Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

Abstract

We show the David-Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As one consequence every Wolff snowflake has infinite surface measure.

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

“…Since this inequality does not necessarily hold in an Ahlfors regular NTAdomain we are forced to use an alternative approach based on a result from [28] (see Lemma 2.9 in Section 2). After proving Theorem 1, we in Remark 3.3 also discuss the work of Badger [2] and its applications to absolute continuity of p-harmonic measure and surface area.…”

“…To briefly outline the proof of this proposition we note that in [2] it is shown that a theorem of David-Jerison [11], regarding Lipschitz approximation (on every scale) in bounded Ahlfors regular NTA-domains, can be proved under weaker assumptions. Using this result it follows easily that if Δ(y, 4s) ⊂ Δ(w, 4r) and …”

Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

“…The implication (i) implies (ii) was proved in [4]; (ii) implies (iii) was proved independently in [14,36] as mentioned above; (iii) implies (iv) is trivial; and (iv) implies (i) was proved in [26]. On the other hand, in [7], it was shown that if Ω is an NTA domain with H n (∂Ω) < ∞, then ∂Ω is n-rectifiable and H n | ∂Ω ≪ ω. Moreover, it was also shown in [7] that if Ω is an NTA domain, then ω ≪ H n ≪ ω on A, where A := x ∈ ∂Ω : lim inf r→0 H n (∂Ω ∩ B(x, r)) r n < ∞ .…”

Flat points in zero sets of harmonic polynomials and harmonic measure from two sides

“…The answer to Problem 6.18 is yes under the additional assumption that H n−1 ∂Ω is a Radon measure (for example, if H n−1 (∂Ω) < ∞). For instance, one can verify this assertion by combining a recent result of Kenig, Preiss, and Toro [11] (see Corollary 4.2) with a recent result of Badger [3] (see Theorem 1.2).…”