A generalization of the classical Cesàro-Volterra path integral formula

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Original languageEnglish
Pages (from-to)577-582
Journal / PublicationComptes Rendus Mathematique
Volume347
Issue number9-10
Online published3 Apr 2009
Publication statusPublished - May 2009

Abstract

If a symmetric matrix field e of order three satisfies the Saint Venant compatibility conditions in a simply-connected domain Ω in R3, there then exists a displacement field u of Ω such that e  =  ½ (∇uT+∇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–Volterrapath integral formula inside Ω with endpoint x.
We assume here that the components of the field e are only in L(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”, whichagain provides an explicit solution u to the equation=  ½ (∇uT+∇u) in this case.  To cite this article: P.G. Ciarlet et al., C. R.Acad. Sci. Paris, Ser. I 347 (2009)