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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 Linear function spaces and their duals Approximations and expansions

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 

References

1.Adams, R. A. and Fournier, J. J. F., Sobolev spaces (Elsevier, 2003).Google Scholar
2.Carl, B. and Stephani, I., Entropy, compactness and the approximation of operators (Cambridge University Press, 1990).CrossRefGoogle Scholar
3.Clark, C., An embedding theorem for function spaces, Pac. J. Math. 19 (1965), 243–251.CrossRefGoogle Scholar
4.Clark, C. and Hewgill, D., One can hear whether a drum has finite area, Proc. Am. Math. Soc. 18 (1967), 236–237.CrossRefGoogle Scholar
5.Edmunds, D. E. and Evans, W. D., Spectral theory and differential operators (Clarendon Press, Oxford, 1996).Google Scholar
6.Edmunds, D. E. and Triebel, H., Function spaces, entropy numbers, differential operators (Cambridge University Press, 1996).CrossRefGoogle Scholar
7.König, H., Operator properties of Sobolev imbeddings over unbounded domains, J. Funct. Analysis 24 (1977), 32–51.CrossRefGoogle Scholar
8.König, H., Approximation numbers of Sobolev imbeddings over unbounded domains, J. Funct. Analysis 29 (1978), 74–87.CrossRefGoogle Scholar
9.Kühn, T., Leopold, H.-G., Sickel, W. and Skrzypczak, L., Entropy numbers of embeddings of weighted Besov spaces, II, Proc. Edinb. Math. Soc. 49 (2006), 331–359.CrossRefGoogle Scholar
10.Leopold, H.-G., Embeddings and entropy numbers for general weighted sequence space: the non-limiting case, Georgian Math. J. 7 (2000), 731–743.CrossRefGoogle Scholar
11.Leopold, H.-G., Embeddings for general weighted sequence space and entropy numbers, in Function spaces, differential operators and nonlinear analysis (ed. V. Mustonen and J.Rakosnik), pp. 170–186 (Institute of Mathematics of the Academy of Sciences, Prague, 2000).Google Scholar
12.Maurin, K., Abbildungen von Hilbert-Schmidtschen Typus und Ihre Anwendungen, Math. Scand. 9 (1961), 187–215.CrossRefGoogle Scholar
13.Triebel, H., Fractals and spectra (Birkhäuser, 1997).CrossRefGoogle Scholar
14.Triebel, H., Function spaces and wavelet on domains (European Mathematical Society Publishing House, Zurich, 2008).CrossRefGoogle Scholar
15.Weyl, H., Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendungen auf die Theorie der Hohlraumstrahlung), Math. Annalen 71 (1912), 441–479.CrossRefGoogle Scholar