Calculus of Variations and Geometric Measure Theory

D. Bucur - G. Buttazzo

On the characterization of the compact embedding of Sobolev spaces

created by buttazzo on 02 Dec 2009
modified by bucur on 23 Jan 2018


Accepted Paper

Inserted: 2 dec 2009
Last Updated: 23 jan 2018

Journal: Calc. Var. and PDEs
Pages: 19
Year: 2012


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.

Keywords: shape optimization, Sobolev spaces, compact embedding, torsion functional