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

• Published in: International mathematics research notices 2023-06, Vol.2023 (13), p.10837-10941
• Main Authors: Azzam, Jonas, Garnett, John, Mourgoglou, Mihalis, Tolsa, Xavier
• Format: Article
• Language: English
• ISSN: 1073-7928; 1687-0247
• DOI: 10.1093/imrn/rnab095

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 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 \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 $\varepsilon $-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