On the variational formulation of some stationary second-order mean field games systems

SIAM Journal on Mathematical Analysis

Volumn 50, Issue 1, 2018, Pages 1255-1277

  Mészáros, Alpár Richárd ^a   Silva, Francisco J ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF LIMOGES   (France)

Author keywords

Density constraints; General couplings; Mean Field Games; Variational formulation; Weak solutions

Indexed keywords

MATHEMATICAL MODELS; MATHEMATICAL TECHNIQUES;

DENSITY CONSTRAINTS; GENERAL COUPLING; MEAN FIELD GAMES; VARIATIONAL FORMULATION; WEAK SOLUTION;

BOUNDARY CONDITIONS;

EID: 85043509596     PISSN: 00361410     EISSN: 10957154     Source Type: Journal    
DOI: 10.1137/17M1125960     Document Type: Article

References (44)

1. [ABC17] Y. Achdou, M. Bardi, and M. Cirant, Mean field games models of segregation, Math. Models Methods Appl. Sci., 27 (2017), pp. 75–113.
2. [ABM06] H. Attouch, G. Buttazzo, and G. Michaille, Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization, MPS/SIAM Series on Optimization, SIAM, Philadelphia, PA, 2006.
3. [AD10] Y. Achdou and I. Capuzzo Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), pp. 1136–1162.
4. [Ada75] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York, 1975.
5. [AFG17] N. Almulla, R. Ferreira, and D. Gomes, Two numerical approaches to stationary mean-field games, Dyn. Games Appl., 7 (2017), pp. 657–682.
6. [BAKS16] L. Briceño-Arias, D. Kalise, and F.-J. Silva, Proximal methods for stationary mean field games with local couplings, SIAM J. Control Optim., to appear.
7. [BF16] M. Bardi and E. Feleqi, Nonlinear elliptic systems and mean-field games, Nonlinear Differential Equations Appl., 23 (2016): Art. 44, 32.
8. [BKRS15] V.-I. Bogachev, N.-V. Krylov, M. Röckner, and S.-V. Shaposhnikov, Fokker-Planck-Kolmogorov Equations, Mathematical Surveys and Monographs 207, American Mathematical Society, Providence, RI, 2015.
9. [BOP16] L. Boccardo, L. Orsina, and A. Porretta, Strongly coupled elliptic equations related to mean-field games systems, J. Differential Equations, 261 (2016), pp. 1796–1834.
10. [Bré72] H. Brézis, Intégrales convexes dans les espaces de Sobolev, in Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces, Vol. 13, Jerusalem, 1972, 1972, pp. 9–23.
11. [BS00] J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.
12. [CC16] S. Cacace and F. Camilli, A generalized newton method for homogenization of Hamilton–Jacobi equations, SIAM J. Sci. Comput., 38 (2016), A3589–A3617.
13. [CC17] A. Cesaroni and M. Cirant, Concentration of Ground States in Stationary MFG Systems, preprint, http://arXiv:1705.10741, 2017.
14. [CG15] P. Cardaliaguet and P. J. Graber, Mean field games systems of first order, ESAIM Control Optim. Calc. Var., 21 (2015), pp. 690–722.
15. [CGPSM16] M. Cirant, D. Gomes, E. Pimentel, and H. Sánchez-Morgado, Singular Mean-Filed Games, preprint, http://arXiv:1611.07187, 2016.
16. [Cir15] M. Cirant, Multi-population mean field games systems with Neumann boundary conditions, J. Math. Pures Appl. (9), 103 (2015), pp. 1294–1315.
17. [Cir16] M. Cirant, Stationary focusing mean-field games, Comm. Partial Differential Equations, 41 (2016), pp. 1324–1346.
18. [CLLP12] P. Cardaliaguet, J.-M. Lasry, P.-L. Lions, and A. Porretta, Long time average of mean field games, Netw. Heterog. Media, 7 (2012), pp. 279–301.
19. [CLLP13] P. Cardaliaguet, J.-M. Lasry, P.-L. Lions, and A. Porretta, Long time average of mean field games with a nonlocal coupling, SIAM J. Control Optim., 51 (2013), pp. 3558–3591.
20. [CMS16] P. Cardaliaguet, A. R. Mészáros, and F. Santambrogio, First order mean field games with density constraints: Pressure equals price, SIAM J. Control Optim., 54 (2016), pp. 2672–2709.
21. [CV16] M. Cirant and G. Verzini, Bifurcation and segregation in quadratic two-populations mean field games systems, ESAIM Control Optim. Calc. Var., 23 (2017), pp. 1145–1177.
22. [Dac08] B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd ed., Applied Mathematical Sciences 78, Springer, New York, 2008.
23. [Dal14] J. Dalphin, Some characterizations of a uniform ball property, in Congrès SMAI 2013, ESAIM Proc. Surveys 45, EDP Sci., Les Ulis, 2014, pp. 437–446.
24. [EG92] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
25. [Fel13] E. Feleqi, The derivation of ergodic mean field game equations for several populations of players, Dyn. Games Appl., 3 (2013), pp. 523–536.
26. [FG16] R. Ferreira and D.-A. Gomes, Existence of Weak Solutions to Stationary Mean-Field Games through Variational Inequalities, http://arxiv.org/abs/1512.05828,2016.
27. [GM12] M. Giaquinta and L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Edizioni della Normale, Pisa, 2012.
28. [GM15] D.-A. Gomes and H. Mitake, Existence for stationary mean-field games with congestion and quadratic Hamiltonians, Nonlinear Differential Equations Appl., 22 (2015), pp. 1897–1910.
29. [GNP17] D.-A. Gomes, L. Nurbekyan, and M. Prazeres, One-dimensional stationary mean-field games with local coupling, Dynamic Games and Applications, to appear, (2017).
30. [GPSM12] D.-A. Gomes, G.-E. Pires, and H. Sánchez-Morgado, A-priori estimates for stationary mean-field games, Netw. Heterog. Media, 7 (2012), pp. 303–314.
31. [GPV14] D.-A. Gomes, S. Patrizi, and V. Voskanyan, On the existence of classical solutions for stationary extended mean field games, Nonlinear Anal., 99 (2014), pp. 49–79.
32. [GSM14] D. Gomes and H. Sánchez Morgado, A stochastic Evans-Aronsson problem, Trans. Amer. Math. Soc., 366 (2014), pp. 903–929.
33. [GT83] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983.
34. [LL06] 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.
35. [LL07] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), pp. 229–260.
36. [MS15] 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., 104 (2015), pp. 1135–1159.
37. [Nit11] R. Nittka, Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains, J. Differential Equations, 251 (2011), pp. 860–880.
38. [PV17] E.-A. Pimentel and V. Voskanyan, Regularity for second order stationary mean-field games, Indiana Univ. Math. J., 66 (2017), pp. 1–22.
39. [Roc70] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.
40. [RW98] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften 317 [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1998.
41. [San12] F. Santambrogio, A modest proposal for MFG with density constraints, Netw. Heterog. Media, 7 (2012), pp. 337–347.
42. [Sch66] L. Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX–X, nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966.
43. [Sta65] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), pp. 189–258.
44. [Tru73] N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Sc. Norm. Super. Pisa (3), 27 (1973), pp. 265–308.