Hilbert Complexes with Mixed Boundary Conditions -- Part 2: Elasticity Complex

TL;DR

This paper proves the closedness and compactness of the elasticity Hilbert complex with mixed boundary conditions on Lipschitz domains, extending previous results to include higher Sobolev orders and more general boundary conditions.

Contribution

It extends existing results on elasticity complexes by establishing compact embeddings and regular decompositions for mixed boundary conditions on Lipschitz domains.

Findings

Proves the elasticity Hilbert complex is closed and compact with mixed boundary conditions.

Establishes higher Sobolev order regularity results.

Extends previous work on de Rham and elasticity complexes to more general boundary conditions.

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 proved as well. This paper extends recent results on the de Rham Hilbert complex with mixed boundary conditions from [11] and recent results on the elasticity Hilbert complex with empty or full boundary conditions from [15].