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Comptes Rendus Mathématique
Volume 342, n° 11
pages 887-891 (juin 2006)
Doi : 10.1016/j.crma.2006.03.026
Received : 15 Mars 2006 ;  accepted : 16 Mars 2006
On Saint Venantʼs compatibility conditions and Poincaréʼs lemma
Sur les conditions de compatibilité de Saint Venant et le lemme de Poincaré
 

Cherif Amrouche a , Philippe G. Ciarlet , Liliana Gratie c , Srinivasan Kesavan d
a Laboratoire de mathématiques appliquées, université de Pau et des pays de lʼAdour, avenue de lʼUniversité, 64000 Pau, France 
b Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong 
c Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong 
d The Institute of Mathematical Sciences, CIT Campus Taramani, Chennai-600113, India 

Abstract

Saint Venantʼs 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   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  . Another objective is to demonstrate that Saint Venantʼs theorem is in fact nothing but the matrix analog of Poincaréʼs lemma. To cite this article: C. Amrouche et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

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Résumé

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   par le second auteur et P. Ciarlet, Jr. en 2005. Un objectif de cette Note est dʼétendre cette caractérisation aux champs de matrices dont les composantes sont seulement dans  . Un autre objectif est de démontrer que le théorème de Saint Venant nʼest autre que lʼanalogue matriciel du lemme de Poincaré. Pour citer cet article : C. Amrouche et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

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