Matthew Badger

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

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.

Harmonic polynomials and tangent measures of harmonic measure

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.

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

Let

Multiscale analysis of 1-rectifiable measures: necessary conditions

\Omega \subset \mathbb{R}^{n+1}

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

,

BEURLING'S CRITERION AND EXTREMAL METRICS FOR FUGLEDE MODULUS

n\geq 2

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

, be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that

Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps

\partial\Omega

Two sufficient conditions for rectifiable measures

is

Quasisymmetry and rectifiability of quasispheres

n