We construct bounded domains D not equal to a ball in n ≥ 3 dimensional Euclidean space, R, for which ∂D is homeomorphic to a sphere under a quasiconformal mapping of R and such that n - 1 dimensional Hausdorff measure equals harmonic measure on ∂D.

Let $\Omega \subset {\mathbb{R}}^{n},n\ge 3$, and let $p,1<p<\infty ,p\ne 2$, be given. In this paper we study the dimension of $p$-harmonic measures that arise from non-negative solutions to the $p$-Laplace equation, vanishing on a portion of $\partial \Omega $, in the setting of $\delta $-Reifenberg flat domains. We prove, for $p\ge n$, that there exists $\tilde{\delta}=\tilde{\delta}(p,n)>0$ small such that if $\Omega $ is a $\delta $-Reifenberg flat domain with $\delta <\tilde{\delta}$, then $p$-harmonic measure is concentrated on a set of $\sigma $-finite ${H}^{n-1}$-measure. We prove, for $p\ge n$, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$-harmonic measure...

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