Hilbert complexes with mixed boundary conditions—Part 2: Elasticity complex

Abstract

We show that the elasticity Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are also proved. This paper extends recent results on the de Rham Hilbert complex with mixed boundary conditions from Pauly and Schomburg (2021, 2022) and recent results on the elasticity Hilbert complex with empty or full boundary conditions from Pauly and Zulehner (2020, 2022).

Hilbert Complexes with Mixed Boundary Conditions -- Part 3: Biharmonic Complexes

“…Higher Sobolev order results are also proved. This paper extends recent results on the de Rham and elasticity Hilbert complexes with mixed boundary conditions from [3,4] and results on the biharmonic Hilbert complex with empty or full boundary conditions from [6].…”

“…We shall follow in close lines the rationale from [3,4]. The main results comprise regular decompositions and regular potentials, compact embeddings, Helmholtz decompositions, closed ranges, Friedrichs/Poincaré type estimates, and bases of the corresponding cohomology groups (generalised Dirichlet/Neumann tensors).…”

“…Throughout this paper, let Ω ⊂ R 3 be a bounded strong Lipschitz domain with boundary Γ, decomposed into two parts Γ t and Γ n := Γ \ Γ t with some relatively open and strong Lipschitz boundary part Γ t ⊂ Γ. We shall consequently use the notations, methods, and results from our corresponding papers for the de Rham complex [3], for the elasticity complex [7,4], and for the biharmonic complexes [6]. In particular, we recall [3, Section 2, Section 3] including the notion of extendable domains.…”

“…Proof of Lemma 3.10 and Corollary 3.11. According to [3], [4], and [7], cf. [1,2], let (U ℓ , ϕ ℓ ) be a partition of unity for Ω, i.e.,…”

Hilbert complexes with mixed boundary conditions part 3: Biharmonic complexes

“…As explained in detail in [1,2], all these Hilbert complexes share the same geometric structure where A 0 and A 1 are densely defined and closed (unbounded) linear operators between Hilbert spaces H 𝓁 . The corresponding domain Hilbert complex is denoted by The goal of this article is to show that the previous biharmonic Hilbert complexes are compact, which is proved by using regular decompositions of the domains of definition of the respective operators as a crucial tool.…”