Hamilton-Jacobi equations on metric measure spaces
| Project/Area Number |
20K22315
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Multi-year Fund |
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| Review Section |
0201:Algebra, geometry, analysis, applied mathematics,and related fields
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| Research Institution | Okinawa Institute of Science and Technology Graduate University |
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Principal Investigator |
ZHOU Xiaodan 沖縄科学技術大学院大学, 距離空間上の解析ユニット, 准教授 (10871494)
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| Project Period (FY) |
2020-09-11 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| Budget Amount *help |
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| Keywords | eikonal equation / metric measure spaces / viscosity solution / discontinuous data / Heisenberg group / h-quasiconvex functions / metric measure space / h-quasiconvexity / Hamilton-Jacobi equation / differential games / viscosity solutions / metric spaces / HJ equations / well-posedness |
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| Outline of Research at the Start |
Motivated by the rapid developments of optimal transport, control theory, data sciences etc., there is a growing interest in studying the nonlinear PDEs on metric measure spaces. We propose to focus on first order Hamilton-Jacobi equations and investigate the well-posedness on metric spaces.
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| Outline of Annual Research Achievements |
The first project studies the discontinuous eikonal equation in metric measure spaces. Besides uniqueness and existence results for the associated Dirichlet boundary, we obtain the regularity of the unique solution under suitable assumptions.
The second project is concerned with a PDE approach to horizontally quasiconvex functions in the Heisenberg group based on a nonlinear second order elliptic operator. We discuss sufficient conditions and necessary conditions for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to the associated elliptic equation.
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Report
(4 results)
Research Products
(15 results)
