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On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains

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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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Authors and Affiliations

  1. IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    Martin Costabel

  2. Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia

    Alan McIntosh

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  1. Martin Costabel
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  2. Alan McIntosh
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Correspondence to Martin Costabel.

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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

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