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 (aka 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.

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• $L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets
• On the uniform rectifiability of AD regular measures with bounded Riesz transform operator: the case of codimension 1

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• Extrapolation of the Dirichlet problem for elliptic equations with complex coefficients
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• The Green function with pole at infinity applied to the study of the elliptic measure
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• Parabolic NTA domains in ${\mathbb{R}}^2$
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• On the $A_{\infty }$ condition for elliptic operators in 1-sided NTA domains satisfying the capacity density condition
• Absolute continuity of degenerate elliptic measure