Compact Sobolev embeddings and torsion functions

Lorenzo Brasco (1) , Berardo Ruffini (2)

1 I2M - Institut de Mathématiques de Marseille
2 X - École polytechnique

For a general open set, we characterize the compactness of the embedding W 1,p 0 → L q 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 details.

Mots clés: Compact embedding; torsional rigidity; Hardy inequalities

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Lorenzo Brasco, Berardo Ruffini. Compact Sobolev embeddings and torsion functions. 2015. ⟨hal-01163464⟩