2020 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 – 2022-03-31
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| Keywords | eikonal equation / metric spaces / Hamilton-Jacobi equation / viscosity solution |
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| Outline of Annual Research Achievements |
We show the equivalence between two well-known notions of solutions to the eikonal equation and a more general class of Hamilton-Jacobi equations in complete and rectifiably connected metric spaces. Moreover, we introduce a simple definition called Monge solution and show the equivalence of all three solutions by using the induced intrinsic (path) metric for the associated Dirichlet boundary problem. Regularity of solutions related to the Euclidean semi-concavity is discussed as well. This result has been published in the Journal of Differential Equations.
Furthermore, we extend the definition of Monge solution to eikonal equations with discontinuous data and achieve the existence and comparison principle.
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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
We completed the first project of showing the equivalence between known solutions to eikonal equations and provided an alternative definition of solution called Monge solution. The simple formulation of Monge solution makes it easy to verify and can lead to many potential applications in metric spaces. We can use this notion in our following projects of studying eikonal equation with discontinuous data and time-dependent Hamilton-Jacobi equations.
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| Strategy for Future Research Activity |
We will focus on the following two projects in the next step:
1) Study the viscosity solutions to discontinuous Hamilton-Jacobi equations in metric measure spaces
We plan to define an appropriate notion of solutions, taking into consideration the measure associated with the space. By adapting the conventional viscosity solution techniques and tools from measure theory, we intend to establish the well-posedness on the general setting.
2) Study the time-dependent Hamilton-Jacobi equations on metric spaces
We also plan to investigate the extension of Monge solution to the time-dependent Hamilton-Jacobi equations and study the equivalence between this definition and other solutions. It is of our interest as well to study the regularity of viscosity solutions.
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| Causes of Carryover |
Due to the travel limit caused by the pandemic, all travels including my visit to other universities and hosting visitors at OIST are cancelled. Hence, this part of the budget remains unused in the previous fiscal year. We plan to organize more online workshop or seminars to facilitate the communication. The budget can be used to improve the meeting facility and honorarium payment for speakers. With the potential resuming of travel in the near future, we will use this part of remaining budget for academic travel as well.
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