The selection problem for some first-order stationary Mean-field games

Diogo A. Gomes; Hiroyoshi Mitake; Kengo Terai; Diogo A. Gomes; Hiroyoshi Mitake; Kengo Terai

doi:10.3934/nhm.2020019

Networks and Heterogeneous Media

2020, Volume 15, Issue 4: 681-710. doi: 10.3934/nhm.2020019

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The selection problem for some first-order stationary Mean-field games

1.
King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900. Saudi Arabia
2.
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

Received: 01 August 2019 Revised: 01 June 2020 Published: 26 August 2020
35A01, 49L25, 91A07

Here, we study the existence and the convergence of solutions for the vanishing discount MFG problem with a quadratic Hamiltonian. We give conditions under which the discounted problem has a unique classical solution and prove convergence of the vanishing-discount limit to a unique solution up to constants. Then, we establish refined asymptotics for the limit. When those conditions do not hold, the limit problem may not have a unique solution and its solutions may not be smooth, as we illustrate in an elementary example. Finally, we investigate the stability of regular weak solutions and address the selection problem. Using ideas from Aubry-Mather theory, we establish a selection criterion for the limit.
- Mean field games,
- Hamilton-Jacobi equation,
- selection problem,
- vanishing discount,
- asymptotic analysis
Citation: Diogo A. Gomes, Hiroyoshi Mitake, Kengo Terai. The selection problem for some first-order stationary Mean-field games[J]. Networks and Heterogeneous Media, 2020, 15(4): 681-710. doi: 10.3934/nhm.2020019

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Abstract

Here, we study the existence and the convergence of solutions for the vanishing discount MFG problem with a quadratic Hamiltonian. We give conditions under which the discounted problem has a unique classical solution and prove convergence of the vanishing-discount limit to a unique solution up to constants. Then, we establish refined asymptotics for the limit. When those conditions do not hold, the limit problem may not have a unique solution and its solutions may not be smooth, as we illustrate in an elementary example. Finally, we investigate the stability of regular weak solutions and address the selection problem. Using ideas from Aubry-Mather theory, we establish a selection criterion for the limit.

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