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

Abstract

In this paper, we study the set of absolute continuity of p-harmonic measure, µ, and (n − 1)−dimensional Hausdorff measure, H n−1 , on locally flat domains in R n , n ≥ 2. We prove that for fixed p with 2 < p < ∞ there exists a Reifenberg flat domain Ω ⊂ R n , n ≥ 2 with H n−1 (∂Ω) < ∞ and a Borel set K ⊂ ∂Ω such that µ(K) > 0 = H n−1 (K) where µ is the p-harmonic measure associated to a positive weak solution to p-Laplace equation in Ω with continuous boundary value zero on ∂Ω. We also show that there exists such a domain for which the same result holds when p is fixed with 2 − η < p < 2 for some η > 0 provided that n ≥ 3.This work is a generalization of a recent result of Azzam, Mourgoglou, and Tolsa when the measure µ is harmonic measure at x, ω = ω x , associated to the Laplace equation, i.e when p = 2.

Absolute continuity of harmonic measure for domains with lower regular boundaries

“…However, in [AMT15, Theorem 1.2], the second and third authors along with Tolsa (using a deep result of Wolff [Wol95]) constructed a two-sided NTA domain Ω with H d (∂Ω) < ∞ but ω Ω H d | ∂Ω . See also [A16,LN12] for the p-harmonic version of these results.…”