Hamilton-Jacobi equations on metric measure spaces

度量测度空间上的 Hamilton-Jacobi 方程

• 批准号: 20K22315
• 负责人: ZHOU Xiaodan
• 金额: $ 1.83万
• 依托单位: Okinawa Institute of Science and Technology Graduate University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Research Activity Start-up
• 财政年份: 2020
• 资助国家: 日本
• 起止时间: 2020-09-11 至 2024-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-20K22315/
• 关键词: eikonal equation; metric measure space; viscosity solution; Heisenberg group; h-quasiconvexity; Hamilton-Jacobi equation; metric measure spaces; differential games; viscosity solutions; metric spaces; HJ equations; well-posedness

The first project studies the eikonal equation in metric measure spaces, where the inhomogeneous term is allowed to be discontinuous, unbounded and merely p-integrable in the domain. Generalizing the notion of Monge solutions in metric space, we establish uniqueness and existence results for the associated Dirichlet boundary problem.The second project is concerned with a PDE-based approach to the horizontally quasiconvex envelope of a given continuous function in the Heisenberg group. We obtain the uniqueness and existence of viscosity solutions to the Dirichlet boundary problem for the nonlocal Hamilton-Jacobi equation.

第一个项目研究了公制度量空间中的艾科纳尔方程，其中允许不均匀的项是不均匀的，无限的，并且仅在域中p-积分。 概括了Monge溶液在度量空间中的概念，我们为相关的Dirichlet边界问题建立了独特性和存在结果。第二个项目与Heisenberg组的给定连续功能的水平Quasiconvex信封的基于PDE的方法有关。我们获得了非局部汉密尔顿 - 雅各比方程的Dirichlet边界问题的粘度解的独特性和存在。

项目成果

Horizontal quasiconvex envelope in the Heisenberg group

海森堡群中的水平拟凸包络

• 发表时间: 2021
• 作者: Masako Tamaki;Zhiyan Wang;Tyler Barnes-Diana;DeeAnn Guo;Aaron V. Berard;Edward Walsh;Takeo Watanabe;Yuka Sasaki.;Reiichiro Kawai;Liron Speyer;Masahiro Morimoto;Taiji Marugame;Xiaodan Zhou
• 通讯作者: Xiaodan Zhou

Horizontally quasiconvex envelope in the Heisenberg group

• DOI: 10.4171/rmi/1417
• 发表时间: 2022-05
• 期刊: Revista Matemática Iberoamericana
• 作者: Antoni Kijowski;Qing Liu;Xiaodan Zhou

DISCONTINUOUS EIKONAL EQUATIONS IN METRIC MEASURE SPACES

• DOI: 10.1090/tran/9294
• 发表时间: 2024-09-25
• 期刊: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
• 影响因子: 1.3
• 作者: Liu，Qing;Shanmugalingam，Nageswari;Zhou，Xiaodan
• 通讯作者: Zhou，Xiaodan

Characterization of horizontal quasiconvexity in the Heisenberg group and applications

海森堡群中水平拟凸性的表征及应用

• 发表时间: 2022
• 作者: Liu Qing;Zhou Xiaodan;Xiaodan Zhou;Xiaodan Zhou;Xiaodan Zhou;Xiaodan Zhou
• 通讯作者: Xiaodan Zhou

Horizontal convex envelope in the Heisenberg group

海森堡群中的水平凸包络

• 发表时间: 2020
• 作者: Xiaodan Zhou