On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains
Martin Costabel, Alan G.R. McIntosh · 2013
We study integral operators related to a regularized version of the classical Poincaré path integral and the adjoint class generalizing Bogovskiĭ’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 Bogovskiĭ-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 ∞ functions. 2000 Mathematics Subject Classification. Primary 35B65, 35C15; Secondary 58J10, 47G30 Key words and phrases. Exterior derivative, differential forms, Lipschitz domain, Sobolev