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

Abstract

International audienceLet Ω be a domain in R^N, i.e., a bounded and connected open subset of R^N 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 L^2(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. Necas, is also a key for proving other basic results, such as, among others, the surjectivity of the divergence operator acting from H^1_0(Ω) into L^2_0(Ω), a “weak” form of the Poincare 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 L^2(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

On the J.L. Lions Lemma and Its Applications to the Maxwell-Stokes Type Problem and the Korn Inequality

“…Remark 3.1. The authors of [1] derived this theorem in L 2 -framework, for example, in (a), in the sense that f ∈ H −1 (Ω) and ∇ f ∈ H −1 (Ω) implies f ∈ L 2 (Ω). Therefore, our Theorem 3.1 is an improvement of [1].…”

“…The authors of [1] derived this theorem in L 2 -framework, for example, in (a), in the sense that f ∈ H −1 (Ω) and ∇ f ∈ H −1 (Ω) implies f ∈ L 2 (Ω). Therefore, our Theorem 3.1 is an improvement of [1]. This improvement is necessary to consider in applications to the Maxwell-Stokes problem containing pcurlcurl equation in Section 4 and the Korn inequality in Section 5.…”

On the curl operator and some characterizations of matrix fields in Lipschitz domains

“…First, we recall some geometry notations. We denote by | property is stated in [6], [17] and proved in [29]. Also, let Ω contained in R 3 be a bounded and connected open set, we recall that Ω is pseudo-Lipschitz if for any point x on the boundary ∂Ω there exist an integer r(x) equal to 1 or 2 and a strictly positive real number ρ 0 such that for all real numbers ρ with 0 < ρ < ρ 0 , the intersection of Ω with the ball with center x and radius ρ, has r(x) connected components, each one being Lipschitz.…”

“…where the vector fields grad q T j are the elements of the basis of K T (Ω) (see [3]). Then, the usual De Rham's Theorem (see [6]) implies that there exists p ∈ H 1 (Ω), unique up to an additive constant, such that…”

Nečas–Lions lemma revisited: An Lp‐version of the generalized Korn inequality for incompatible tensor fields

“…Furthermore, for 1 < p < ∞ , the Lions lemma resp. Nečas estimate (Theorem 1) applied to () yields a variant of Korn's second inequality in W 1, p (Ω) from which, in turn, the first Korn's inequality (with boundary conditions) can be deduced 35–38 using an indirect argument together with the compactness of the dual spaces …”