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

Abstract

Let $\Omega \subset {\mathbb{R}}^{n+1}$, $n\ge 2$, be 1-sided NTA domain (aka 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.

References in corpus (8)

• A new characterization of chord-arc domains
• Rectifiability of harmonic measure
• $A_{\infty }$ implies NTA for a class of variable coefficient elliptic operators
• Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability
• $L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets
• Uniform domains with rectifiable boundaries and harmonic measure
• Rectifiability of harmonic measure in domains with porous boundaries
• Absolute continuity between the surface measure and harmonic measure implies rectifiability

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• Rectifiability of harmonic measure
• $A_{\infty }$ implies NTA for a class of variable coefficient elliptic operators
• Rectifiability, interior approximation and Harmonic Measure
• Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets
• Tangent measures of elliptic harmonic measure and applications
• Boundary rectifiability and elliptic operators with $W^{1,1}$ coefficients
• Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part II: The large constant case
• Uniform rectifiability and elliptic operators satisfying a Carleson measure condition
• $L^2$-boundedness of gradients of single layer potentials and uniform rectifiability
• Lipschitz decompositions of domains with bilaterally flat boundaries