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A modest proposal for MFG with density constraints

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  • 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.
    Mathematics Subject Classification: Primary: 91A23; Secondary: 49K15, 49K20, 49L20, 35F25.


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  • [1]

    L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191-246.


    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.


    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.


    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.


    G. Dal Maso, "An Introduction to $\Gamma-$Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.


    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.


    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.


    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.


    J.-M. Lasry and P.-L. Lions, Mean-field games, Japan. J. Math, 2 (2007), 229-260.


    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.


    B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Net. Het. Media, 6 (2011), 485-519.


    B. Maury and J. Venel, "Handling of Contacts in Crowd Motion Simulations," Traffic and Granular Flow, Springer, 2007.


    R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179.doi: 10.1006/aima.1997.1634.


    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.


    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.


    C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58, AMS, Providence, RI, 2003.


    C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.

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