Comptes Rendus
Mathematical Problems in Mechanics
On Saint Venant's compatibility conditions and Poincaré's lemma
[Sur les conditions de compatibilité de Saint Venant et le lemme de Poincaré]
Comptes Rendus. Mathématique, Volume 342 (2006) no. 11, pp. 887-891.

Le théorème de Saint Venant constitue une caractérisation classique de champs de matrices réguliers comme des champs de tenseurs de déformation linéarisés. Ce théorème a été étendu aux champs de matrices avec des composantes dans L2 par le second auteur et P. Ciarlet, Jr. en 2005. Un objectif de cette Note est d'étendre cette caractérisation aux champs de matrices dont les composantes sont seulement dans H−1. Un autre objectif est de démontrer que le théorème de Saint Venant n'est autre que l'analogue matriciel du lemme de Poincaré.

Saint Venant's theorem constitutes a classical characterization of smooth matrix fields as linearized strain tensor fields. This theorem has been extended to matrix fields with components in L2 by the second author and P. Ciarlet, Jr. in 2005. One objective of this Note is to further extend this characterization to matrix fields whose components are only in H−1. Another objective is to demonstrate that Saint Venant's theorem is in fact nothing but the matrix analog of Poincaré's lemma.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.03.026

Cherif Amrouche 1 ; Philippe G. Ciarlet  ; Liliana Gratie 2 ; Srinivasan Kesavan 3

1 Laboratoire de mathématiques appliquées, université de Pau et des pays de l'Adour, avenue de l'Université, 64000 Pau, France
2 Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
3 The Institute of Mathematical Sciences, CIT Campus Taramani, Chennai–600113, India
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Cherif Amrouche; Philippe G. Ciarlet; Liliana Gratie; Srinivasan Kesavan. On Saint Venant's compatibility conditions and Poincaré's lemma. Comptes Rendus. Mathématique, Volume 342 (2006) no. 11, pp. 887-891. doi : 10.1016/j.crma.2006.03.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.03.026/

[1] C. Amrouche, P.G. Ciarlet, L. Gratie, S. Kesavan, New formulations of linearized elasticity problems, based on extensions of Donati's theorem, C. R. Acad. Sci. Paris, Ser. I, in press

[2] C. Amrouche, P.G. Ciarlet, L. Gratie, S. Kesavan, On the characterizations of matrix fields as linearized strain tensor fields, J. Math. Pures Appl., in press

[3] P.G. Ciarlet; P. Ciarlet Another approach to linearized elasticity and a new proof of Korn's inequality, Math. Models Methods Appl. Sci., Volume 15 (2005), pp. 259-271

[4] R. Dautray; J.-L. Lions Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2: Functional and Variational Methods, Springer, 1988

[5] G. Geymonat, F. Krasucki, Some remarks on the compatibility conditions in elasticity, Accad. Naz. Sci. XL Mem. Math. Appl., in press

[6] G. Geymonat; F. Krasucki Beltrami's solutions of general equilibrium equations in continuum mechanics, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006), pp. 359-363

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

[8] S. Kesavan On Poincaré's and J.-L. Lions' lemmas, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 27-30

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