On Bogovski\uı and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains

Abstract

We study integral operators related to a regularized version of the classical Poincaré path integral and the adjoint class generalizing Bogovski\uı's integral operator, acting on differential forms in $R^n$. We prove that these operators are pseudodifferential operators of order -1. The Poincaré-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 Poincaré-type operators) and with full Dirichlet boundary conditions (using Bogovski\uı-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.