JournalsjemsVol. 19, No. 4pp. 967–981

A new characterization of chord-arc domains

  • Jonas Azzam

    Universitat Autònoma de Barcelona, Spain
  • Steve Hofmann

    University of Missouri, Columbia, USA
  • José María Martell

    Universidad Autónoma de Madrid, Spain
  • Kaj Nyström

    Uppsala University, Sweden
  • Tatiana Toro

    University of Washington, Seattle, USA
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Abstract

We show that if ΩRn+1\Omega \subset \mathbb{R}^{n+1}, n1n\geq 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\Omega implies the existence of exterior corkscrew points at all scales, so that in fact, Ω\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.

Cite this article

Jonas Azzam, Steve Hofmann, José María Martell, Kaj Nyström, Tatiana Toro, A new characterization of chord-arc domains. J. Eur. Math. Soc. 19 (2017), no. 4, pp. 967–981

DOI 10.4171/JEMS/685