|
Abstract
:
The Riemann mapping theorem, one of the most celebrated
results in complex analysis, states that any bounded, connected
and simply connected domain D in the plane can be
conformally mapped onto the unit disk. It has long been
understood that there are subtle connections between the
smoothness of the conformal transformation near the
boundary of the domain and the degree of regularity of the domain
under discussion.
This talk addresses the issue of global regularity of the conformal map on Sobolev-Besov scales, in the case when the domain is allowed to have a Lipschitz graph-like boundary. The approach taken relies on tools from Harmonic Analysis and PDE's, in particular on boundary layer potentials and sharp estimates for the Dirichlet Laplacian with Besov data in Lipschitz domains. |