Calculus of Variations and Geometric Measure Theory
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L. Brasco - B. Ruffini

Compact Sobolev embeddings and torsion functions

created by brasco on 18 May 2015
modified on 22 Aug 2016


Accepted Paper

Inserted: 18 may 2015
Last Updated: 22 aug 2016

Journal: Ann. Inst. H. Poincaré Anal. Non Linéaire
Pages: 29
Year: 2015


For a general open set, we characterize the compactness of the embedding for the homogeneous Sobolev space $\mathcal{D}^{1,p}_0\hookrightarrow L^q$ 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 torsional Hardy inequality that we investigate in details.

Keywords: compact embedding, Torsional rigidity, Hardy inequalities


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