In this paper we study evolutive first order Mean Field Games in the Heisenberg group; each agent can move in the whole space but it has to follow “horizontal” trajectories which are given in terms of the vector fields generating the group and the kinetic part of the cost depends only on the horizontal velocity. The Hamiltonian is not coercive in the gradient term and the coefficients of the first order term in the continuity equation may have a quadratic growth at infinity. The main results of this paper are two: the former is to establish the existence of a weak solution to the Mean Field Game systems while the latter is to represent this solution following the Lagrangian formulation of the Mean Field Games. We also provide some generalizations to Heisenberg-type structures.
Bibliographical noteKAUST Repository Item: Exported on 2022-05-25
Acknowledged KAUST grant number(s): OSR-2017-CRG6-3452.01
Acknowledgements: The first and the second authors are members of GNAMPA-INdAM and were partially supported also by the research project of the University of Padova “Mean-Field Games and Nonlinear PDEs”, by the Fondazione Cariparo Project “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games” and by KAUST project OSR-2017-CRG6-3452.01. The third author has been partially funded by the ANR project ANR-16-CE40-0015-01.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.
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