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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 16
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Research Article

The elasticity complex: compact embeddings and regular decompositions

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Pages 4393-4421 | Received 06 Jul 2020, Accepted 30 Sep 2021, Published online: 16 Sep 2022
 

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.

Maths:

Disclosure statement

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

Notes

1 A closer inspection of the respective proof of [Citation1, Lemma 3.22] shows immediately that even the embedding H˚S(Rot,Ω)HS0,1(divDiv,Ω)LS2(Ω) is compact.

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