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2019 Tangent measures of elliptic measure and applications
Jonas Azzam, Mihalis Mourgoglou
Anal. PDE 12(8): 1891-1941 (2019). 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 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).

Citation

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Jonas Azzam. Mihalis Mourgoglou. "Tangent measures of elliptic measure and applications." Anal. PDE 12 (8) 1891 - 1941, 2019. https://doi.org/10.2140/apde.2019.12.1891

Information

Received: 15 August 2017; Revised: 19 October 2018; Accepted: 30 November 2018; Published: 2019
First available in Project Euclid: 14 December 2019

zbMATH: 07143406
MathSciNet: MR4023971
Digital Object Identifier: 10.2140/apde.2019.12.1891

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

Rights: Copyright © 2019 Mathematical Sciences Publishers

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