First order mean field games with density constraints: Pressure equals price

SIAM Journal on Control and Optimization

Volumn 54, Issue 5, 2016, Pages 2672-2709

  Cardaliaguet, Pierre ^a   Mészáros, Alpár R ^b   Santambrogio, Filippo ^c  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSITÉ PARIS SUD   (France)

Author keywords

Benamou Brenier; BV functions; Convex duality; Hamilton Jacobi; Incompressible Euler; Regularity

Indexed keywords

OPTIMIZATION;

BENAMOU-BRENIER; BV FUNCTIONS; CONVEX DUALITY; HAMILTON-JACOBI; INCOMPRESSIBLE EULER; REGULARITY;

COSTS;

EID: 84992539178     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/15M1029849     Document Type: Article

References (30)

1. L. AMBROSIO AND A. FIGALLI, On the regularity of the pressure field of Brenier's weak solutions to incompressible Euler equations, Calc. Var. Partial Differential Equations, 31 (2008), pp. 497-509.
2. L. AMBROSIO AND A. FIGALLI, Geodesics in the space of measure-preserving maps and plans, Arch. Ration. Mech. Anal., 194 (2009), pp. 421-462.
3. L. AMBROSIO, N. GIGLI, AND G. SAVARÉ, Gradient Flows in Metric Spaces and in the Space of Probability Measure, Birkhäuser, 2008.
4. J.-D. BENAMOU AND Y. BRENIER, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375-393.
5. J.-D. BENAMOU AND G. CARLIER, Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations, J. Optim. Theory Appl., 167 (2015), pp. 1-26.
6. L. BRASCO, G. CARLIER, AND F. SANTAMBROGIO, Congested traffic dynamics, weak flows and very degenerate elliptic equations, J. Math. Pures Appl. (9), 93 (2010), pp. 652-671.
7. Y. BRENIER, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. Math., 52 (1999), pp. 411-452.
8. Y. BRENIER, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), pp. 805-808.
9. Y. BRENIER, Polar factorization and monotone rearrangement of vector valued functions, Comm. Pure Appl. Math., 44 (1991), pp. 375-417.
10. M. BURGER, M. DI FRANCESCO, P. MARKOWICH, AND M.-T. WOLFRAM, Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), pp. 1311-1333.
11. P. CARDALIAGUET, Weak solutions for first order mean field games with local coupling, in Analysis and Geometry in Control Theory and Its Applications, Springer INdAM Ser. 11, Springer, Cham, Switzerland, 2015, pp. 111-158.
12. P. CARDALIAGUET, G. CARLIER, AND B. NAZARET, Geodesics for a class of distances in the space of probability measures, Calc. Var. Partial Differential Equations, 48 (2013), pp. 395-420.
13. P. CARDALIAGUET AND P.J. GRABER, Mean field games systems of first order, ESAIM Control Optim. Calc. Var., 21 (2015), pp. 690-722.
14. P. CARDALIAGUET, P.J. GRABER, A. PORRETTA, AND D. TONON, Second order mean field games with degenerate diffusion and local coupling, NoDEA Nonlinear Differential Equations Appl., 22 (2015), pp. 1287-1317.
15. P. CARDALIAGUET, A. PORRETTA, AND D. TONON, Sobolev regularity for the first order Hamilton-Jacobi equation, Calc. Var. Partial Differential Equations, 54 (2015), pp. 3037-3065.
16. G. CARLIER, C. JIMENEZ, AND F. SANTAMBROGIO, Optimal transportation with traffic congestion and Wardrop equilibria, SIAM J. Control Optim., 47 (2008), pp. 1330-1350, https://doi.org/10.1137/060672832.
17. I. EKELAND AND R. TEMAM, Convex Analysis and Variational Problems, Stud. Math. Appl. 1, North-Holland, Amsterdam, Oxford, Elsevier, New York, 1976; translated from the French.
18. D.A. GOMES AND H. MITAKE, Existence for stationary mean-field games with congestion and quadratic Hamiltonians, NoDEA Nonlinear Differential Equations Appl., 22 (2015), pp. 1897-1910.
19. D.A. GOMES AND V. VOSKANYAN, Short-time existence of solutions for mean-field games with congestion, J. Lond. Math. Soc. (2), 92 (2015), pp. 778-799.
20. P.J. GRABER, Optimal control of first-order Hamilton-Jacobi equations with linearly bounded Hamiltonian, Appl. Math. Optim., 70 (2014), pp. 185-224.
21. M. HUANG, R.P. MALHAMÉ, AND P.E. CAINES, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), pp. 221-252.
22. J.-M. LASRY AND P.-L. LIONS, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), pp. 619-625.
23. 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), pp. 679-684.
24. J.-M. LASRY AND P.-L. LIONS, Mean field games, Jpn. J. Math., 2 (2007), pp. 229-260.
25. P.-L. LIONS, Cours au Collège de France, http://www.college-de-france.fr.
26. R.J. MCCANN, A convexity principle for interacting gases, Adv. Math., 128 (1997), pp. 153-179.
27. A.R. MÉSZÁROS AND F.J. SILVA, A variational approach to second order mean field games with density constraints: the stationary case, J. Math. Pures Appl. (9), 104 (2015), pp. 1135-1159.
28. F. SANTAMBROGIO, A modest proposal for MFG with density constraints, Netw. Heterog. Media, 7 (2012), pp. 337-347.
29. F. SANTAMBROGIO, Optimal Transport for Applied Mathematicians. Calculus of variations, PDEs, and modeling, Progr. Nonlinear Differential Equations Appl. 87, Birkäuser/Springer, Cham, Switzerland, 2015.
30. C. VILLANI, Topics in Optimal Transportation, Grad. Stud. Math., AMS, Providence, RI, 2003.