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

Abstract

Let $\Omega \subset {\mathbb{R}}^{n+1}$, $n\ge 2$, be an open set with Ahlfors-David 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 non-symmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^{\ast }$ 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 $L^{\ast }v=0$ in $\Omega$ is $\epsilon$-approximmable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^{\ast }v=0$ 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 $L^{\ast }$-harmonic measures. We also prove that if $L$-harmonic measure and $L^{\ast }$-harmonic measure satisfy a weak $A_{\infty }$-type condition, then $\partial \Omega$ is $n$-uniformly rectifiable. In the process we obtain a version of 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.

In this version we extend the main theorem to the non-symmetric case without assuming the domain to be uniform. In this situation new arguments are required. In particular, we need to prove a version of the ACF formula for a fairly wide class of uniformly elliptic operators with lower order terms, which may be of independent interest. We also prove two additional theorems and changed the title

References in corpus (1)

• $A_{\infty }$ implies NTA for a class of variable coefficient elliptic operators

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• $A_{\infty }$ implies NTA for a class of variable coefficient elliptic operators
• Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part II: The large constant case
• Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets
• BMO solvability and absolute continuity of harmonic measure
• Uniform rectifiability and $\epsilon$-approximability of harmonic functions in $L^p$