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

Giuseppe Geymonat,Françoise Krasucki

doi:10.3934/cpaa.2009.8.295

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.

Highlights

Beltrami completeness condition in L2 settingLet Ω be a general bounded Lipschitz connected but eventually multiply connected domain in R3 with boundary ∂Ω
The aim of this paper is to prove ”a further possible instance of this”: a general Hodge decomposition for symmetric matrix fields analogous to the classical Hodge decomposition for vector fields; for this we study the linear elasticity complex for general Lipschitz domains in various Sobolev spaces settings
Let us explicitly remark that these decompositions of L2 Ω; Ms3ym in mutually orthogonal subspaces are the counterpart of the decompositions of Helmoltz type for vector fields in L2 (Ω; R3)

Summary

Beltrami completeness condition in L2 setting

Let Ω be a general bounded Lipschitz connected but eventually multiply connected domain in R3 with boundary ∂Ω. Green’s formula, one has only to prove that the sum in (2.20) is direct For this let be S ∈ Ker(Div; L2) and let us remark that there exists a unique u ∈ Y := {v ∈ H1(Ω; R3), v|γ0 = 0 and v|γq ∈ rig, q = 1, . One can give a variational characterization of a basis of Y From this proposition and from theorem 2.1 one deduces the following decompositions of L2 Ω; Ms3ym in mutually orthogonal subspaces: (2.21) L2 Ω; Ms3ym = ∇s(H01 Ω; R3 ) ⊕⊥ CU RL CU RL (H2 Ω; Ms3ym ) ⊕⊥ Y and from (2.12). Let us explicitly remark that these decompositions of L2 Ω; Ms3ym in mutually orthogonal subspaces are the counterpart of the decompositions of Helmoltz type for vector fields in L2 (Ω; R3). In order to obtain a complete decomposition of Hodge type we need the results of the section

Saint Venant compatibility condition in L2 setting

The complete Hodge decomposition