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

Martin Costabel (1) , Alan Mcintosh (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 CMA - Centre for Mathematics and its Applications, MSI

We study integral operators related to a regularized version of the classical Poincaré 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 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\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.

Mots clés: Exterior derivative; differential forms; Lipschitz domain; Sobolev spaces; pseudodifferential operator

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Martin Costabel, Alan Mcintosh. On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains. Mathematische Zeitschrift, 2010, 265 (2), pp.297-320. ⟨10.1007/s00209-009-0517-8⟩. ⟨hal-00311594v2⟩