Compact Sobolev embeddings and torsion functions

Lorenzo Brasco; Berardo Ruffini

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arXiv · 1506.04371

Compact Sobolev embeddings and torsion functions

Lorenzo Brasco·Berardo Ruffini

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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.

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Lorenzo Brasco, Berardo Ruffini. 2015-06-14. Compact Sobolev embeddings and torsion functions. https://arxiv.org/abs/1506.04371

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