Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions

Abstract

In this work we introduce the use of powerful tools from geometric measure theory (GMT) to study problems related to the size and structure of sets of mutual absolute continuity for the harmonic measure ω + of a domain Ω = Ω + ⊂ R n and the harmonic measure ω − of Ω − , Ω − = int(Ω c ), in dimension n ≥ 3.

Regularity and free boundary regularity for the $p$-Laplace operator in Reifenberg flat and Ahlfors regular domains

“…Let μ ± be the measures corresponding to u ± as in Lemma 2. The case p = 2 of Proposition 6.1 is due to [21]. Moreover, using Lemma 2.5, as well as the blow-up argument in Theorem 4, we believe that one can essentially copy the proof of Theorem 4.1 and Corollary 4.1 for harmonic functions in [21] with slight adjustments.…”

“…Moreover, using Lemma 2.5, as well as the blow-up argument in Theorem 4, we believe that one can essentially copy the proof of Theorem 4.1 and Corollary 4.1 for harmonic functions in [21] with slight adjustments. For example in [21] the authors quote a result of Hardt and Simon in order to show, for a harmonic function v in R n , that |∇v| = 0 somewhere on {x : v(x) = 0}. If v is p-harmonic in R n , for some 1 < p < ∞, then this statement follows easily from Lemma 2.4 and a barrier type argument.…”

“…Section 5 and Section 6 are devoted to the proof of Theorem 3 and Theorem 4, respectively. In Section 6 we also make some closing remarks concerning work in [21] and [3].…”

“…Closing remarks. We note that a problem related to Theorem 4 was considered in [21] and [3]. Indeed, suppose Ω + is an NTA-domain with parameters M, r 0 = ∞, and that Ω − = R n \ Ω + is also an NTA-domain with parameters M, r 0 = ∞.…”

Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

“…Up to now, the main contribution on this objective was the work of Kenig, Preiss, and Toro [21], whose result we paraphrase below.…”