# A compatible-incompatible decomposition of symmetric tensors in Lp with application to elasticity

created by scala on 09 Mar 2014
modified by maggiani on 15 Mar 2017

[BibTeX]

Mathematical Methods in Applied Sciences

Inserted: 9 mar 2014
Last Updated: 15 mar 2017

Year: 2014

Abstract:

We prove the Saint-Venant compatibility conditions in $L^p$ in a simply connected domain, with $p \in (1,+\infty)$. Moreover we use the Helhmoltz decomposition to deduce that every symmetric $L^p$ tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of the gradient of a displacement, and in an incompatible part, which is the incompatibility of a certain divergence-free tensor. We observe that if the displacement satisfies a Dirichlet condition on the boundary, the decomposition is unique. We apply these results to provide an alternative proof of some classical Korn inequalities, which are very important in elasticity