$\varepsilon$-Approximability of Harmonic Functions in $L^p$ Implies Uniform Rectifiability

TL;DR

This paper demonstrates that the $L^p$ $ extit{ε}$-approximability of harmonic functions on certain domains implies the boundary's uniform rectifiability, providing a new characterization that enhances understanding of geometric measure theory.

Contribution

It establishes a novel link between harmonic function approximation properties and the geometric regularity of domain boundaries, specifically uniform rectifiability.

Findings

$L^p$ $ extit{ε}$-approximability implies uniform rectifiability

Provides a new characterization of uniform rectifiability

Complements recent geometric measure theory results

Abstract

Suppose that $\Omega \subset \mathbb{R}^{n+1}$, $n \ge 2$, is an open set satisfying the corkscrew condition with an $n$-dimensional ADR boundary, $\partial \Omega$. In this note, we show that if harmonic functions are $\varepsilon$-approximable in $L^p$ for any $p > n/(n-1)$, then $\partial \Omega$ is uniformly rectifiable. Combining our results with those in [HT] (Hofmann-Tapiola) gives us a new characterization of uniform rectifiability which complements the recent results in [HMM] (Hofmann-Martell-Mayboroda), [GMT] (Garnett-Mourgoglou-Tolsa) and [AGMT] (Azzam-Garnett-Mourgoglou-Tolsa).