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

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