The elasticity complex: compact embeddings and regular decompositions

Abstract

We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincaré type estimates, Helmholtz-type decompositions, regular decompositions, regular potentials, finite cohomology groups, and, most importantly, new compact embedding results. Our results hold for general bounded strong Lipschitz domains of arbitrary topology and rely on a general functional analysis framework (FA-ToolBox). Moreover, we present a simple technique to prove the compact embeddings based on regular decompositions/potentials and Rellich's selection theorem, which can be easily adapted to any Hilbert complex.

Keywords:

• elasticity complex
• Hilbert complexes
• Helmholtz decompositions
• compact embeddings
• Friedrichs/Poincaré type estimates
• regular decompositions
• cohomology groups

Maths:

• 35G15
• 58A14

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A closer inspection of the respective proof of [1, Lemma 3.22] shows immediately that even the embedding $\mathring{\mathcal{H}}{}_{\mathbb{S}}^{}(\mathrm{Rot},\mathrm{\Omega })\cap \mathcal{H}{}_{\mathbb{S}}^{0,-1}(\mathrm{div}\mathrm{Div},\mathrm{\Omega })\hookrightarrow \mathcal{L}{}_{\mathbb{S}}^2\left(\mathrm{\Omega }\right)$ is compact.