Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 3 (2019), 737-787.

Concentration of ground states in stationary mean-field games systems

Annalisa Cesaroni and Marco Cirant

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/apde.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We provide the existence of classical solutions to stationary mean-field game systems in the whole space  N , with coercive potential and aggregating local coupling, under general conditions on the Hamiltonian. The only structural assumption we make is on the growth at infinity of the coupling term in terms of the growth of the Hamiltonian. This result is obtained using a variational approach based on the analysis of the nonconvex energy associated to the system. Finally, we show that in the vanishing viscosity limit, mass concentrates around the flattest minima of the potential. We also describe the asymptotic shape of the rescaled solutions in the vanishing viscosity limit, in particular proving the existence of ground states, i.e., classical solutions to mean-field game systems in the whole space without potential, and with aggregating coupling.

Article information

Source
Anal. PDE, Volume 12, Number 3 (2019), 737-787.

Dates
Received: 16 August 2017
Revised: 25 May 2018
Accepted: 29 June 2018
First available in Project Euclid: 25 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.apde/1540432869

Digital Object Identifier
doi:10.2140/apde.2019.12.737

Mathematical Reviews number (MathSciNet)
MR3864209

Zentralblatt MATH identifier
06986452

Subjects
Primary: 35J50: Variational methods for elliptic systems
Secondary: 49N70: Differential games 35J47: Second-order elliptic systems 91A13: Games with infinitely many players 35B25: Singular perturbations

Keywords
ergodic mean-field games semiclassical limit concentration-compactness method mass concentration elliptic systems variational methods

Citation

Cesaroni, Annalisa; Cirant, Marco. Concentration of ground states in stationary mean-field games systems. Anal. PDE 12 (2019), no. 3, 737--787. doi:10.2140/apde.2019.12.737. https://projecteuclid.org/euclid.apde/1540432869


Export citation

References

  • S. Agmon, “The $L\sb{p}$ approach to the Dirichlet problem, I: Regularity theorems”, Ann. Scuola Norm. Sup. Pisa $(3)$ 13:4 (1959), 405–448.
  • A. Arapostathis, A. Biswas, and J. Carroll, “On solutions of mean field games with ergodic cost”, J. Math. Pures Appl. $(9)$ 107:2 (2017), 205–251.
  • M. Bardi and F. S. Priuli, “Linear-quadratic $N$-person and mean-field games with ergodic cost”, SIAM J. Control Optim. 52:5 (2014), 3022–3052.
  • G. Barles and J. Meireles, “On unbounded solutions of ergodic problems in $\mathbb{R}^m$ for viscous Hamilton–Jacobi equations”, Comm. Partial Differential Equations 41:12 (2016), 1985–2003.
  • G. Barles, A. Porretta, and T. T. Tchamba, “On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton–Jacobi equations”, J. Math. Pures Appl. $(9)$ 94:5 (2010), 497–519.
  • J. M. Borwein and J. D. Vanderwerff, Convex functions: constructions, characterizations and counterexamples, Encyclopedia of Math. and Its Appl. 109, Cambridge Univ. Press, 2010.
  • H. Brézis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals”, Proc. Amer. Math. Soc. 88:3 (1983), 486–490.
  • A. Briani and P. Cardaliaguet, “Stable solutions in potential mean field game systems”, Nonlinear Differential Equations Appl. 25:1 (2018), art. id. 1.
  • P. Cardaliaguet and P. J. Graber, “Mean field games systems of first order”, ESAIM Control Optim. Calc. Var. 21:3 (2015), 690–722.
  • A. Cesaroni and M. Cirant, “Introduction to variational methods for viscous ergodic mean-field games with local coupling”, lecture notes, Istituto Nazionale di Alta Matematica, 2017, https://tinyurl.com/cesaindam.
  • M. Cirant, “On the solvability of some ergodic control problems in $\mathbb{R}^d$”, SIAM J. Control Optim. 52:6 (2014), 4001–4026.
  • M. Cirant, “Multi-population mean field games systems with Neumann boundary conditions”, J. Math. Pures Appl. $(9)$ 103:5 (2015), 1294–1315.
  • M. Cirant, “Stationary focusing mean-field games”, Comm. Partial Differential Equations 41:8 (2016), 1324–1346.
  • M. Cirant, “On the existence of oscillating solutions in non-monotone mean-field games”, preprint, 2017.
  • M. Cirant and D. Tonon, “Time-dependent focusing mean-field games: the sub-critical case”, J. Dynam. Differential Equations (online publication April 2018).
  • D. A. Gomes and E. Pimentel, “Local regularity for mean-field games in the whole space”, Minimax Theory Appl. 1:1 (2016), 65–82.
  • D. A. Gomes, E. A. Pimentel, and V. Voskanyan, Regularity theory for mean-field game systems, Springer, 2016.
  • D. A. Gomes, L. Nurbekyan, and M. Prazeres, “One-dimensional stationary mean-field games with local coupling”, Dyn. Games Appl. 8:2 (2018), 315–351.
  • 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. 6:3 (2006), 221–251.
  • N. Ichihara, “Recurrence and transience of optimal feedback processes associated with Bellman equations of ergodic type”, SIAM J. Control Optim. 49:5 (2011), 1938–1960.
  • N. Ichihara, “The generalized principal eigenvalue for Hamilton–Jacobi–Bellman equations of ergodic type”, Ann. Inst. H. Poincaré Anal. Non Linéaire 32:3 (2015), 623–650.
  • J.-M. Lasry and P.-L. Lions, “Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints, I: The model problem”, Math. Ann. 283:4 (1989), 583–630.
  • J.-M. Lasry and P.-L. Lions, “Jeux à champ moyen, I: Le cas stationnaire”, C. R. Math. Acad. Sci. Paris 343:9 (2006), 619–625.
  • J.-M. Lasry and P.-L. Lions, “Jeux à champ moyen, II: Horizon fini et contrôle optimal”, C. R. Math. Acad. Sci. Paris 343:10 (2006), 679–684.
  • J.-M. Lasry and P.-L. Lions, “Mean field games”, Jpn. J. Math. 2:1 (2007), 229–260.
  • P.-L. Lions, “The concentration-compactness principle in the calculus of variations: the locally compact case, I”, Ann. Inst. H. Poincaré Anal. Non Linéaire 1:2 (1984), 109–145.
  • A. R. Mészáros and F. J. Silva, “On the variational formulation of some stationary second-order mean field games systems”, SIAM J. Math. Anal. 50:1 (2018), 1255–1277.
  • G. Metafune, D. Pallara, and A. Rhandi, “Global properties of invariant measures”, J. Funct. Anal. 223:2 (2005), 396–424.
  • A. Porretta, “On the weak theory for mean field games systems”, Boll. Unione Mat. Ital. 10:3 (2017), 411–439.