Tangent measures of elliptic measure and applications

Azzam, Jonas; Mourgoglou, Mihalis

doi:10.2140/apde.2019.12.1891

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Tangent measures of elliptic measure and applications

Jonas Azzam and Mihalis Mourgoglou

Vol. 12 (2019), No. 8, 1891–1941

DOI: 10.2140/apde.2019.12.1891

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 $ω \subset ℝ^{n + 1}$ , $n \geq 2$ :

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 $n$ .
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.
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).

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Keywords

harmonic measure, elliptic measure, capacity density condition, $\Delta$-regular domains, tangent measures, absolute continuity, rectifiability

Mathematical Subject Classification 2010

Primary: 31A15, 28A75, 28A78, 28A33

Milestones

Received: 15 August 2017

Revised: 19 October 2018

Accepted: 30 November 2018

Published: 28 October 2019

Authors

Jonas Azzam
School of Mathematics University of Edinburgh Edinburgh United Kingdom

Mihalis Mourgoglou
Departamento de Matemáticas Universidad del País Vasco Leioa Spain	Ikerbasque, Basque Foundation for Science Bilbao Spain

Vol. 12, No. 8, 2019