@article{8bf0785b2fde4186a03fc7804d4499a7, title = "On a Vector Version of a Fundamental Lemma of J. L. Lions", abstract = "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(∂jvi + ∂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. 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.", keywords = "Donati compatibility conditions, J. L. Lions lemma, Ne{\v c}as inequality, Saint-Venant compatibility conditions", author = "CIARLET, {Philippe G.} and Maria MALIN and Cristinel MARDARE", year = "2018", month = jan, doi = "10.1007/s11401-018-1049-5", language = "English", volume = "39", pages = "33--46", journal = "Chinese Annals of Mathematics. Series B", issn = "0252-9599", publisher = "Springer", number = "1", }