On Bogovski\u{\i} and regularized Poincar\'e integral operators for de Rham complexes on Lipschitz domains

TL;DR

This paper analyzes integral operators related to the Poincaré and Bogovski operators, proving they are pseudodifferential of order -1 and demonstrating their applications in regularity and cohomology of de Rham complexes on Lipschitz domains.

Contribution

It introduces a regularized version of Poincaré and Bogovski integral operators, proving their pseudodifferential nature and applying them to establish regularity and cohomology results on Lipschitz domains.

Findings

Operators are pseudodifferential of order -1

Polynomials are mapped to polynomials by these operators

Cohomology spaces can be represented by smooth functions

Abstract

We study integral operators related to a regularized version of the classical Poincar\'e path integral and the adjoint class generalizing Bogovski\u{\i}'s integral operator, acting on differential forms in $R^n$. We prove that these operators are pseudodifferential operators of order -1. The Poincar\'e-type operators map polynomials to polynomials and can have applications in finite element analysis. For a domain starlike with respect to a ball, the special support properties of the operators imply regularity for the de Rham complex without boundary conditions (using Poincar\'e-type operators) and with full Dirichlet boundary conditions (using Bogovski\u{\i}-type operators). For bounded Lipschitz domains, the same regularity results hold, and in addition we show that the cohomology spaces can always be represented by $C^\infty$ functions.