Tangent measures of elliptic 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 nonsymmetric) elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains $\omega \subset {ℝ}^{n+1}$, $n\ge 2$:

1. We extend the results of Kenig, Preiss, and Toro (J. Amer. Math. Soc. 22:3 (2009), 771–796) by showing mutual absolute continuity of interior and exterior elliptic measures for 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 (J. Reine Agnew. Math. 596 (2006), 1–44) and show that $VMO$ equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always supported on the zero sets of 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 (Trans. Amer. Math. Soc. 369:8 (2017), 5711–5745).