# American Institute of Mathematical Sciences

June  2012, 7(2): 337-347. doi: 10.3934/nhm.2012.7.337

## A modest proposal for MFG with density constraints

 1 Laboratoire de Mathématiques d'Orsay, Faculté de Sciences, Université Paris-Sud, 91405 Orsay cedex, France

Received  November 2011 Revised  March 2012 Published  June 2012

We consider a typical problem in Mean Field Games: the congestion case, where in the cost that agents optimize there is a penalization for passing through zones with high density of agents, in a deterministic framework. This equilibrium problem is known to be equivalent to the optimization of a global functional including an $L^p$ norm of the density. The question arises as to produce a similar model replacing the $L^p$ penalization with an $L^\infty$ constraint, but the simplest approaches do not give meaningful definitions. Taking into account recent works about crowd motion, where the density constraint $\rho\leq 1$ was treated in terms of projections of the velocity field onto the set of admissible velocity (with a constraint on the divergence) and a pressure field was introduced, we propose a definition and write a system of PDEs including the usual Hamilton-Jacobi equation coupled with the continuity equation. For this system, we analyze an example and propose some open problems.
Citation: Filippo Santambrogio. A modest proposal for MFG with density constraints. Networks and Heterogeneous Media, 2012, 7 (2) : 337-347. doi: 10.3934/nhm.2012.7.337
##### References:
 [1] L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191-246. [2] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. [3] J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002. [4] G. Buttazzo, C. Jimenez and E. Oudet, An optimization problem for mass transportation with congested dynamics, SIAM J. Control Optim., 48 (2009), 1961-1976. doi: 10.1137/07070543X. [5] G. Dal Maso, "An Introduction to $\Gamma-$Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. [6] E. De Giorgi, New problems on minimizing movements, in "Boundary Value Problems for PDE and Applications" (eds. C. Baiocchi and J. L. Lions), RMA Res. Notes Appl. Math., 29, Masson, Paris, (1993), 81-98. [7] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [8] J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018. [9] J.-M. Lasry and P.-L. Lions, Mean-field games, Japan. J. Math, 2 (2007), 229-260. [10] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Mat. Mod. Meth. Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799. [11] B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Net. Het. Media, 6 (2011), 485-519. [12] B. Maury and J. Venel, "Handling of Contacts in Crowd Motion Simulations," Traffic and Granular Flow, Springer, 2007. [13] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. [14] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. [15] S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Disc. Cont. Dyn. Systems, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427. [16] C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58, AMS, Providence, RI, 2003. [17] C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.

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##### References:
 [1] L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191-246. [2] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. [3] J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002. [4] G. Buttazzo, C. Jimenez and E. Oudet, An optimization problem for mass transportation with congested dynamics, SIAM J. Control Optim., 48 (2009), 1961-1976. doi: 10.1137/07070543X. [5] G. Dal Maso, "An Introduction to $\Gamma-$Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. [6] E. De Giorgi, New problems on minimizing movements, in "Boundary Value Problems for PDE and Applications" (eds. C. Baiocchi and J. L. Lions), RMA Res. Notes Appl. Math., 29, Masson, Paris, (1993), 81-98. [7] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [8] J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018. [9] J.-M. Lasry and P.-L. Lions, Mean-field games, Japan. J. Math, 2 (2007), 229-260. [10] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Mat. Mod. Meth. Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799. [11] B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Net. Het. Media, 6 (2011), 485-519. [12] B. Maury and J. Venel, "Handling of Contacts in Crowd Motion Simulations," Traffic and Granular Flow, Springer, 2007. [13] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. [14] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. [15] S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Disc. Cont. Dyn. Systems, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427. [16] C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58, AMS, Providence, RI, 2003. [17] C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.
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