2023 Fiscal Year Final Research Report
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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| Keywords | Eikonal equation / viscosity solution / metric spaces / Heisenberg group / differential games / convexity |
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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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| Free Research Field |
Analysis on metric spaces
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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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