Feneuil Joseph, Li Linhan, Mayboroda Svitlana
Mathematical Sciences Institute, Australian National University, Acton, ACT Australia.
School of Mathematics, The University of Edinburgh, Edinburgh, UK.
Math Ann. 2024;389(3):2637-2727. doi: 10.1007/s00208-023-02715-6. Epub 2023 Sep 5.
In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function behaves like a distance function to the boundary, in the sense that is the density of a Carleson measure, where is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa's -number of uniformly rectifiable sets. We believe that the method can be applied to many other problems at the intersection of PDE and geometric measure theory, and in particular, we are able to derive a generalization of the classical F. and M. Riesz theorem to the same class of elliptic operators as above.
在本文中，我们证明了对于单侧弦弧区域上具有非光滑系数的一类最优椭圆算子，当且仅当格林函数在如下意义下表现得像到边界的距离函数时，该区域的边界是一致可求长的：即 是卡尔松测度的密度，其中 是适应于该区域边界的正则化距离。我们证明中的主要要素是一种与托尔斯关于一致可求长集的 -数兼容的冠分解。我们相信该方法可以应用于偏微分方程和几何测度理论交叉领域的许多其他问题，特别是，我们能够将经典的F. 和M. 里斯定理推广到与上述相同类别的椭圆算子。
