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

TL;DR

This paper establishes the mathematical properties of biharmonic Hilbert complexes with mixed boundary conditions, including closedness, compactness, and higher Sobolev regularity, extending previous work on related complexes.

Contribution

It extends existing results by proving compact embeddings and regularity for biharmonic complexes with mixed boundary conditions on Lipschitz domains.

Findings

Biharmonic Hilbert complex is closed and compact.

Compact embeddings are established using functional analysis.

Higher Sobolev order regularity results are proved.

Abstract

We show that the biharmonic 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 of the authors on the de Rham and elasticity Hilbert complexes with mixed boundary conditions and results of Pauly and Zulehner on the biharmonic Hilbert complex with empty or full boundary conditions.