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On a lemma of Jacques-Louis Lions and its relation to other fundamental results. (English. French summary) Zbl 1344.46027

Summary: Let \(\varOmega\) be a domain in \(\mathbb{R}^N\), i.e., a bounded and connected open subset of \(\mathbb{R}^N\) with a Lipschitz-continuous boundary \(\partial\varOmega\), the set \(\varOmega\) being locally on the same side of \(\partial\varOmega\). A fundamental lemma, due to Jacques-Louis Lions [E. Magenes and G. Stampacchia, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 12, 247–358 (1958; Zbl 0082.09601)], provides a characterization of the space \(L^2(\Omega)\), as the space of all distributions on \(\varOmega\) whose gradient is in the space \(\mathbf{H}^{- 1}(\Omega)\). This lemma, which provides in particular a short proof of a crucial inequality due to J. Nečas [“Equations aux dérivées partielles”, Presses de l’Université de Montréal (1965)], is also a key for proving other basic results, such as, among others, the surjectivity of the divergence operator acting from \(\mathbf{H}_0^1(\Omega)\) into \(L_0^2(\Omega)\), a “weak” form of the Poincaré lemma or a “simplified version” of the de Rham theorem, each of which provides sufficient conditions insuring that a vector field in \(\mathbf{H}^{- 1}(\Omega)\) is the gradient of a function in \(L^2(\Omega)\).{ } The main objective of this paper is to establish an “equivalence theorem”, which asserts that J. L. Lions’s 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. Bogovskij [Sov. Math., Dokl. 20, 1094–1098 (1979; Zbl 0499.35022); translation from Dokl. Akad. Nauk SSSR 248, 1037–1040 (1979)] 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’s lemma, the known “direct” proofs of which for a general domain are notoriously difficult.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E05 Lattices of continuous, differentiable or analytic functions
74B05 Classical linear elasticity
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