# Green function vs geometry

-
Svitlana Mayboroda, University of Minnesota

In this talk we will discuss connections between the geometric and PDE properties of sets. The emphasis is on quantifiable, global results which yield true equivalence between the geometric and PDE notions in very rough scenarios, including domains and equations with singularities and structural complexity. The main result establishes that in all dimensions $d<n$, a $d$-dimensional set in $\mathbb R^n$ is regular (rectifiable) if and only if the Green function for elliptic operators is well approximated by affine functions (distance to the hyperplanes). To the best of our knowledge, this is the first free boundary result of this type for lower dimensional sets and the first free boundary result in the classical case $d=n-1$ without restrictions on the coefficients of the equation.