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

Diogo A. Gomes; Hiroyoshi Mitake; Kengo Terai

doi:10.3934/nhm.2020019

Article Contents

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

This issue Previous Article Efficient numerical methods for gas network modeling and simulation Next Article

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: August 2019

Revised: June 2020

Early access: August 2020

Published: December 2020

D. Gomes was partially supported by King Abdullah University of Science and Technology (KAUST) baseline funds and KAUST OSR-CRG2017-3452. H. Mitake was partially supported by the JSPS grants: KAKENHI #19K03580, #19H00639, #17KK0093, #20H01816. K. Terai was supported by King Abdullah University of Science and Technology (KAUST) through the Visiting Student Research Program (VSRP) and by the JSPS grants: KAKENHI #20J10824

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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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Mathematics Subject Classification: 35A01, 49L25, 91A07.

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Figure 1. Density $m$ for (2.2) which exhibits areas with no agents

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Figure 2. Two distinct solutions, $\hat u$ and $\tilde u$ , of the Hamilton-Jacobi equation in (2.2). Their gradients differ only when $m$ vanishes

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