# American Institute of Mathematical Sciences

January  2009, 8(1): 295-309. doi: 10.3934/cpaa.2009.8.295

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

 1 Laboratoire de Mécanique et Génie Civil,CNRS UMR 5508, Université Montpellier II, Case courier 048, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France 2 Institut de Mathématique et Modélisation de Montpellier, CNRS UMR 5149, Université Montpellier II,Case courier 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

Received  February 2008 Revised  July 2008 Published  October 2008

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.
Citation: Giuseppe Geymonat, Françoise Krasucki. Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains. Communications on Pure & Applied Analysis, 2009, 8 (1) : 295-309. doi: 10.3934/cpaa.2009.8.295
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