Green functions and smooth distances

TL;DR

This paper establishes a characterization of boundary rectifiability for certain elliptic operators on Chord-Arc domains through the behavior of Green functions, linking PDE properties with geometric measure theory.

Contribution

It introduces a new criterion connecting Green function behavior to boundary rectifiability, utilizing a corona decomposition compatible with Tolsa's alpha-numbers.

Findings

Boundary of domain is uniformly rectifiable iff Green function behaves like a distance function.

Develops a corona decomposition method compatible with geometric measure theory.

Generalizes the classical F. and M. Riesz theorem to a broader class of elliptic operators.

Abstract

In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function $G$ behaves like a distance function to the boundary, in the sense that $\Big|\frac{\nabla G(X)}{G(X)}-\frac{\nabla D(X)}{D(X)}\Big|^2D(X) dX$ is the density of a Carleson measure, where $D$ is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa's $\alpha$-number of uniformly rectifiable sets. We believe that the method can be applied to many other problems at the intersection of PDE and geometric measure theory, and in particular, we are able to derive a generalization of the classical F. and M. Riesz theorem to the same class of elliptic operators as above.