On a lemma of Jacques-Louis Lions and its relation to other fundamental results

Abstract

Let Ω be a domain in RN, i.e., a bounded and connected open subset of RN with a Lipschitz-continuous boundary ∂Ω, the set Ω being locally on the same side of ∂Ω. A fundamental lemma, due to Jacques-Louis Lions, provides a characterization of the space L2(Ω), as the space of all distributions on Ω whose gradient is in the space H−1(Ω). This lemma, which provides in particular a short proof of a crucial inequality due to J. Nečas, is also a key for proving other basic results, such as, among others, the surjectivity of the divergence operator acting from H01(Ω) into L02(Ω), a “weak” form of the Poincaré lemma or a “simplified version” of de Rham theorem, each of which provides sufficient conditions insuring that a vector field in H−1(Ω) is the gradient of a function in L2(Ω). The main objective of this paper is to establish an “equivalence theorem”, which asserts that J.L. Lions lemma is in effect equivalent to a number of other fundamental properties, which include in particular the ones mentioned above. The key for proving this theorem is a specific “approximation lemma”, itself one of these equivalent results, which appears to be new to the best of our knowledge. Some of these equivalent properties can be given an independent, i.e., “direct”, proof, such as for instance the constructive proof by M.E. Bogovskii of the surjectivity of the divergence operator. Therefore, the proof of any one of such properties provides, by way of our equivalence theorem, a means of proving J.L. Lions lemma, the known “direct” proofs of which for a general domain are notoriously difficult.