Article Dans Une Revue Calculus of Variations and Partial Differential Equations Année : 2012

On the characterization of the compact embedding of Sobolev spaces

Dorin Bucur
Giuseppe Buttazzo
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Résumé

For every positive regular Borel measure, possibly infinite valued, vanishing on all sets of p-capacity zero, we characterize the compactness of the embedding W1, p(RN) boolean AND L-p(R-N, mu) hooked right arrow 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 Plya and Szego (Isoperimetric inequalities in mathematical physics, Annals of mathematics studies, 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.

Dates et versions

hal-00956619 , version 1 (07-03-2014)

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Dorin Bucur, Giuseppe Buttazzo. On the characterization of the compact embedding of Sobolev spaces. Calculus of Variations and Partial Differential Equations, 2012, 44 (3-4), pp.455-475. ⟨10.1007/s00526-011-0441-8⟩. ⟨hal-00956619⟩
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