Akman, Murat and Badger, Matthew and Hofmann, Steve and Martell, José María (2019) 'Rectifiability and elliptic measures on 1sided NTA domains with AhlforsDavid regular boundaries.' Transactions of the American Mathematical Society, 369 (8). pp. 57115745. ISSN 00029947

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Abstract
Let $\Omega\subset\mathbb{R}^{n+1}$, $n \geq 2$, be 1sided NTA domain (aka uniform domain), i.e.~a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that $\partial\Omega$ is ndimensional AhlforsDavid 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 chordarc 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.
Item Type:  Article 

Uncontrolled Keywords:  NTA domains; 1sided NTA domains; uniform domains; AhlforsDavid regular sets; rectifiability; harmonic measure; elliptic measure; surface measure; linearly approximability; elliptic operators 
Divisions:  Faculty of Science and Health Faculty of Science and Health > Mathematical Sciences, Department of 
SWORD Depositor:  Elements 
Depositing User:  Elements 
Date Deposited:  25 Jul 2019 13:59 
Last Modified:  06 Jan 2022 14:01 
URI:  http://repository.essex.ac.uk/id/eprint/25003 
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