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

Institute for Mathematics and Computer Science, Tsuda University, 2-1-1 Tsuda, Kodaira, Tokyo, 187-8577 Japan

This contribution is part of the Special Issue: Critical values in nonlinear pdes – Special Issue dedicated to Italo Capuzzo Dolcetta
Guest Editor: Fabiana Leoni
Link: www.aimspress.com/mine/article/5754/special-articles

Special Issues: Critical values in nonlinear pdes - Special Issue dedicated to Italo Capuzzo Dolcetta

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.
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Keywords systems of Hamilton-Jacobi equations; Mather measures; vanishing discount

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

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