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 |
Granted (Fiscal Year 2022)
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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 space / viscosity solution / Heisenberg group / h-quasiconvexity / Hamilton-Jacobi equation / metric measure spaces / 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 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.
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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 first project extends our previous results of eikonal equations on metric measure spaces with continuous inhomogeneous term. The main results are all completed and the paper is under final revision for submission.
The second project studies Hamilton-Jacobi equations on the Heisenberg group and its application to horizontally quasiconvex function. The paper has been accepted for publication at Rev. Mat. Iberoam.
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| Strategy for Future Research Activity |
We will continue the study of viscosity solutions to Hamilton-Jacobi equations in general metric measure spaces and Heisenberg group in the following two aspects.
1. Study the solution to eikonal equation with more general inhomogeneous term and on more general conditions for the metric space. Extend the method and results to other classes of Hamilton-Jacobi equations.
2. Study second-order characterization of horizontally quasiconvex functions in the Heisenberg group and its application to properties including convexity preserving of horizontal mean curvature flow equations.
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Report
(3 results)
Research Products
(15 results)
