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Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq2$, be an open set withAhlfors-David regular boundary that satisfies the corkscrew condition. Weconsider a uniformly elliptic operator $L$ in divergence form associated with amatrix $A$ with real, merely bounded and possibly non-symmetric 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 formassociated with the transpose matrix of $A$, then $\partial\Omega$ is uniformly$n$-rectifiable if and only if every bounded solution of $Lu=0$ and everybounded solution of $L^*v=0$ in $\Omega$ is $\varepsilon$-approximmable if andonly 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 isgiven in terms of the so-called $S<N$ estimates, and another in terms of asuitable corona decomposition involving $L$-harmonic and $L^*$-harmonicmeasures. We also prove that if $L$-harmonic measure and $L^*$-harmonic measuresatisfy a weak $A_\infty$-type condition, then $\partial \Omega$ is$n$-uniformly rectifiable. In the process we obtain a version ofAlt-Caffarelli-Friedman monotonicity formula for a fairly wide class ofelliptic operators which is of independent interest and plays a fundamentalrole in our arguments.

Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability via an ACF monotonicity formula