Abstract
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 \({\mathbb{R}^n}\) . 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 \({\fancyscript{C}^{\infty}}\) functions.
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Costabel, M., McIntosh, A. On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains. Math. Z. 265, 297–320 (2010). https://doi.org/10.1007/s00209-009-0517-8
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DOI: https://doi.org/10.1007/s00209-009-0517-8