Uniform Rectifiability, Elliptic Measure, Square Functions, and ε-Approximability Via an ACF Monotonicity Formula
| Autor: | Mihalis Mourgoglou, Jonas Azzam, John Garnett, Xavier Tolsa |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | International Mathematics Research Notices. |
| ISSN: | 1687-0247 1073-7928 |
| DOI: | 10.1093/imrn/rnab095 |
| Popis: | 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 |
| Databáze: | OpenAIRE |
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