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Nonlinear elliptic systems and mean-field games

Bardi, Martino and Feleqi, Ermal 2016. Nonlinear elliptic systems and mean-field games. Nonlinear Differential Equations and Applications NoDEA 23 (4) 10.1007/s00030-016-0397-7

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We consider a class of quasilinear elliptic systems of PDEs consisting of N Hamilton–Jacobi–Bellman equations coupled with N divergence form equations, generalising to N > 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of sufficient conditions for the existence of solutions to these systems: either the Hamiltonians are required to behave at most linearly for large gradients, as it occurs when the controls of the agents are bounded, or they must grow faster than linearly and not oscillate too much in the space variables, in a suitable sense. We show the connection of these systems with the classical strongly coupled systems of Hamilton–Jacobi–Bellman equations of the theory of N-person stochastic differential games studied by Bensoussan and Frehse. We also prove the existence of Nash equilibria in feedback form for some N-person games.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Springer
ISSN: 1021-9722
Date of First Compliant Deposit: 1 December 2016
Date of Acceptance: 9 July 2016
Last Modified: 23 Jul 2020 02:07

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