| 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 / viscosity solution / metric spaces / Heisenberg group / differential games / convexity / eikonal equation / metric measure spaces / discontinuous data / h-quasiconvex functions / metric measure space / h-quasiconvexity / Hamilton-Jacobi equation / viscosity solutions / 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 Final Research Achievements |
First, showing the equivalence of solutions of eikonal equations in metric spaces by introducing a new simple notions called Monges solution. Exploiting the notion of Monge solution and using it to study discontinuous eikonal equations in metric measure spaces which produces even new results in the Euclidean spaces. Second, constructing a two-person continuous-time game in a geodesic space and show that the value function is the unique solution of the Hamilton-Jacobi equation. Our result develops, in a general geometric setting, the classical connection between differential games and the viscosity solutions to possibly nonconvex Hamilton-Jacobi equations. Third, using first-order and second-order PDE-based approaches to study the horizontally quasiconvex (h-quasiconvex for short) functions in the Heisenberg group and apply the characterizations to construct h-quasiconvex envelope and study the h-convexity preserving property for horizontal curvature flow in the Heisenberg group.
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| Academic Significance and Societal Importance of the Research Achievements |
Although several notions of viscosity solutions to the HJ equations on metric spaces have been introduced, our research reveals intrinsic connections between numerous results on HJ equations in general settings and has great potential to be applied in other fields.
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