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

Mészáros, Alpár Richárd; Silva, Francisco J.

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


Authors

Francisco J. Silva



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

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