Compact Sobolev embeddings and torsion functions

Lorenzo Brasco; Berardo Ruffini

doi:10.1016/j.anihpc.2016.05.005

JournalsaihpcVol. 34, No. 4pp. 817–843

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, France
Berardo 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 $D_{0}^{1, p} ↪ L^{q}$ in terms of the summability of its torsion function. In particular, for $1 \leq q < p$ 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 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