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Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries


Authors: Murat Akman, Matthew Badger, Steve Hofmann and José María Martell
Journal: Trans. Amer. Math. Soc. 369 (2017), 5711-5745
MSC (2010): Primary 28A75, 28A78, 31A15, 31B05, 35J25, 42B37, 49Q15
DOI: https://doi.org/10.1090/tran/6927
Published electronically: April 24, 2017
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Abstract: Let $ \Omega \subset \mathbb{R}^{n+1}$, $ n\geq 2$, be a 1-sided NTA domain (also known as a uniform domain), i.e., a domain which satisfies interior corkscrew and Harnack chain conditions, and assume that $ \partial \Omega $ is $ n$-dimensional Ahlfors-David regular. We characterize the rectifiability of $ \partial \Omega $ in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that $ \partial \Omega $ can be covered $ \mathcal {H}^n$-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of $ \Omega $ and to the fact that $ \partial \Omega $ possesses exterior corkscrew points in a qualitative way $ \mathcal {H}^n$-a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition.


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Additional Information

Murat Akman
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, E-28049 Madrid, Spain
Address at time of publication: Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720
Email: makman@msri.org

Matthew Badger
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: matthew.badger@uconn.edu

Steve Hofmann
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: hofmanns@missouri.edu

José María Martell
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, E-28049 Madrid, Spain
Email: chema.martell@icmat.es

DOI: https://doi.org/10.1090/tran/6927
Keywords: NTA domains, 1-sided NTA domains, uniform domains, Ahlfors-David regular sets, rectifiability, harmonic measure, elliptic measure, surface measure, linearly approximability, elliptic operators
Received by editor(s): July 9, 2015
Received by editor(s) in revised form: January 7, 2016
Published electronically: April 24, 2017
Additional Notes: The first and last authors have been supported in part by the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554), and they acknowledge that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT. The second author was partially supported by an NSF postdoctoral fellowship, DMS 1203497, and by NSF grant DMS 1500382. The third author was partially supported by NSF grant DMS 1361701.
Article copyright: © Copyright 2017 American Mathematical Society