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
,
:
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
.
We generalize the work of Kenig and Toro (J. Reine Agnew. Math. 596
(2006), 1–44) and show that
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.
In a uniform domain that satisfies the capacity density condition and
whose boundary is locally finite and has a.e. positive lower
-Hausdorff
density, we show that if the elliptic measure is absolutely continuous with
respect to
-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).
