A variational approach to second order mean field games with density constraints: The stationary case

Mészáros, Alpár Richárd; Silva, Francisco J.

Abstract

In this paper we study second order stationary Mean Field Game systems under density constraints on a bounded domain . We show the existence of weak solutions for power-like Hamiltonians with arbitrary order of growth. Our strategy is a variational one, i.e. we obtain the Mean Field Game system as the optimality condition of a convex optimization problem, which has a solution. When the Hamiltonian has a growth of order , the solution of the optimization problem is continuous which implies that the problem constraints are qualified. Using this fact and the computation of the subdifferential of a convex functional introduced by Benamou and Brenier (see [1]), we prove the existence of a solution of the MFG system. In the case where the Hamiltonian has a growth of order , the previous arguments do not apply and we prove the existence by means of an approximation argument.

Citation

Mészáros, A. R., & Silva, F. J. (2015). A variational approach to second order mean field games with density constraints: The stationary case. Journal de Mathématiques Pures et Appliquées, 104(6), 1135-1159. https://doi.org/10.1016/j.matpur.2015.07.008

• Journal Article Type: Article
• Online Publication Date: Jul 1, 2015
• Publication Date: Dec 1, 2015
• Deposit Date: Oct 1, 2019
• Publicly Available Date: Feb 28, 2020
• Journal: Journal de Mathématiques Pures et Appliquées
• Print ISSN: 0021-7824
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 104
• Issue: 6
• Pages: 1135-1159
• DOI: https://doi.org/10.1016/j.matpur.2015.07.008
• Related Public URLs: https://arxiv.org/abs/1502.06026