On the characterization of the compact embedding of Sobolev spaces

Details On the characterization of the compact embedding of Sobolev spaces On the characterization of the compact embedding of Sobolev spaces

2009-12-02

arxiv.org/abs/0912.0357v1

Mathematics

Optimization and Control

19 pages

Scientific paper

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.

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