American Institute of Mathematical Sciences

Advanced
June  2012, 7(2): 303-314. doi: 10.3934/nhm.2012.7.303

A-priori estimates for stationary mean-field games

Diogo A. Gomes 1, , Gabriel E. Pires 1, and  Héctor Sánchez-Morgado 2,

1. 

Departamento de Matemática and CAMGSD, IST Avenida Rovisco Pais, Lisboa, Portugal, Portugal
2. 

Instituto de Matem, Universidad Nacional Aut, M, Mexico

Received  November 2011 Revised  March 2012 Published  June 2012

In this paper we establish a new class of a-priori estimates for stationary mean-field games which have a quasi-variational structure. In particular we prove $W^{1,2}$ estimates for the value function $u$ and that the players distribution $m$ satisfies $\sqrt{m}\in W^{1,2}$. We discuss further results for power-like nonlinearities and prove higher regularity if the space dimension is 2. In particular we also obtain in this last case $W^{2,p}$ estimates for $u$.
Keywords: a-priori estimates, Mean field games, variational and quasi-variational methods..
Mathematics Subject Classification: 35J47, 49L25, 49N7.
Citation: Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303
References:
[1]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods,, SIAM J. Numer. Anal., 48 (2010), 1136. doi: 10.1137/090758477. Google Scholar

[2]

Julien Salomon, Aimée Lachapelle and Gabriel Turinici, Computation of mean field equilibria in economics,, Math. Models Methods Appl. Sci., 20 (2010), 567. doi: 10.1142/S0218202510004349. Google Scholar

[3]

F. Camilli, Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem,, SIAM J. Control Opt., 50 (2012), 77. doi: 10.1137/100790069. Google Scholar

[4]

F. Cagnetti, D. Gomes and H. V. Tran, Adjoint methods for obstacle problems and weakly coupled systems of PDE,, submitted., (). Google Scholar

[5]

F. Cagnetti, D. Gomes and H. V. Tran, Aubry-Mather measures in the non convex setting,, submitted., (). Google Scholar

[6]

Lawrence C. Evans and Charles K. Smart, Adjoint methods for the infinity Laplacian partial differential equation,, Arch. Ration. Mech. Anal., 201 (2011), 87. doi: 10.1007/s00205-011-0399-x. Google Scholar

[7]

Lawrence C. Evans, Some new PDE methods for weak KAM theory,, Calc. Var. Partial Differential Equations, 17 (2003), 159. doi: 10.1007/s00526-002-0164-y. Google Scholar

[8]

Lawrence C. Evans, Further PDE methods for weak KAM theory,, Calc. Var. Partial Differential Equations, 35 (2009), 435. doi: 10.1007/s00526-008-0214-1. Google Scholar

[9]

L. C. Evans, Adjoint and compensated compactness methods for Hamilton-Jacobi PDE,, Arch. Ration. Mech. Anal., 197 (2010), 1053. doi: 10.1007/s00205-010-0307-9. Google Scholar

[10]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649. doi: 10.1016/S0764-4442(97)84777-5. Google Scholar

[11]

A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens,, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043. Google Scholar

[12]

A. Fathi, Orbite hétéroclines et ensemble de Peierls,, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1213. Google Scholar

[13]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik,, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267. doi: 10.1016/S0764-4442(98)80144-4. Google Scholar

[14]

D. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games,, Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308. Google Scholar

[15]

D. Gomes, J. Mohr and R. R. Souza, Mean-field limit of a continuous time finite state game,, preprint, (2011). Google Scholar

[16]

D. Gomes, A stochastic analogue of Aubry-Mather theory,, Nonlinearity, 15 (2002), 581. doi: 10.1088/0951-7715/15/3/304. Google Scholar

[17]

D. Gomes and H Sanchez-Morgado, On the stochastic Evans-Aronsson problem,, preprint, (2011). Google Scholar

[18]

O. Gueant, "Mean Field Games and Applications to Economics,", Ph.D. Thesis, (2009). Google Scholar

[19]

O. Gueant, A reference case for mean field games models,, J. Math. Pures Appl. (9), 92 (2009), 276. Google Scholar

[20]

Minyi Huang, Peter E. Caines and Roland P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria,, IEEE Trans. Automat. Control, 52 (2007), 1560. doi: 10.1109/TAC.2007.904450. Google Scholar

[21]

Minyi Huang, Roland P. Malhamé and Peter E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle,, Commun. Inf. Syst., 6 (2006), 221. Google Scholar

[22]

Jean-Michel Lasry and Pierre-Louis Lions, Jeux à champ moyen. I. Le cas stationnaire,, C. R. Math. Acad. Sci. Paris, 343 (2006), 619. doi: 10.1016/j.crma.2006.09.019. Google Scholar

[23]

Jean-Michel Lasry and Pierre-Louis Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal,, C. R. Math. Acad. Sci. Paris, 343 (2006), 679. doi: 10.1016/j.crma.2006.09.018. Google Scholar

[24]

Jean-Michel Lasry and Pierre-Louis Lions, Mean field games,, Jpn. J. Math., 2 (2007), 229. Google Scholar

[25]

Jean-Michel Lasry and Pierre-Louis Lions, "Mean Field Games,", Cahiers de la Chaire Finance et Développement Durable, (2007). Google Scholar

[26]

Jean-Michel Lasry, Pierre-Louis Lions and O. Guéant, Application of mean field games to growth theory,, preprint, (2010). Google Scholar

[27]

Jean-Michel Lasry, Pierre-Louis Lions and O. Guéant, Mean field games and applications,, in, 2003 (2011), 205. Google Scholar

[28]

J. Mather, Action minimizing invariant measure for positive definite Lagrangian systems,, Math. Z, 207 (1991), 169. doi: 10.1007/BF02571383. Google Scholar

[29]

Ricardo Mañé, On the minimizing measures of Lagrangian dynamical systems,, Nonlinearity, 5 (1992), 623. Google Scholar

[30]

Kaizhi Wang, Action minimizing stochastic invariant measures for a class of Lagrangian systems,, Commun. Pure Appl. Anal., 7 (2008), 1211. Google Scholar

show all references

References:
[1]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods,, SIAM J. Numer. Anal., 48 (2010), 1136. doi: 10.1137/090758477. Google Scholar

[2]

Julien Salomon, Aimée Lachapelle and Gabriel Turinici, Computation of mean field equilibria in economics,, Math. Models Methods Appl. Sci., 20 (2010), 567. doi: 10.1142/S0218202510004349. Google Scholar

[3]

F. Camilli, Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem,, SIAM J. Control Opt., 50 (2012), 77. doi: 10.1137/100790069. Google Scholar

[4]

F. Cagnetti, D. Gomes and H. V. Tran, Adjoint methods for obstacle problems and weakly coupled systems of PDE,, submitted., (). Google Scholar

[5]

F. Cagnetti, D. Gomes and H. V. Tran, Aubry-Mather measures in the non convex setting,, submitted., (). Google Scholar

[6]

Lawrence C. Evans and Charles K. Smart, Adjoint methods for the infinity Laplacian partial differential equation,, Arch. Ration. Mech. Anal., 201 (2011), 87. doi: 10.1007/s00205-011-0399-x. Google Scholar

[7]

Lawrence C. Evans, Some new PDE methods for weak KAM theory,, Calc. Var. Partial Differential Equations, 17 (2003), 159. doi: 10.1007/s00526-002-0164-y. Google Scholar

[8]

Lawrence C. Evans, Further PDE methods for weak KAM theory,, Calc. Var. Partial Differential Equations, 35 (2009), 435. doi: 10.1007/s00526-008-0214-1. Google Scholar

[9]

L. C. Evans, Adjoint and compensated compactness methods for Hamilton-Jacobi PDE,, Arch. Ration. Mech. Anal., 197 (2010), 1053. doi: 10.1007/s00205-010-0307-9. Google Scholar

[10]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649. doi: 10.1016/S0764-4442(97)84777-5. Google Scholar

[11]

A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens,, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043. Google Scholar

[12]

A. Fathi, Orbite hétéroclines et ensemble de Peierls,, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1213. Google Scholar

[13]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik,, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267. doi: 10.1016/S0764-4442(98)80144-4. Google Scholar

[14]

D. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games,, Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308. Google Scholar

[15]

D. Gomes, J. Mohr and R. R. Souza, Mean-field limit of a continuous time finite state game,, preprint, (2011). Google Scholar

[16]

D. Gomes, A stochastic analogue of Aubry-Mather theory,, Nonlinearity, 15 (2002), 581. doi: 10.1088/0951-7715/15/3/304. Google Scholar

[17]

D. Gomes and H Sanchez-Morgado, On the stochastic Evans-Aronsson problem,, preprint, (2011). Google Scholar

[18]

O. Gueant, "Mean Field Games and Applications to Economics,", Ph.D. Thesis, (2009). Google Scholar

[19]

O. Gueant, A reference case for mean field games models,, J. Math. Pures Appl. (9), 92 (2009), 276. Google Scholar

[20]

Minyi Huang, Peter E. Caines and Roland P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria,, IEEE Trans. Automat. Control, 52 (2007), 1560. doi: 10.1109/TAC.2007.904450. Google Scholar

[21]

Minyi Huang, Roland P. Malhamé and Peter E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle,, Commun. Inf. Syst., 6 (2006), 221. Google Scholar

[22]

Jean-Michel Lasry and Pierre-Louis Lions, Jeux à champ moyen. I. Le cas stationnaire,, C. R. Math. Acad. Sci. Paris, 343 (2006), 619. doi: 10.1016/j.crma.2006.09.019. Google Scholar

[23]

Jean-Michel Lasry and Pierre-Louis Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal,, C. R. Math. Acad. Sci. Paris, 343 (2006), 679. doi: 10.1016/j.crma.2006.09.018. Google Scholar

[24]

Jean-Michel Lasry and Pierre-Louis Lions, Mean field games,, Jpn. J. Math., 2 (2007), 229. Google Scholar

[25]

Jean-Michel Lasry and Pierre-Louis Lions, "Mean Field Games,", Cahiers de la Chaire Finance et Développement Durable, (2007). Google Scholar

[26]

Jean-Michel Lasry, Pierre-Louis Lions and O. Guéant, Application of mean field games to growth theory,, preprint, (2010). Google Scholar

[27]

Jean-Michel Lasry, Pierre-Louis Lions and O. Guéant, Mean field games and applications,, in, 2003 (2011), 205. Google Scholar

[28]

J. Mather, Action minimizing invariant measure for positive definite Lagrangian systems,, Math. Z, 207 (1991), 169. doi: 10.1007/BF02571383. Google Scholar

[29]

Ricardo Mañé, On the minimizing measures of Lagrangian dynamical systems,, Nonlinearity, 5 (1992), 623. Google Scholar

[30]

Kaizhi Wang, Action minimizing stochastic invariant measures for a class of Lagrangian systems,, Commun. Pure Appl. Anal., 7 (2008), 1211. Google Scholar

[1]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[2]

Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583
[3]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383
[4]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165
[5]

Nobuyuki Kenmochi. Parabolic quasi-variational diffusion problems with gradient constraints. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 423-438. doi: 10.3934/dcdss.2013.6.423
[6]

Laura Scrimali. Mixed behavior network equilibria and quasi-variational inequalities. Journal of Industrial & Management Optimization, 2009, 5 (2) : 363-379. doi: 10.3934/jimo.2009.5.363
[7]

Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101
[8]

Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277
[9]

Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks & Heterogeneous Media, 2012, 7 (2) : 315-336. doi: 10.3934/nhm.2012.7.315
[10]

Haisen Zhang. Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities. Mathematical Control & Related Fields, 2014, 4 (3) : 365-379. doi: 10.3934/mcrf.2014.4.365
[11]

Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2523-2538. doi: 10.3934/dcds.2015.35.2523
[12]

Samir Adly, Tahar Haddad. On evolution quasi-variational inequalities and implicit state-dependent sweeping processes. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020105
[13]

Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions, Alessio Porretta. Long time average of mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 279-301. doi: 10.3934/nhm.2012.7.279
[14]

Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173
[15]

Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311
[16]

Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks. Networks & Heterogeneous Media, 2019, 14 (3) : 537-566. doi: 10.3934/nhm.2019021
[17]

Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence. Journal of Dynamics & Games, 2019, 6 (3) : 221-239. doi: 10.3934/jdg.2019016
[18]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
[19]

Maria Alessandra Ragusa, Atsushi Tachikawa. Estimates of the derivatives of minimizers of a special class of variational integrals. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1411-1425. doi: 10.3934/dcds.2011.31.1411
[20]

Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences