Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains

In 1999 M. Eastwood has used the general construction known as the Bernstein-Gelfand-Gelfand (BGG) resolution to prove, at least in smooth situation, the equivalence of the linear elasticity complex and of the de Rham complex in $\mathbf{R}^{3}$. The main objective of this paper is to study the linear elasticity complex for general Lipschitz domains in $\mathbf{R}^{3}$ and deduce a complete Hodge orthogonal decomposition for symmetric matrix fields in $L^{2}$, counterpart of the Hodge decomposition for vector fields. As a byproduct one obtains that the finite dimensional terms of this Hodge decomposition can be interpreted in homological terms as the corresponding terms for the de Rham complex if one takes the homology with value in $RIG\cong \mathbf{R}^{6}$ as in the (BGG) resolution.