Compact Sobolev embeddings and torsion functions

TL;DR

This paper characterizes when Sobolev embeddings are compact based on torsion functions, revealing that for certain exponents, continuity and compactness are equivalent, and introduces a torsional Hardy inequality.

Contribution

It provides a new characterization of compact Sobolev embeddings using torsion functions and develops a torsional Hardy inequality for general open sets.

Findings

Embedding is continuous iff it is compact for 1 ≤ q < p.

Compactness characterized by torsion function summability.

Introduces and analyzes a torsional Hardy inequality.

Abstract

For a general open set, we characterize the compactness of the embedding $W^{1,p}_0\hookrightarrow L^q$ in terms of the summability of its torsion function. In particular, for $1\le q<p$ we obtain that the embedding is continuous if and only if it is compact. The proofs crucially exploit a {\it torsional Hardy inequality} that we investigate in details.