Comptes Rendus
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.

Published online:
DOI: 10.1016/j.crma.2004.11.021

Srinivasan Kesavan 1

1 The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai – 600 113, India
     author = {Srinivasan Kesavan},
     title = {On {Poincar\'e's} and {J.L.} {Lions'} lemmas},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {27--30},
     publisher = {Elsevier},
     volume = {340},
     number = {1},
     year = {2005},
     doi = {10.1016/j.crma.2004.11.021},
     language = {en},
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PB  - Elsevier
DO  - 10.1016/j.crma.2004.11.021
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Srinivasan Kesavan. 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.

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[3] G. Duvaut; J.L. Lions; G. Duvaut; J.L. Lions Les inéquations en mécanique et en physique, Inequalities in Mechanics and Physics, Dunod, 1972 (English translation, 1976, Springer-Verlag)

[4] V. Girault; P.A. Raviart Finite Element Methods for Navier–Stokes Equations, Springer-Verlag, 1986

[5] J. Nečas Les méthodes directes en théorie des équations elliptiques, Masson, 1967

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