Hilbert Complexes with Mixed Boundary Conditions -- Part 1: De Rham Complex

TL;DR

This paper establishes the closedness and compactness of the de Rham Hilbert complex with mixed boundary conditions on Lipschitz domains, providing key functional analysis results and higher-order Sobolev embeddings.

Contribution

It proves the closedness and compactness of the de Rham complex with mixed boundary conditions, including higher Sobolev order results, using abstract functional analysis and regular decompositions.

Findings

The de Rham Hilbert complex with mixed boundary conditions is closed.

The complex is compact on bounded Lipschitz domains.

Higher Sobolev order regularity results are established.

Abstract

We show that the de Rham 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.