𝜖-approximability of harmonic functions in 𝐿^{𝑝} implies uniform rectifiability

Bortz, Simon; Tapiola, Olli

doi:10.1090/proc/14394

$\varepsilon$-approximability of harmonic functions in $L^p$ implies uniform rectifiability
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by Simon Bortz and Olli Tapiola PDF

Proc. Amer. Math. Soc. 147 (2019), 2107-2121 Request permission

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 paper, 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 of Hofmann and Tapiola [arXiv:1710.05528] gives us a new characterization of uniform rectifiability which complements the recent results of Garnet et al. [Duke Math. J. 167 (2018), no. 8, 1473–1524], and Hofmann et al. [Duke Math. J. 165 (2016), no. 12, 2331–2389].

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