Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part I: The small constant case

TL;DR

This paper explores the relationship between elliptic measure absolute continuity and boundary rectifiability for elliptic operators satisfying a Carleson measure condition, using geometric measure theory techniques in a small constant setting.

Contribution

It establishes the equivalence of absolute continuity and uniform rectifiability for a class of elliptic operators with small Carleson measure coefficients, advancing the understanding of PDE and geometric set properties.

Findings

Proves the equivalence under small Carleson measure conditions

Develops techniques linking PDE solutions to geometric measure theory

Sets the stage for extending results to general cases in Part II

Abstract

The present paper, along with its sequel, 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. This paper addresses the free boundary problem under the assumption of smallness of the Carleson measure of the coefficients. Part II of this work develops an extrapolation argument to bootstrap this result to the general case. The ideas in Part I constitute a novel application of techniques developed in geometric measure theory. They highlight the synergy between several areas. The ideas developed in this paper are well suited to study singularities arising in variational problems in a geometric setting.