Geodesic
Compact Sobolev embeddings and torsion functions
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 817-843

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For a general open set, we characterize the compactness of the embedding for the homogeneous Sobolev space D01,p↪Lq in terms of the summability of its torsion function. In particular, for 1≤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.

DOI : 10.1016/j.anihpc.2016.05.005
Classification : 46E35, 35P30, 39B72
Keywords: Compact embedding, Torsional rigidity, Hardy inequalities 
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     title = {Compact {Sobolev} embeddings and torsion functions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {817--843},
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Brasco, Lorenzo; Ruffini, Berardo. Compact Sobolev embeddings and torsion functions. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 817-843. doi: 10.1016/j.anihpc.2016.05.005

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