# On the characterization of the compact embedding of Sobolev spaces

created by buttazzo on 02 Dec 2009

[BibTeX]

Submitted Paper

Inserted: 2 dec 2009

Pages: 19
Year: 2009

Abstract:

For every positive regular Borel measure, possibly infinite valued, vanishing on all sets of $p$-capacity zero, we characterize the compactness of the embedding $W^{1,p}(*R*^N)\cap L^p (*R*^N,\mu)\hr L^q(*R*^N)$ 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 Szegö 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.