Potential maps for de Rahm complexes on Lipshitz domains
Speaker:Prof. Alan McIntosh
Australian National University
Canberra, Australia
Date & Time:11 Aug 2010 (Wednesday) 16:00 - 17:00
Venue:J307
Organized by:Department of Mathematics

Abstract

On a domain which is starlike with respect to a ball, integral operators related to the classical Poincaré path integral serve as potential operators for de Rham complexes without boundary conditions, and the dual class generalizing Bogovskii-type operators work for de Rham complexes with full Dirichlet boundary conditions. Such operators were introduced by Mitrea, and further studied by Mitrea, Mitrea and Monniaux. In joint work with Costabel, we prove that these operators are pseudodifferential operators of order -1, and thus obtain further regularity results for these de-Rham complexes, for example in Hardy spaces.

In recent work with Costabel and Taggart, we turn our attention to unbounded special Lipschitz domains, and adapt operators constructed by Chang, Krantz and Stein to construct potential maps for de Rham complexes in this case, thus obtaining similar regularity results.

Biography

Name: Prof. Alan McIntosh
Position: Professor
Affiliation: Australia National University
Research interests:
Harmonic Analysis, Operator Theory and Partial Differential Equations: Singular integrals and square function estimates on Lipschitz surfaces, with applications to boundary value problems for partial differential equations; scattering theory for Maxwell's equations on irregular domains; spectral theory and functional calculi of operators in Banach spaces; Clifford analysis; heat kernel bounds and functional calculi of elliptic partial differential operators.