Uniform Rectifiability, Elliptic Measure, Square Functions, and ε-Approximability Via an ACF Monotonicity Formula
Uniform Rectifiability, Elliptic Measure, Square Functions, and ε-Approximability Via an ACF Monotonicity Formula
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通过 ACF 单调性公式计算均匀可整流性、椭圆测度、平方函数和 ε 逼近性
DOI：
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发表时间：
2021
影响因子：
1
通讯作者：
X. Tolsa
中科院分区：
文献类型：
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作者：
Jonas Azzam;John W. Garnett;Mihalis Mourgoglou;X. Tolsa
设$\Omega \subset{{\mathbb{R}^{n+1}$，$n\geq 2$，是一个具有Ahlfors正则边界且满足螺旋条件的开集.我们考虑一个一致椭圆算子L$的发散形式与矩阵A$与真实的，只是有界的，可能是非对称的系数，这也是局部Lipschitz和满足适当的Carleson型估计。本文证明了：如果L^*$是A的转置矩阵的发散型算子，则$\partial \Omega $是一致n$-可求长的当且仅当$Lu=0$的每一个有界解和$L^*v=0$在$\Omega $中的每一个有界解是$\partial $-可逼近的当且仅当$Lu=0$的每一个有界解并且$L^*v=0$在$\Omega $中的每一个有界解都满足一个适当的平方函数Carleson测度估计。此外，我们得到了两个额外的一致可求直的准则。一个是在所谓的“$S<N$”估计方面，另一个是在一个合适的电晕分解涉及$L$-调和和$L^*$-调和措施。证明了如果L$-调和测度和L^*$-调和测度满足弱A\infty $-型条件，则$\partial \Omega $是n$-一致可求长的.在这个过程中，我们得到了一个版本的Alt-Caffarelli-Friedman单调性公式的一个相当广泛的一类椭圆算子，这是独立的利益，并在我们的论点中发挥了重要作用。
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 “$S<N$” estimates, and another in terms of a suitable corona decomposition involving $L$-harmonic and $L^*$-harmonic measures. We also prove that if $L$-harmonic measure and $L^*$-harmonic measure satisfy a weak $A_\infty $-type condition, then $\partial \Omega $ is $n$-uniformly rectifiable. In the process we obtain a version of the 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.
具有 Ahlfors-David 正则边界的单边 NTA 域上的可修正性和椭圆测度
影响因子：
1.3
作者：
Akman, Murat;Badger, Matthew;Hofmann, Steve;Martell, Jose Maria
通讯作者：
Martell, Jose Maria