Title: Tangent measures of elliptic harmonic measure and applications,
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.