$A_{\infty }$ implies NTA for a class of variable coefficient elliptic operators

Abstract

We consider a certain class of second order, variable coefficient divergence form elliptic operators, in a uniform domain $\Omega$ with Ahlfors regular boundary, and we show that the $A_{\infty }$ property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact NTA (and hence chord-arc). The converse was already known, and follows from work of Kenig and Pipher.

arXiv admin note: text overlap with arXiv:1605.07291 by other authors

References in corpus (6)

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• Extrapolation of the Dirichlet problem for elliptic equations with complex coefficients
• Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators
• Transference of scale-invariant estimates from Lipschitz to Non-tangentially accessible to Uniformly rectifiable domains
• Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets
• Tangent measures of elliptic harmonic measure and applications
• Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part II: The large constant case
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• $L^2$-boundedness of gradients of single layer potentials and uniform rectifiability
• Absolute continuity of degenerate elliptic measure
• A Green function characterization of uniformly rectifiable sets of any codimension
• On the $A_{\infty }$ condition for elliptic operators in 1-sided NTA domains satisfying the capacity density condition