On the characterization of the compact embedding of Sobolev spaces
نویسندگان
چکیده
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 (R ) ∩ L(R , μ) ↪→ L(R ) 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 [14] 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. AMS Subject Classification (2000): 49Q10, 49J45, 46E35, 47A10, 74P05
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