On a Vector Version of a Fundamental Lemma of J. L. Lions


Philippe G. CIARLET,Maria MALIN,Cristinel MARDARE.On a Vector Version of a Fundamental Lemma of J. L. Lions[J].Chinese Annals of Mathematics B,2018,39(1):33~46
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Philippe G. CIARLET; Maria MALIN;Cristinel MARDARE


This work was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No.9041738-CityU 100612).
Abstract: Let $\Omega$ be a bounded and connected open subset of $\mathbb{R}^N$ with a Lipschitz-continuous boundary, the set $\Omega$ being locally on the same side of $\partial \Omega$. A vector version of a fundamental lemma of J. L. Lions, due to C.Amrouche, the first author, L.\ Gratie and S. Kesavan, asserts that any vector field $\bm{v} = (v_i) \in (\mathcal{D}'(\Omega))^N$, such that all the components $\frac12 (\partial_j v_i + \partial_i v_j ) , \, 1 \leq i, \, j \leq N$, of its symmetrized gradient matrix field are in the space $H^{-1} (\Omega)$, is in effect in the space $(L^2(\Omega))^N$. The objective of this paper is to show that this vector version of J. L. Lions lemma is equivalent to a certain number of other properties of interest by themselves. These include in particular a vector version of a well-known inequality due to J.Ne\v{c}as, weak versions of the classical Donati and Saint-Venant compatibility conditions for a matrix field to be the symmetrized gradient matrix field of a vector field, or a natural vector version of a fundamental surjectivity property of the divergence operator.


J. L. Lions lemma, Nev{c}as inequality, Donati compatibilityconditions, Saint-Venant compatibility conditions


46F05, 47A05, 74B05
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