Compact Sobolev embeddings and torsion functions
Lorenzo Brasco
Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy; Institut de Mathématiques de Marseille, Aix-Marseille Université, Marseille, FranceBerardo Ruffini
Institut Montpelliérain Alexander Grothendieck, University of Montpellier, CNRS, 34095 Montpellier Cedex 5, France
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Abstract
For a general open set, we characterize the compactness of the embedding for the homogeneous Sobolev space in terms of the summability of its torsion function. In particular, for 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 detail.
Cite this article
Lorenzo Brasco, Berardo Ruffini, Compact Sobolev embeddings and torsion functions. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 4, pp. 817–843
DOI 10.1016/J.ANIHPC.2016.05.005