A generalization of the classical Cesàro–Volterra path integral formula

Abstract

If a symmetric matrix field e of order three satisfies the Saint Venant compatibility conditions in a simply-connected domain Ω in R 3 , there then exists a displacement field u of Ω such that e = 1 2 ( ∇ u T + ∇ u ) in Ω. If the field e is sufficiently smooth, the displacement u ( x ) at any point x ∈ Ω can be explicitly computed as a function of e and CURL e by means of a Cesàro–Volterra path integral formula inside Ω with endpoint x. We assume here that the components of the field e are only in L 2 ( Ω ) , in which case the classical path integral formula of Cesàro and Volterra becomes meaningless. We then establish the existence of a “Cesàro–Volterra formula with little regularity”, which again provides an explicit solution u to the equation e = 1 2 ( ∇ u T + ∇ u ) in this case. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).