  The present paper establishes equivalence between uniform rectifiability of
the boundary of a domain and the property that the Green function for elliptic
operators is well approximated by affine functions (distance to the
hyperplanes). The results are novel in a variety of ways, in particular (1)
this is the first time the underlying property of the control of the Green
function by affine functions, or by the distance to the boundary, in the sense
of the Carleson prevalent sets, appears in the literature; the "direct" result
established here is new even in the half space; (2) the results are optimal,
providing a full characterization of uniform rectifiability under the
(standard) mild topological assumptions; (3) to the best of the authors'
knowledge, this is the first free boundary result applying to all elliptic
operators, without any restriction on the coefficients (the direct one assumes
the standard, and necessary, Carleson measure condition); (4) our theorems
apply to all domains, with possibly lower dimensional boundaries: this is the
first free boundary result in higher co-dimensional setting and as such, the
first PDE characterization of uniform rectifiability for a set of dimension
$d$, $d<n-1$, in ${\mathbb{R}}^n$. The paper offers a general way to deal with
related issues considerably beyond the scope of the aforementioned theorem,
including the question of approximability of the gradient of the Green
function, and the comparison of the Green function to a certain version of the
distance to the original set rather than distance to the hyperplanes.
