ε-APPROXIMABILITY OF HARMONIC FUNCTIONS IN Lp IMPLIES UNIFORM RECTIFIABILITY.

Suppose that Ω ⊂ Rn+1, n ≥ 2, is an open set satisfying the corkscrew condition with an n-dimensional ADR boundary, ∂Ω. In this paper, we show that if harmonic functions are ε-approximable in Lp for any p > n/(n - 1), then ∂Ω 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]. [ABSTRACT FROM AUTHOR]