Connectivity conditions and boundary Poincar\'e inequalities

TL;DR

This paper explores the relationships between geometric conditions of open sets with regular boundaries and boundary inequalities, establishing new implications and providing counterexamples in the context of harmonic analysis and geometric measure theory.

Contribution

It proves that certain geometric conditions imply the Harnack chain condition and that 2-sided chord-arc domains support weak boundary Poincaré inequalities, also providing a counterexample.

Findings

Local John and exterior corkscrew conditions imply Harnack chain condition.

2-sided chord-arc domains support weak boundary Poincaré inequalities.

Counterexample of a set with boundary Poincaré inequality but not a chord-arc domain.

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 Poincar\'e 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 $1$-Poincar\'e 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 $1$-Poincar\'e inequality but $\Omega$ is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincar\'e theories.