Tangent measures of elliptic harmonic measure and applications

Jonas Azzam, Mihalis Mourgoglou

Research output: Contribution to journalArticlepeer-review

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 non-symmetric) elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains Ω⊆ℝn+1, n≥2:
(1) We extend the results of Kenig, Preiss, and Toro [KPT09] 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 [KT06] and show that VMO equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always 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 [ABHM17].
Original languageEnglish
Pages (from-to)1891-1941
Number of pages50
JournalAnalysis and PDE
Volume12
Issue number8
DOIs
Publication statusPublished - 28 Oct 2019

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