Contributors: Dipartimento di Matematica Padova; Università degli Studi di Padova = University of Padua (Unipd); Department of Information Engineering Padova (DEI); Université de Rennes (UR); Institut de Recherche Mathématique de Rennes (IRMAR); Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes); Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest; Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro); prot. BIRD 187147, Università degli Studi di Padova; ANR-16-CE40-0015-01, Agence Nationale de la Recherche; ANR-16-CE40-0015,MFG,Jeux Champs Moyen(2016); ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)
نبذة مختصرة : International audience ; We study first order evolutive Mean Field Games where the Hamiltonian is non-coercive. This situation occurs, for instance, when some directions are "forbidden" to the generic player at some points. We establish the existence of a weak solution of the system via a vanishing viscosity method and, mainly, we prove that the evolution of the population's density is the push-forward of the initial density through the flow characterized almost everywhere by the optimal trajectories of the control problem underlying the Hamilton-Jacobi equation. As preliminary steps, we need that the optimal trajectories for the control problem are unique (at least for a.e. starting points) and that the optimal controls can be expressed in terms of the horizontal gradient of the value function.
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