论文标题
绿色功能和域的近似，具有所有维度的均匀整流边界
Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions
论文作者
论文摘要
本文在域边界的均匀重新覆盖和椭圆算子的绿色函数的属性之间建立了等效性，该属性通过仿射函数（到超高平面的距离）很好地近似。结果以多种方式是新颖的，尤其是（1）这是仿射函数控制绿色函数的基本特性，或者是从Carleson普遍存在的意义上讲，出现在文献中。即使在半空间中，这里建立的“直接”结果是新的。 （2）结果是最佳的，在（标准）轻度拓扑假设下提供了均匀重差性的完整表征； （3）据作者所知，这是适用于所有椭圆运算符的第一个自由边界结果，而无需任何限制系数（直接的一个假定标准，并且是必要的，Carleson测量条件）； （4）我们的定理适用于所有域，可能具有较低的尺寸边界：这是第一个自由边界导致较高的共同维度设置，因此，对于一组尺寸$ d $，$ d <n-1 $，在$ {\ mathbb {r}}}^n $中，对一组尺寸$ d $，$ d <n-1 $的第一个PDE表征均匀地表征。该论文提供了一种一般处理相关问题的一般方法，超出了上述定理的范围，包括绿色函数梯度的近似性的问题，以及将绿色函数与特定版本的距离与原始集合的特定版本进行比较，而不是与超涂层的距离。
The present paper establishes equivalence between uniform rectifiability of the boundary of a domain and the property that the Green function for elliptic operators is well approximated by affine functions (distance to the hyperplanes). The results are novel in a variety of ways, in particular (1) this is the first time the underlying property of the control of the Green function by affine functions, or by the distance to the boundary, in the sense of the Carleson prevalent sets, appears in the literature; the "direct" result established here is new even in the half space; (2) the results are optimal, providing a full characterization of uniform rectifiability under the (standard) mild topological assumptions; (3) to the best of the authors' knowledge, this is the first free boundary result applying to all elliptic operators, without any restriction on the coefficients (the direct one assumes the standard, and necessary, Carleson measure condition); (4) our theorems apply to all domains, with possibly lower dimensional boundaries: this is the first free boundary result in higher co-dimensional setting and as such, the first PDE characterization of uniform rectifiability for a set of dimension $d$, $d<n-1$, in ${\mathbb{R}}^n$. The paper offers a general way to deal with related issues considerably beyond the scope of the aforementioned theorem, including the question of approximability of the gradient of the Green function, and the comparison of the Green function to a certain version of the distance to the original set rather than distance to the hyperplanes.
