A Cesàro-Volterra formula with little regularity

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 (∇u^T + ∇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 (∇u^T + ∇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.