# American Institute of Mathematical Sciences

doi: 10.3934/nhm.2020019

## 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 Published  August 2020

Fund Project: 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

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.

Citation: Diogo A. Gomes, Hiroyoshi Mitake, Kengo Terai. The selection problem for some first-order stationary Mean-field games. Networks & Heterogeneous Media, doi: 10.3934/nhm.2020019
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Density $m$ for (2.2) which exhibits areas with no agents
Two distinct solutions, $\hat u$ and $\tilde u$, of the Hamilton-Jacobi equation in (2.2). Their gradients differ only when $m$ vanishes