Uniform rectifiability and elliptic operators satisfying a Carleson measure condition

TL;DR

This paper proves that for certain elliptic PDEs with coefficients satisfying a Carleson measure condition, the boundary's geometric property of uniform rectifiability is equivalent to the absolute continuity of elliptic measure, linking PDE solutions to geometric measure theory.

Contribution

It establishes the equivalence between absolute continuity of elliptic measure and uniform rectifiability under optimal Carleson measure conditions, extending previous results to a broader class of operators.

Findings

Proved the equivalence under small Carleson measure constants.

Developed a novel geometric measure theory technique.

Introduced an extrapolation argument for scale-invariant estimates.

Abstract

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. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. We split our proof on two main steps. In the first one 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 the second step 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.