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

  title={On the absolute continuity of p-harmonic measure and surface measure in Reifenberg flat domains},
  author={Murat Akman},
  journal={arXiv: Analysis of PDEs},
  • M. Akman
  • Published 23 September 2015
  • Mathematics
  • arXiv: Analysis of PDEs
In this paper, we study the set of absolute continuity of p-harmonic measure, $\mu$, and $(n-1)-$dimensional Hausdorff measure, $\mathcal{H}^{n-1}$, on locally flat domains in $\mathbb{R}^{n}$, $n\geq 2$. We prove that for fixed $p$ with $2 0=\mathcal{H}^{n-1}(K)$ where $\mu$ is the p-harmonic measure associated to a positive weak solution to p-Laplace equation in $\Omega$ with continuous boundary value zero on $\partial\Omega$. We also show that there exists such a domain for which the same… 
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The first author was partially supported by NSF RTG grant 0838212. The second author was supported by NSF grants DMS-1101244 and DMS-1361701. The third author was supported in part by MINECO Grant