The Poisson-Dirichlet problem in domains with Ahlfors regular boundary

TL;DR

This paper discusses recent advances in solving the Poisson-Dirichlet problem in complex domains with boundaries that have Ahlfors regularity, extending previous results beyond Lipschitz domains and strong regularity conditions.

Contribution

It introduces new results on well-posedness of the Poisson-Dirichlet problem with boundary data in Besov spaces for domains with Ahlfors regular boundaries, broadening the scope of prior work.

Findings

Well-posedness established for Besov boundary data in Ahlfors regular domains

Generalization beyond Lipschitz domains and regular coefficients

Extension to fractional smoothness classes

Abstract

We present an announcement of some recent results concerning well-posedness of the Poisson-Dirichlet problem with boundary data in Besov spaces with fractional smoothness. This is a far-reaching generalization as previously known theorems concerning well-posedness of the Poisson problem in such intermediate smoothness classes were mostly restricted to the context of Lipschitz domains and coefficients satisfying strong regularity assumptions.