Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition

The present paper establishes the correspondence between properties of solutions a class PDEs and geometry sets in Euclidean space. We settle question whether (quantitative) absolute continuity elliptic measure with respect to surface uniform rectifiability boundary are equivalent, an optimal divergence form operators satisfying suitable Carleson condition domains Ahlfors regular boundaries. result can be viewed as quantitative analogue Wiener criterion adapted singular $$L^p$$ data case. first step is taken Part I, where we considered case which desired on coefficients holds sufficiently small constant, using novel application techniques developed geometric theory. In II establish final result, that is, “large constant case”. key elements powerful extrapolation argument, provides general pathway self-improve scale-invariant estimates, new mechanism transfer domain its subdomains.