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 = (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. 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.
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This work was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. 9041738-CityU 100612).
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Ciarlet, P.G., Malin, M. & Mardare, C. On a vector version of a fundamental Lemma of J. L. Lions. Chin. Ann. Math. Ser. B 39, 33–46 (2018). https://doi.org/10.1007/s11401-018-1049-5
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DOI: https://doi.org/10.1007/s11401-018-1049-5
Keywords
- J. L. Lions lemma
- Nečas inequality
- Donati compatibility conditions
- Saint-Venant compatibility conditions