Horizontally quasiconvex envelope in the Heisenberg group
Horizontally quasiconvex envelope in the Heisenberg group
复制标题
DOI：
10.4171/rmi/1417
复制
发表时间：
2022-05
期刊：
影响因子：
--
通讯作者：
Antoni Kijowski;Qing Liu;Xiaodan Zhou
中科院分区：
文献类型：
--
作者：
Antoni Kijowski;Qing Liu;Xiaodan Zhou
研究了Heisenberg群中连续函数的水平拟凸（简称h-拟凸）包络的偏微分方程方法.利用一阶非局部Hamilton-Jacobi方程的粘性子解给出了上连续h-拟凸函数的一个特征.我们还通过迭代非局部算子来构造连续函数的相应包络。证明了非局部Hamilton-Jacobi方程Dirichlet边值问题粘性解的存在性和唯一性。我们的方法的应用程序的h-凸船体的海森堡群中的一个给定的集合进行了讨论。
This paper is concerned with a PDE-based approach to the horizontally quasiconvex (h-quasiconvex for short) envelope of a given continuous function in the Heisenberg group. We provide a characterization for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to a first-order nonlocal Hamilton-Jacobi equation. We also construct the corresponding envelope of a continuous function by iterating the nonlocal operator. One important step in our arguments is to prove the uniqueness and existence of viscosity solutions to the Dirichlet boundary problems for the nonlocal Hamilton-Jacobi equation. Applications of our approach to the h-convex hull of a given set in the Heisenberg group are discussed as well.