Introduction to Variational Methods for Viscous Ergodic Mean-Field Games with Local Coupling

  • Annalisa CesaroniEmail author
  • Marco Cirant
Part of the Springer INdAM Series book series (SINDAMS, volume 33)


We collect in these notes some results on the existence and uniqueness of classical solutions to viscous ergodic Mean-Field Game systems with local coupling. We present in particular some methods and ideas based on convex optimization techniques and elliptic regularity.


Ergodic Mean-Field Games Elliptic systems Variational methods 



This work has been supported by the INdAM Intensive Period “Contemporary Research in elliptic PDEs and related topics”. The authors are partially supported by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games”.


  1. 1.
    S. Agmon, The L p approach to the Dirichlet problem. I. Regularity theorems. Ann. Scuola Norm. Sup. Pisa (3) 13, 405–448 (1959)Google Scholar
  2. 2.
    M. Bardi, E. Feleqi, The derivation of ergodic mean field game equations for several populations of players. Dyn. Games Appl. 3(4), 523–536 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    M. Bardi, E. Feleqi, Nonlinear elliptic systems and mean field games. NoDEA Nonlinear Differ. Equ. Appl. 23, 23–44 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    G. Barles, P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations. SIAM J. Math. Anal. 32(6), 1311–1323 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    J.-D. Benamou, G. Carlier, F. Santambrogio, Variational Mean Field Games, ed. by Bellomo, Degond, Tadmor. Active Particles, vol. 1 (Springer, Berlin, 2017)Google Scholar
  6. 6.
    A. Bensoussan, Perturbation Methods in Optimal Control (John Wiley & Sons, Hoboken, 1988)zbMATHGoogle Scholar
  7. 7.
    A. Bensoussan, J. Y. Frehse, Mean Field Games and Mean Field Type Control (Springer, Berlin, 2013)CrossRefGoogle Scholar
  8. 8.
    J. Borwein, J. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples (Cambridge University Press, Cambridge, 2010)CrossRefGoogle Scholar
  9. 9.
    A. Briani, P. Cardaliaguet, Stable solutions in potential mean field game systems. Nonlinear Differ. Equ. Appl. 25, 1 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    P. Cardaliaguet, Notes on mean-field games (2011). Available at cardaliaguet/MFG20130420.pdfGoogle Scholar
  11. 11.
    P. Cardaliaguet, P.-J. Graber, Mean field games systems of first order. ESAIM Control Optim. Calc. Var. 21, 690–722 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    P. Cardaliaguet, P.J. Graber, A. Porretta, D. Tonon, Second order mean field games with degenerate diffusion and local coupling. NoDEA Nonlinear Differ. Equ. Appl. 22(5), 1287–1317 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    P. Cardaliaguet, A.R. Mészáros, F. Santambrogio, First order mean field games with density constraints: pressure equals price. SIAM J. Control. Optim. 54(5), 2672–2709 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    P. Cardaliaguet, A. Porretta, D. Tonon, A segregation problem in multi-population mean field games, in Advances in Dynamic and Mean Field Games. ISDG 2016. Annals of the International Society of Dynamic Games ed. by J. Apaloo, B. Viscolani, vol. 15 (Birkhäuser, Basel, 2017), pp. 49–70CrossRefGoogle Scholar
  15. 15.
    P. Cardaliaguet, F. Delarue, J.-M. Lasry, P.-L. Lions, The master equation and the convergence problem in mean field games. Preprint arXiv:1509.02505Google Scholar
  16. 16.
    A. Cesaroni, M. Cirant, Concentration of ground states in stationary mean-field games systems. Anal. PDE 12(3), 737–787 (2019)MathSciNetCrossRefGoogle Scholar
  17. 17.
    A. Cesaroni, M. Cirant, S. Dipierro, M. Novaga, E. Valdinoci, On stationary fractional mean field games. J. Math. Pures Appl. 122, 1–22 (2019)MathSciNetCrossRefGoogle Scholar
  18. 18.
    M. Cirant, On the solvability of some ergodic control problems in \({\mathbb R}^d\). SIAM J. Control Optim. 52(6), 4001–4026 (2014)Google Scholar
  19. 19.
    M. Cirant, Multi-population mean field games systems with Neumann boundary conditions. J. Math. Pures Appl. (9) 103(5), 1294–1315 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Cirant, A generalization of the Hopf-Cole transformation for stationary mean field games systems. C.R. Math. 353(9), 807–811 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    M. Cirant, Stationary focusing mean-field games. Commun. Part. Diff. Eq. 41(8), 1324–1346 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    M. Cirant, D. Tonon, Time-dependent focusing mean-field games: the sub-critical case. J. Dyn. Diff. Equat. 31(1), 49–79 (2019)MathSciNetCrossRefGoogle Scholar
  23. 23.
    M. Cirant, G. Verzini, Bifurcation and segregation in quadratic two-populations mean field games systems. ESAIM Control Optim. Calc. Var. 23, 1145–1177 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    I. Ekeland, R. Temam, Convex Analysis and Variational Problems (North-Holland Publishing Co., Amsterdam-Oxford, 1976)zbMATHGoogle Scholar
  25. 25.
    E. Feleqi, The derivation of ergodic mean field game equations for several populations of players. Dyn. Games Appl. 3(4), 523–536 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. (Springer-Verlag, Berlin, 2001)zbMATHGoogle Scholar
  27. 27.
    D.A. Gomes, J. Saude, Mean field games models–a brief survey. Dyn. Games Appl. 4(2), 110–154 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    D.A. Gomes, S. Patrizi, V. Voskanyan, On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal. 99, 49–79 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    D.A. Gomes, L. Nurbekyan, M. Prazeres, One-dimensional stationary mean-field games with local coupling. Dyn. Games Appl. 8(2), 315–351 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    D.A. Gomes, E.A. Pimentel, V. Voskanyan, Regularity Theory for Mean-Field Game Systems (Springer, Berlin, 2016)CrossRefGoogle Scholar
  31. 31.
    M. Huang, R. Malhamé, P. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–251 (2006)MathSciNetzbMATHGoogle Scholar
  32. 32.
    R. Khasminskii, Stochastic stability of differential equations, in Stochastic Modelling and Applied Probability, vol. 66 (Springer, Berlin, 2012)CrossRefGoogle Scholar
  33. 33.
    J.-M. Lasry, P.-L. Lions, Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. Math. Ann. 283(4), 583–630 (1989)MathSciNetCrossRefGoogle Scholar
  34. 34.
    J.-M. Lasry, P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(10), 679–684 (2006)CrossRefGoogle Scholar
  35. 35.
    J.-M. Lasry, P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619–625 (2006)CrossRefGoogle Scholar
  36. 36.
    J.-M. Lasry, P.-L. Lions, Mean field games. Japan. J. Math. 2, 229–260 (2007)MathSciNetCrossRefGoogle Scholar
  37. 37.
    P.-L. Lions, Cours au Collége de France,
  38. 38.
    A.R. Mészáros, F.J. Silva, A variational approach to second order mean field games with density constraints: the stationary case. J. Math. Pures Appl. (9) 104(6), 1135–1159 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    E. Pimentel, V. Voskanyan, Regularity for second-order stationary mean-field games. Indiana Univ. Math. J. 66, 1–22 (2017)MathSciNetCrossRefGoogle Scholar
  40. 40.
    R.T. Rockafellar, Integral functionals, normal integrands and measurable selections. Lect. Notes Math. 543, 157–207 (1976)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Dipartimento di Scienze StatisticheUniversità di PadovaPadovaItaly
  2. 2.Dipartimento di Matematica Tullio Levi CivitaUniversità di PadovaPadovaItaly

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