Tangent measures of elliptic harmonic measure and applications

TL;DR

This paper develops new methods for analyzing tangent measures of elliptic harmonic measures in arbitrary domains, revealing their structure and implications for boundary regularity and rectifiability, especially under VMO conditions.

Contribution

It introduces a novel approach to tangent measures of elliptic measures, extending prior results to non-symmetric operators and general domains, and links measure properties to boundary rectifiability.

Findings

Mutual absolute continuity implies tangent measures are flat and have dimension n.

VMO equivalence leads tangent measures to be elliptic polynomials.

Absolute continuity of elliptic measure implies boundary rectifiability.

Abstract

Tangent measure and blow-up methods, are powerful tools for understanding the relationship between the infinitesimal structure of the boundary of a domain and the behavior of its harmonic measure. We introduce a method for studying tangent measures of elliptic measures in arbitrary domains associated with (possibly non-symmetric) elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains $ \Omega \subset \mathbb{R}^{n+1}$: 1. We extend the results of Kenig, Preiss, and Toro [KPT09] by showing mutual absolute continuity of interior and exterior elliptic measures for {\it any} domains implies the tangent measures are a.e. flat and the elliptic measures have dimension $n$. 2. We generalize the work of Kenig and Toro [KT06] and show that VMO equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always elliptic polynomials. 3. In a uniform domain that satisfies the capacity density condition and whose boundary is locally finite and has a.e. positive lower $n$-Hausdorff density, we show that if the elliptic measure is absolutely continuous with respect to $n$-Hausdorff measure then the boundary is rectifiable. This generalizes the work of Akman, Badger, Hofmann, and Martell [ABHM17]. Finally, we generalize one of the main results of [Bad11] by showing that if $\omega$ is a Radon measure for which all tangent measures at a point are harmonic polynomials vanishing at the origin, then they are all homogeneous harmonic polynomials.