Remarks on a lemma by Jacques-Louis Lions

Chérif Amrouche (1) , Philippe G. Ciarlet (2) , Cristinel Mardare (3)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 HKU - Department of Mathematics
3 LJLL - Laboratoire Jacques-Louis Lions

Let Ω be a bounded and connected open subset of R^Nwith a Lipschitz-continuous boundary ∂Ω, the set Ω being locally on one side of ∂Ω. It is shown in this Note that a fundamental characterization of the space L^2(Ω) due to Jacques-Louis Lions is in effect equivalent to a variety of other properties. One of the keys for establishing these equivalences is a specific “approximation lemma”, itself one of these equivalent properties.

Mots clés: surjectivity of the divergence operator; Necas inequality; de Rham theorem; weak Poincare lemma; J.L. Lions lemma

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

Chérif Amrouche, Philippe G. Ciarlet, Cristinel Mardare. Remarks on a lemma by Jacques-Louis Lions. Comptes Rendus. Mathématique, 2014, 352 (9), pp.691-695. ⟨10.1016/j.crma.2014.08.003⟩. ⟨hal-01077274⟩