Tangent measures of elliptic measure and applications (Q2279784)

The authors consider elliptic measures of domains in \(\mathbb{R}^{n}\), \(n\geq 3\), associated with elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. They study the implications of the relationship between the elliptic measures of two complementary domains on the geometry of their common boundaries. A method for studying tangent measures of elliptic measures is introduced and several known results are generalized. In the first main result of the paper, it is shown that mutual absolute continuity between the elliptic measures of two complementary domains implies that the tangent measures are a.e. flat and the elliptic measures have dimension \(n-1\), extending the results of \textit{C. Kenig} et al. [J. Am. Math. Soc. 22, No. 3, 771--796 (2009; Zbl 1206.28002)]. In the second result, they show that the VMO equivalence of doubling elliptic measures implies that the tangent measures are supported on the zero sets of elliptic polynomials, generalizing the work of \textit{C. Kenig} and \textit{T. Toro} [J. Reine Angew. Math. 596, 1--44 (2006; Zbl 1106.35147)]. Finally, assuming that a uniform domain satisfies the capacity density condition and its boundary is locally finite and has a.e. positive lower (\(n-1\))-Hausdorff density, they show that absolute continuity of an elliptic measure with respect to (\(n-1\))-Hausdorff measure implies that the boundary is rectifiable, generalizing the work of \textit{M. Akman} et al. [Trans. Am. Math. Soc. 369, No. 8, 5711--5745 (2017; Zbl 1373.31003)].