Uniform Rectifiability, Elliptic Measure, Square Functions, and ε-Approximability Via an ACF Monotonicity Formula

Let Omega subset of Rn+1, n >= 2, be an open set with Ahlfors regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator L in divergence form associated with a matrix A with real, merely bounded and possibly nonsymmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if L* is the operator in divergence form associated with the transpose matrix of A, then partial derivative Omega is uniformly n-rectifiable if and only if every bounded solution of Lu = 0 and every bounded solution of L*v = 0 in Omega is epsilon-approximable if and only if every bounded solution of Lu = 0 and every bounded solution of L*v = 0 in Omega satisfies a suitable square-function Carleson measure estimate. Moreover, we obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called "S < N" estimates, and another in terms of a suitable corona decomposition involving L-harmonic and L*-harmonic measures. We also prove that if L-harmonic measure and L*-harmonic measure satisfy a weak A(infinity)-type condition, then partial derivative Omega is n-uniformly rectifiable. In the process we obtain a version of the Alt-Caffarelli-Friedman monotonicity formula for a fairly wide class of elliptic operators which is of independent interest and plays a fundamental role in our arguments.