Vol. 12, No. 8, 2019

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Tangent measures of elliptic measure and applications

Jonas Azzam and Mihalis Mourgoglou

Vol. 12 (2019), No. 8, 1891–1941
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 n+1 , n 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).
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