Singular sets for harmonic measure on locally flat domains with locally finite surface measure

Abstract

A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in high dimensions if we relax the Ahlfors regularity assumption by showing that, for each $d>1$, there exists a Reifenberg flat domain $\Omega\subset \mathbb{R}^{d+1}$ with $\mathcal{H}^{d}(\partial\Omega)<\infty$ and a subset $E\subset \partial \Omega$ with positive harmonic measure yet zero $\mathcal{H}^{d}$-measure. In particular, this implies that a classical theorem of F. and M. Riesz theorem fails in higher dimensions for this type of domains.