Abstract
Here, we investigate the existence of solutions to a stationary mean-field game model introduced by J.-M. Lasry and P.-L. Lions. This model features a quadratic Hamiltonian and congestion effects. The fundamental difficulty of potential singular behavior is caused by congestion. Thanks to a new class of a priori bounds, combined with the continuation method, we prove the existence of smooth solutions in arbitrary dimensions.
Article PDF
Similar content being viewed by others
References
Amann H., Crandall M.G.: On some existence theorems for semi-linear elliptic equations. Indiana Univ. Math. J. 27(5), 779–790 (1978)
Achdou, Y.: Finite difference methods for mean-field games. In: Hamilton–Jacobi Equations: Approximations, Numerical Analysis and Applications, pp. 1–47. Springer, Berlin (2013)
Cardaliaguet, P.: Notes on mean-field games (2011)
Cardaliaguet P., Lasry J.-M., Lions P.-L., Porretta A.: Long time average of mean-field games. Netw. Heterog. Media 7(2), 279–301 (2012)
Dieudonné, J.: Foundations of modern analysis. Academic Press, New York-London, 1969. Enlarged and corrected printing, Pure and Applied Mathematics, vol. 10-I
Gomes D.A., Iturriaga R., Sánchez-Morgado H., Yu Y.: Mather measures selected by an approximation scheme. Proc. Am. Math. Soc. 138(10), 3591–3601 (2010)
Gomes D.A., Pires, G.E., Sánchez-Morgado, H.: A-priori estimates for stationary mean-field games. Netw. Heterog. Media 7(2), 303–314 (2012)
Gomes, D.A., Pimentel, E., Sanchez-Morgado, H.: Time dependent mean-field games in the superquadratic case (2013). arXiv:1311.6684
Gomes, D.A., Pimentel, E., Sanchez-Morgado, H.: Time dependent mean-field games in the subquadratic case. Commun. Partial Differ. Equ. (2014, to appear)
Gomes DA, Patrizi S, Voskanyan V (2014) On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal. 99:49–79
Gomes D.A., Saúde J.: Mean Field Games Models—A Brief Survey. Dyn. Games Appl. 4(2), 110–154 (2014)
Gomes D.A., Sánchez-Morgado H.: A stochastic Evans–Aronsson problem. Trans. Am. Math. Soc. 366(2), 903–929 (2014)
Gueant, O.: Existence and uniqueness result for mean-field games with congestion effect on graphs (preprint)
Huang M., Caines P.E., Malhamé R.P.: Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized \({\epsilon}\)-Nash equilibria. IEEE Trans. Autom. Control 52(9), 1560–1571 (2007)
Huang M., Malhamé R.P., Caines P.E.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–251 (2006)
Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619–625 (2006)
Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(10), 679–684 (2006)
Lasry J.-M., Lions P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)
Lasry, J.-M., Lions, P.-L.: Mean field games. Cahiers de la Chaire Finance et Développement Durable (2007)
Lasry, J.-M., Lions, P.-L., Gueant, O.: Mean field games and applications. Paris-Princeton lectures on Mathematical Finance (2010)
Lions, P.-L.: College de France course on mean-field games (2007–2011)
Porretta, A.: Weak solutions to Fokker–Planck equations and mean-field games (2013, preprint)
Porretta A.: On the planning problem for the mean field games system. Dyn. Games Appl. 4(2), 231–256 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
DG was partially supported by KAUST SRI, Uncertainty Quantification Center in Computational Science and Engineering, CAMGSD-LARSys through FCT-Portugal and by grants PTDC/MAT-CAL/0749/2012. HM was partially supported by the JST program to disseminate the tenure tracking system, and KAKENHI #15K17574, #26287024.
Rights and permissions
About this article
Cite this article
Gomes, D.A., Mitake, H. Existence for stationary mean-field games with congestion and quadratic Hamiltonians. Nonlinear Differ. Equ. Appl. 22, 1897–1910 (2015). https://doi.org/10.1007/s00030-015-0349-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00030-015-0349-7