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

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Original languageEnglish
Pages (from-to)33-46
Journal / PublicationChinese Annals of Mathematics. Series B
Issue number1
Online published6 Jan 2018
Publication statusPublished - Jan 2018


Let Ω be a bounded and connected open subset of ℝN with a Lipschitz-continuous boundary, the set Ω being locally on the same side of Ω. 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 v = (vi) ∈ (D'(Ω))N, such that all the components 1/2(jv+ ivj), 1 ≤ i, j N, of its symmetrized gradient matrix field are in the space H−1(Ω), is in effect in the space (L2(Ω))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. Nečas, 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.

Research Area(s)

  • Donati compatibility conditions, J. L. Lions lemma, Nečas inequality, Saint-Venant compatibility conditions