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On the characterization of the compact embedding of Sobolev spaces

Bucur, D. and Buttazzo, G. (2009) On the characterization of the compact embedding of Sobolev spaces. arXiv .

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SUMMARY For every positive regular Borel measure, possibly infinite valued, vanishing on all sets of $p$-capacity zero, we characterize the compactness of the embedding $W^{1,p}({\bf R}^N)\cap L^p ({\bf R}^N,\mu)\hr L^q({\bf R}^N)$ in terms of the qualitative behavior of some characteristic PDE. This question is related to the well posedness of a class of geometric inequalities involving the torsional rigidity and the spectrum of the Dirichlet Laplacian introduced by Polya and Szeg\"o in 1951. In particular, we prove that finite torsional rigidity of an arbitrary domain (possibly with infinite measure), implies the compactness of the resolvent of the Laplacian.

Item Type: Article
Additional Information: Imported from arXiv
Subjects: Area01 - Scienze matematiche e informatiche > MAT/05 - Analisi matematica
Divisions: Dipartimenti (from 2013) > DIPARTIMENTO DI MATEMATICA
Depositing User: dott.ssa Sandra Faita
Date Deposited: 05 Aug 2013 12:46
Last Modified: 05 Aug 2013 12:46
URI: http://eprints.adm.unipi.it/id/eprint/1327

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