Green functions and smooth distances

Abstract

AbstractIn 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 functionGbehaves like a distance function to the boundary, in the sense that$$\left| \frac{\nabla G(X)}{G(X)}-\frac{\nabla D(X)}{D(X)}\right| ^2D(X) dX$$∇G(X)G(X)-∇D(X)D(X)2D(X)dXis the density of a Carleson measure, whereDis 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$$\alpha $$α-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.