We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^d$ under two simple geometric conditions: The Dirichlet boundary part is Ahlfors--David regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the Neumann boundary part. This improves upon all existing results even in the case of pure Dirichlet or Neumann boundary conditions. We also treat elliptic systems with lower order terms. As a side product we establish new regularity results for the fractional powers of the Laplacian with boundary conditions in our geometric setup.

中文翻译：

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局部均匀域上的加藤平方根问题

我们在两个简单的几何条件下，在 $\mathbb{R}^d$ 中的开集和可能无界集上以散度形式获得了具有混合边界条件的二阶椭圆算子的加藤平方根估计：狄利克雷边界部分是 Ahlfors--大卫正则和本着局部统一域精神的定量连通性属性在诺依曼边界部分附近成立。即使在纯 Dirichlet 或 Neumann 边界条件的情况下，这也改进了所有现有结果。我们还处理具有低阶项的椭圆系统。作为副产品，我们在几何设置中为具有边界条件的拉普拉斯算子的分数幂建立新的正则性结果。