  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.
