HamiltonJacobi equations on metric measure spaces
Project/Area Number 
20K22315

Research Category 
GrantinAid for Research Activity Startup

Allocation Type  Multiyear Fund 
Review Section 
0201:Algebra, geometry, analysis, applied mathematics,and related fields

Research Institution  Okinawa Institute of Science and Technology Graduate University 
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)

Project Period (FY) 
20200911 – 20220331

Project Status 
Granted (Fiscal Year 2020)

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)

Keywords  eikonal equation / metric spaces / HamiltonJacobi equation / viscosity solution / HJ equations / wellposedness / metric measure spaces 
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 HamiltonJacobi equations and investigate the wellposedness on metric spaces.

Outline of Annual Research Achievements 
We show the equivalence between two wellknown notions of solutions to the eikonal equation and a more general class of HamiltonJacobi 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 semiconcavity 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.

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 timedependent HamiltonJacobi equations.

Strategy for Future Research Activity 
We will focus on the following two projects in the next step: 1) Study the viscosity solutions to discontinuous HamiltonJacobi 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 wellposedness on the general setting. 2) Study the timedependent HamiltonJacobi equations on metric spaces We also plan to investigate the extension of Monge solution to the timedependent HamiltonJacobi 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.

Report
(1 results)
Research Products
(5 results)