Connectivity conditions and boundary Poincaré inequalities

Abstract

Inspired by recent work of Mourgoglou and the second named author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincaré inequalities in open sets $\Omega \subset {\mathbb{R}}^{n+1}$, with codimension $1$ Ahlfors--David regular boundaries. First, we prove that if $\Omega$ satisfies both the local John condition and the exterior corkscrew condition, then $\Omega$ also satisfies the Harnack chain condition (and hence, is a chord-arc domain). Second, we show that if $\Omega$ is a $2$-sided chord-arc domain, then the boundary $\partial \Omega$ supports a Heinonen--Koskela type weak $1$-Poincaré inequality. We also construct an example of a set $\Omega \subset {\mathbb{R}}^{n+1}$ such that the boundary $\partial \Omega$ is Ahlfors--David regular and supports a weak boundary $1$-Poincaré inequality but $\Omega$ is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincaré theories.

40 pages, 5 figures. v3: accepted version; updated grant information and picture formats. To appear in Analysis & PDE

References in corpus (3)

• A new characterization of chord-arc domains
• Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem
• Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions

Cited by in corpus (1)

• Interactions between quantitative rectifiability, singular integrals, and boundary value problems for harmonic functions