Partial Differential Equations
On Poincaré's and J.L. Lions' lemmas
Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 27-30.

Let Ω be a bounded, connected and simply connected open subset of RN with a Lipschitz continuous boundary. It is shown that an irrotational vector field whose components are in H−1(Ω) is the gradient of a function in L2(Ω). It is also shown that this generalization of a classical lemma of Poincaré is equivalent to a well-known lemma of J.L. Lions.

Soit Ω un ouvert borné de RN connexe et simplement connexe à frontière lipschitzienne. On montre qu'un champ vectoriel à composantes dans H−1(Ω) dont le rotationnel est nul est le gradient d'une fonction dans L2(Ω). On montre que cette généralisation d'un lemme classique de Poincaré est equivalent à un lemme très connu de J.L. Lions.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2004.11.021
Kesavan, Srinivasan 1

1 The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai – 600 113, India
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Kesavan, Srinivasan. On Poincaré's and J.L. Lions' lemmas. Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 27-30. doi : 10.1016/j.crma.2004.11.021. http://www.numdam.org/articles/10.1016/j.crma.2004.11.021/

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