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

We study the set of absolute continuity of p-harmonic measure μ associated to a positive weak solution to the p-Laplace equation with continuous zero boundary values and.n-1/-dimensional Hausdorffmeasure n-1 on locally flat domains in space. We prove that when n ≥ 2 and 2 < p < ∞ and when n ≥ 3 and 2-η < p < 2 for some η > 0 there exist locally flat domains Ω ⊂ ℝn with locally finite perimeter and Borel sets E ⊂ ∂Ω such that μ(E) > 0 = n-1(E).