Hausdorff dimension and $\sigma$ finiteness of $p-$harmonic measures in space when $p\ge n$

Abstract

In this paper we study a p harmonic measure, associated with a positive p harmonic function \hat{u} defined in an open set O, subset of R^n, and vanishing on a portion Γof boundary of O. If p>n we show that this p harmonic measure is concentrated on a set of σ- finite H^{n-1} measure while if p=n the same conclusion holds provided Γis uniformly fat in the sense of n capacity.