A Cesàro-Volterra formula with little regularity

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Original languageEnglish
Pages (from-to)41-60
Journal / PublicationJournal des Mathematiques Pures et Appliquees
Issue number1
Online published29 May 2009
Publication statusPublished - Jan 2010


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 with e as its associated linearized strain tensor, i.e., e = ½ (∇uT + ∇u) in Ω. A classical result, due to Cesàro and Volterra, asserts that, if the field e is sufficiently smooth, the displacement u (x) at any point x ∈ Ω can be explicitly computed as a function of the matrix fields e and CURL e, by means of a path integral inside Ω with endpoint x. 
We assume here that the components of the field e are only in L2 (Ω) (as in the classical variational formulation of three-dimensional linearized elasticity), 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 =  ½ (∇uT + ∇u)  in this case. We also show how the classical Cesàro-Volterra formula can be recovered from the formula with little regularity when the field e is smooth. Interestingly, our analysis also provides as a by-product a variational problem that satisfies all the assumptions of the Lax-Milgram lemma, and whose solution is precisely the unknown displacement field u
It is also shown how such results may be used in the mathematical analysis of "intrinsic" linearized elasticity, where the linearized strain tensor e (instead of the displacement vector u as is customary) is regarded as the primary unknown.  

Research Area(s)

  • Saint-Venant compatibility equations, Poincare's lemma, Cesaro-Volterra formula, Three-dimensional linearized elasticity