On a vector version of a fundamental Lemma of J. L. Lions

Abstract

Let Ω be a bounded and connected open subset of ℝ N 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 = (ui) ∈ (D′(Ω)) N , such that all the components \(\frac{1}{2}({\partial _j}{v_i} + {\partial _i}{v_j})\) , 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. Necas, 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.