A new characterization of chord-arc domains

Abstract

We show that if Ω ⊂ ℝ, n ≥ 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 Ω implies the existence of exterior corkscrew points at all scales, so that in fact, Ω 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. ; The first author was partially supported by NSF RTG grant 0838212. The second author was supported by NSF grants DMS-1101244 and DMS-1361701. The third author was supported in part by MINECO Grant MTM2010-16518, ICMAT Severo Ochoa project SEV-2011-0087. He also acknowledges that the research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The fourth author was partly supported by the Swedish research council VR. The last author was partially supported by the Robert R. & Elaine F. Phelps Professorship in Mathematics.