The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling
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...
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| Format: | Article |
| Language: | English |
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AIMS Press
2021-03-01
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| Series: | Mathematics in Engineering |
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| Online Access: | http://www.aimspress.com/article/doi/10.3934/mine.2021032?viewType=HTML |
| _version_ | 1818622889532850176 |
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| author | Hitoshi Ishii |
| author_facet | Hitoshi Ishii |
| author_sort | Hitoshi Ishii |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-16T18:32:20Z |
| format | Article |
| id | doaj.art-9132be2d5b9e45548456c88c151984e1 |
| institution | Directory Open Access Journal |
| issn | 2640-3501 |
| language | English |
| last_indexed | 2024-12-16T18:32:20Z |
| publishDate | 2021-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematics in Engineering |
| spelling | doaj.art-9132be2d5b9e45548456c88c151984e12022-12-21T22:21:16ZengAIMS PressMathematics in Engineering2640-35012021-03-013412110.3934/mine.2021032The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear couplingHitoshi Ishii 0Institute for Mathematics and Computer Science, Tsuda University, 2-1-1 Tsuda, Kodaira, Tokyo, 187-8577 JapanWe 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.http://www.aimspress.com/article/doi/10.3934/mine.2021032?viewType=HTMLsystems of hamilton-jacobi equationsmather measuresvanishing discount |
| spellingShingle | Hitoshi Ishii The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling Mathematics in Engineering systems of hamilton-jacobi equations mather measures vanishing discount |
| title | The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling |
| title_full | The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling |
| title_fullStr | The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling |
| title_full_unstemmed | The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling |
| title_short | The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling |
| title_sort | vanishing discount problem for monotone systems of hamilton jacobi equations part 1 linear coupling |
| topic | systems of hamilton-jacobi equations mather measures vanishing discount |
| url | http://www.aimspress.com/article/doi/10.3934/mine.2021032?viewType=HTML |
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