Let Ω be a bounded, connected and simply connected open subset of with a Lipschitz continuous boundary. It is shown that an irrotational vector field whose components are in is the gradient of a function in . 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 connexe et simplement connexe à frontière lipschitzienne. On montre qu'un champ vectoriel à composantes dans dont le rotationnel est nul est le gradient d'une fonction dans . 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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Srinivasan Kesavan 1
@article{CRMATH_2005__340_1_27_0, 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}, }
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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.11.021/
[1] Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czech. Math. J., Volume 44 (1994), pp. 109-140
[2] P.G. Ciarlet, P. Ciarlet Jr., Linearized elasticity and Korn's inequality revisited, Preprint No. 14, Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, June 2004
[3] Les inéquations en mécanique et en physique, Inequalities in Mechanics and Physics, Dunod, 1972 (English translation, 1976, Springer-Verlag)
[4] Finite Element Methods for Navier–Stokes Equations, Springer-Verlag, 1986
[5] Les méthodes directes en théorie des équations elliptiques, Masson, 1967
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