A new characterization of chord-arc domains

Abstract

We show that if $\Omega \subset {\mathbb{R}}^{n+1}$, $n\ge 1$, is a uniform domain (also known as a 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of $\Omega$ implies the existence of exterior corkscrew points at all scales, so that in fact, $\Omega$ is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior corkscrew conditions, and an interior Harnack chain condition. We discuss some implications of this result for theorems of F. and M. Riesz type, and for certain free boundary problems.