On the dimension of p-harmonic measure in space

Let ßcRⁿ, n≥3, and let p, 1 > p > ∞, p ≠ 2, be given. In this paper we study the dimension of p -harmonic measures that arise from nonnegative solutions to the p -Laplace equation, vanishing on a portion of ∂ω, in the setting of δ-Reifenberg flat domains. We prove, for p ≥ n, that there exists δ = δ(p, n) > 0 small such that if ω is a δ-Reifenberg flat domain with δ < δ, then p-harmonic measure is concentrated on a set of ω -finite H ⁿ∼ -measure. We prove, for p ≥ n, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of p-harmonic measure is always less than n - 1. We also prove that if 2 < p < n, then there exist Wolff snowflakes such that the Hausdorff dimension of p-harmonic measure is less than n - 1, while if 1 < p < 2, then there exist Wolff snowflakes such that the Hausdorff dimension of p-harmonic measure is larger than n - 1. Furthermore, perturbing off the case p = 2, we derive estimates for the Hausdorff dimension of p-harmonic measure when p is near 2.