Non coercive unbounded first order Mean Field Games: The Heisenberg example

• Published in: Journal of Differential Equations Vol. 309; pp. 809 - 840
• Main Authors: Mannucci, Paola, Marchi, Claudio, Tchou, Nicoletta
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 05.02.2022; Elsevier
• Subjects: Analysis of PDEs; Continuity equation; Degenerate optimal control problem; First order Hamilton-Jacobi equations; Fokker-Planck equation; Heisenberg group; Heisenberg-type groups; Mathematics; Mean Field Games; Noncoercive Hamiltonian; Optimization and Control; 35Q91; Heisenberg group; 49K20; Mean Field Games; Degenerate optimal control problem; Continuity equation; 35F50; Heisenberg-type groups; Fokker-Planck equation; 49L25; First order Hamilton-Jacobi equations; Noncoercive Hamiltonian; degenerate optimal control problem. 2010 AMS Subject classification: 35F50; Heisenbergtype groups; first order Hamilton-Jacobi equations; continuity equation; noncoercive Hamiltonian
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2021.11.029

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.