Complexes from complexes

TL;DR

This paper develops a systematic method to derive new differential complexes from existing ones, analyzing their properties and applications in various fields such as mechanics and relativity.

Contribution

It introduces a procedure to generate new complexes from known complexes and studies their properties, including cohomology, closed ranges, and boundary value problem solutions.

Findings

New complexes have closed ranges and satisfy Hodge decomposition.

Derived complexes exhibit Poincaré inequalities and regular decompositions.

Applications include well-posed boundary value problems in Lipschitz domains.

Abstract

This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old. We relate the cohomology of the output complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincar\'e type inequalities, well-posed Hodge-Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domains.