Communications on Pure and Applied Analysis (CPAA)

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

Pages: 295 - 309, Volume 8, Issue 1, January 2009      doi:10.3934/cpaa.2009.8.295

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Giuseppe Geymonat - Laboratoire de Mécanique et Génie Civil,CNRS UMR 5508, Université Montpellier II, Case courier 048, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France (email)
Françoise Krasucki - Institut de Mathématique et Modélisation de Montpellier, CNRS UMR 5149, Université Montpellier II,Case courier 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France (email)

Abstract: 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.

Keywords:  Beltrami completeness condition, Saint Venant compatibility condition, BGG-resolution
Mathematics Subject Classification:  Primary: 58A14, 74B05; Secondary: 35J55

Received: February 2008;      Revised: July 2008;      Available Online: October 2008.