### Mathematics in Engineering

2021, Issue 4: 1-21. doi: 10.3934/mine.2021032
Research article Special Issues

# The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling

• Received: 30 September 2019 Accepted: 06 February 2020 Published: 21 August 2020
• We establish a convergence theorem for the vanishing discount problem for a weakly coupled system of Hamilton-Jacobi equations. The crucial step is the introduction of Mather measures and their relatives for the system, which we call respectively viscosity Mather and Green-Poisson measures. This is done by the convex duality and the duality between the space of continuous functions on a compact set and the space of Borel measures on it. This is part 1 of our study of the vanishing discount problem for systems, which focuses on the linear coupling, while part 2 will be concerned with nonlinear coupling.

Citation: Hitoshi Ishii. The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling[J]. Mathematics in Engineering, 2021, 3(4): 1-21. doi: 10.3934/mine.2021032

### Related Papers:

• We establish a convergence theorem for the vanishing discount problem for a weakly coupled system of Hamilton-Jacobi equations. The crucial step is the introduction of Mather measures and their relatives for the system, which we call respectively viscosity Mather and Green-Poisson measures. This is done by the convex duality and the duality between the space of continuous functions on a compact set and the space of Borel measures on it. This is part 1 of our study of the vanishing discount problem for systems, which focuses on the linear coupling, while part 2 will be concerned with nonlinear coupling. ###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142 1.0 2.2

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