaces on quasi-bounded domains and the distribution of eigenvalues of related elliptic operators

Part of: Spectral theory and eigenvalue problems Approximations and expansions Linear function spaces and their duals

Published online by Cambridge University Press:  10 July 2013

Hans-Gerd Leopold  and
Leszek Skrzypczak
Hans-Gerd Leopold
Affiliation:
Mathematisches Institut, Freidrich-Schiller Universität, Ernst Abbe Platz 1–2, 07740 Jena, Germany (hans-gerd.leopold@uni-jena.de)
Leszek Skrzypczak
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland (lskrzyp@amu.edu.pl)
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Abstract

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We prove sufficient and necessary conditions for compactness of the Sobolev embeddings of Besov and Triebel–Lizorkin spaces defined on bounded and unbounded uniformly E-porous domains. The asymptotic behaviour of the corresponding entropy numbers is calculated. Some applications to the spectral properties of elliptic operators are described.

Keywords

compact embeddingsBesov and Triebel–Lizorkin spacesquasi-bounded domainselliptic operatorsdistribution of eigenvalues

MSC classification

Secondary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 3 , October 2013 , pp. 829 - 851
Copyright
Copyright © Edinburgh Mathematical Society 2013 

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