2021 Fiscal Year Research-status Report
Hamilton-Jacobi equations on metric measure spaces
| Project/Area Number |
20K22315
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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 – 2023-03-31
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| Keywords | Hamilton-Jacobi equation / metric measure spaces / differential games / viscosity solutions / Heisenberg group |
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| Outline of Annual Research Achievements |
We provide a game-based interpretation of Hamilton-Jacobi-Isaacs equations in metric spaces. Our result develops the classical connection between differential games and the viscosity solutions to possibly nonconvex Hamilton-Jacobi equations.
We propose a notion of Monge solutions to eikonal equations in a metric space with a discontinuous inhomogeneous term. We obtained a comparison principle and existence and proved that it coincides with our previous definitions when the inhomogeneous term is continuous.
We studies a PDE-based approach to the horizontally quasiconvex envelope of a given continuous function in the Heisenberg group. One main result is to prove the uniqueness and existence of viscosity solutions to the Dirichlet boundary problems for the nonlocal Hamilton-Jacobi equation.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The completed paper mentioned in the summary of research achievement is related to the study of time-dependent Hamilton-Jacobi equations. This result extends the classical game-theoretic representation theorem for time-dependent Hamilton-Jacobi equations to metric spaces.
We also obtain well-posedness results on viscosity solutions to eikonal equations with discontinuous space variables in metric measure spaces. These results serve as an extension of the results for our first project when the inhomogeneous term is continuous and lay the foundation for the generalization in the next step.
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| Strategy for Future Research Activity |
We will mainly focus on studying viscosity solutions to discontinuous HJ equations in metric measure spaces and list two explicit questions for the next step.
In Euclidean spaces, two main references on the viscosity solution theory for the HJ equation with measurable dependence on the state variables we plan to compare to are given by Camilli-Siconolfi (2003) and Briani-Davini (2005). We propose to explore an explicit relation between these definitions and ours. Furthermore, we propose to extend the viscosity solutions to general discontinuous HJ equations in metric measure spaces.
Another question we propose to investigate focuses on the eikonal equation. We propose to relax the condition that the inhomogeneous term being essentially bounded to a weaker integrability condition.
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| Causes of Carryover |
Due to the global pandemic, all research visits and onsite conference travel plans have been cancelled or delayed. I will use the remaining amount in research travel and hosting a conference at OIST.
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Research Products
(4 results)
