Uniform rectifiability and elliptic operators satisfying a Carleson measure condition

The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition in uniform domains with Ahlfors regular boundaries. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular Lp data case. The first step is taken in Part I, where we considered the case in which the desired Carleson measure condition on the coefficients holds with sufficiently small constant, using a novel application of techniques developed in geometric measure theory. In Part II we establish the final result, that is, the ¿large constant case¿. The key elements are a powerful extrapolation argument, which provides a general pathway to self-improve scale-invariant small constant estimates, and a new mechanism to transfer quantitative absolute continuity of elliptic measure between a domain and its subdomains.
The first author was partially supported by NSF Grants DMS-1664047 and DMS-2000048. The second author acknowledges financial support from the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S) and Grant MTM PID2019-107914GB-I00. The second author also acknowledges 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 third author was partially supported by the NSF INSPIRE Award DMS 1344235, the NSF RAISE-TAQ Grant DMS 1839077, and the Simons Foundation Grant 563916, SM. The fourth author was partially supported by the Craig McKibben & Sarah Merner Professor in Mathematics, by NSF Grant Numbers DMS-1664867 and and DMS-1954545, and by the Simons Foundation Fellowship 614610. The fifth author was partially supported by NSF Grants DMS-1361823, DMS-1500098, DMS-1664867, DMS-1902756 and by the Institute for Advanced Study.