On Saint Venant's compatibility conditions and Poincaré's lemma

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. To cite this article: C. Amrouche et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). © 2006 Académie des sciences.