On the characterization of the compact embedding of Sobolev spaces

TL;DR

This paper characterizes when the embedding of Sobolev spaces with measures vanishing on sets of zero p-capacity is compact, linking it to PDE behavior and geometric inequalities involving torsional rigidity and the Dirichlet Laplacian.

Contribution

It provides a new characterization of the compactness of Sobolev space embeddings in terms of PDE properties and geometric measures, extending classical results.

Findings

Compactness characterized by PDE behavior

Finite torsional rigidity implies Laplacian resolvent compactness

Measures vanishing on p-capacity zero sets are key

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 $W^{1,p}({\bf R}^N)\cap L^p ({\bf R}^N,\mu)\hr L^q({\bf R}^N)$ 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.