Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability
Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability
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均匀可校正性、椭圆测度、平方函数和 $varepsilon$ 逼近性
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
X. Tolsa
中科院分区：
文献类型：
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作者：
Jonas Azzam;J. Garnett;Mihalis Mourgoglou;X. Tolsa
设$\Omega\subset\mathbb{R}^{n+1}$，$n\geq 2$，是具有Ahlfors-David正则边界的开集.我们考虑一个一致椭圆算子L与一个系数为真实的且仅有界的矩阵A相关联的发散形式，该矩阵也是局部Lipschitz的，且满足适当的Carleson型估计.本文证明了：若A是对称的，且$\Omega$满足螺旋条件，则$\partial\Omega$是一致n$-可求长的当且仅当$Lu=0$的每一个有界解是$\vareps $-可逼近的当且仅当$Lu=0$在$\Omega$中的每一个有界解满足一个适当的平方函数Carleson测度估计.当矩阵$A$是非对称的时，在$\Omega$是一致域并且表示$L$被$L^*$转置的附加假设下，我们证明了$\partial\Omega$是一致n$-可求长的当且仅当$\Omega$中$Lu=0$和$L^*v=0$的每个有界解是$\partial\Omega$-近似的当且仅当$Lu=0$和$L^*v=0$在$\Omega$中的每一个有界解满足一个合适的平方函数Carleson测度估计。
Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, be an open set with Ahlfors-David regular boundary. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real and merely bounded coefficients which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we prove that if $A$ is symmetric and $\Omega$ satisfies the corkscrew condition, then $\partial\Omega$ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ is $\varepsilon$-approximable if and only if every bounded solution of $Lu=0$ in $\Omega$ satisfies a suitable square-function Carleson measure estimate. When the matrix $A$ is non-symmetric, under the additional assumption that $\Omega$ is a uniform domain and denoting the transpose of $L$ by $L^*$, we show that $\partial\Omega$ 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.
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