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 mu associated to a positive weak solution to the p-Laplace equation with continuous zero boundary values and (n - 1)-dimensional Hausdorff measure Hn-1 on locally flat domains in space. We prove that when n >= 2 and 2 < p < infinity and when n >= 3 and 2 - eta < p < 2 for some eta > 0 there exist locally flat domains Omega subset of R-n with locally finite perimeter and Borel sets E subset of partial derivative Omega such that mu (E) > 0 = Hn-1 (E).