Harmonic polynomials and tangent measures of harmonic measure

Abstract

We show that on an NTA domain if each tangent measure to harmonic measure at a point is a polynomial harmonic measure then the associated polynomials are homogeneous. Geometric information for solutions of a two-phase free boundary problem studied by Kenig and Toro is derived.

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

“…Finally we note that the arguments in [21] are further enhanced by Badger in [3]. In order to state these results let VMO (μ + ) denote the space of functions on ∂Ω + that are of vanishing mean oscillation with respect to μ + .…”

“…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 = ∞.…”

Local Set Approximation: Mattila–vuorinen Type Sets, Reifenberg type Sets, and Tangent Sets

“…In [13], Kenig and Toro showed that solutions for a certain two-phase free boundary problem for harmonic measures on two-sided domains in R n are locally well approximated by H in a Hausdorff distance sense. Refined information about the structure and size of the free boundary was obtained by Badger [1,2] by studying the geometry of abstract sets approximated by H and measures on their support. For instance, in [2], Badger showed that, if A ⊆ R n is closed, A is locally well approximated by H, and A looks pointwise on small scales like the zero set of a homogeneous harmonic polynomial, then A = A 1 ∪ A 2 , where A 1 is locally well approximated by G(n, n − 1), while near each x ∈ A 2 the set A looks locally like the zero set of a homogeneous harmonic polynomial of degree at least 2.…”

Singular points of Hölder asymptotically optimally doubling measures

“…Lemma 3.6 ( [Ba2]). If µ is a measure on R n , x ∈ supp (µ), and ν ∈ Tan(µ, x) such that 0 ∈ supp (ν), then Tan(ν, 0) ⊆ Tan(µ, x).…”