#### Vol. 286, No. 1, 2017

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On the absolute continuity of $p$-harmonic measure and surface measure in Reifenberg flat domains

### Murat Akman

Vol. 286 (2017), No. 1, 25–37
##### Abstract

We study the set of absolute continuity of $p$-harmonic measure $\mu$ associated to a positive weak solution to the $p$-Laplace equation with continuous zero boundary values and $\left(n-1\right)$-dimensional Hausdorff measure ${\mathsc{ℋ}}^{n-1}$ on locally flat domains in space. We prove that when $n\ge 2$ and $2 and when $n\ge 3$ and $2-\eta for some $\eta >0$ there exist locally flat domains $\Omega \subset {ℝ}^{n}$ with locally finite perimeter and Borel sets $E\subset \partial \Omega$ such that $\mu \left(E\right)>0={\mathsc{ℋ}}^{n-1}\left(E\right)$.

##### Keywords
Hausdorff dimension of $p$-harmonic measure, harmonic measure, $p$-harmonic measure, nonlinear elliptic PDEs, Hausdorff measure, Hausdorff dimension, singular sets for $p$-harmonic measure, Reifenberg flat domains, NTA domains
##### Mathematical Subject Classification 2010
Primary: 28A75, 28A78, 35J25, 37F35, 35J92, 31A15