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The selection problem for some first-order stationary mean-field games. (English) Zbl 1455.35261

Summary: 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.

MSC:

35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
35A01 Existence problems for PDEs: global existence, local existence, non-existence
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
91A07 Games with infinitely many players
35B40 Asymptotic behavior of solutions to PDEs
35D30 Weak solutions to PDEs
35B35 Stability in context of PDEs
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References:

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