American Institute of Mathematical Sciences

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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]

SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477.  Google Scholar

[2]

Math. Models Methods Appl. Sci., 20 (2010), 567-588. doi: 10.1142/S0218202510004349.  Google Scholar

[3]

SIAM J. Control Opt., 50 (2012), 77-109. 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]

Arch. Ration. Mech. Anal., 201 (2011), 87-113. doi: 10.1007/s00205-011-0399-x.  Google Scholar

[7]

Calc. Var. Partial Differential Equations, 17 (2003), 159-177. doi: 10.1007/s00526-002-0164-y.  Google Scholar

[8]

Calc. Var. Partial Differential Equations, 35 (2009), 435-462. doi: 10.1007/s00526-008-0214-1.  Google Scholar

[9]

Arch. Ration. Mech. Anal., 197 (2010), 1053-1088. doi: 10.1007/s00205-010-0307-9.  Google Scholar

[10]

C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649-652. doi: 10.1016/S0764-4442(97)84777-5.  Google Scholar

[11]

C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046.  Google Scholar

[12]

C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1213-1216.  Google Scholar

[13]

C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270. doi: 10.1016/S0764-4442(98)80144-4.  Google Scholar

[14]

Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308-328.  Google Scholar

[15]

preprint, 2011. Google Scholar

[16]

Nonlinearity, 15 (2002), 581-603. doi: 10.1088/0951-7715/15/3/304.  Google Scholar

[17]

preprint, 2011. Google Scholar

[18]

Ph.D. Thesis, Université Paris Dauphine, Paris, 2009. Google Scholar

[19]

J. Math. Pures Appl. (9), 92 (2009), 276-294.  Google Scholar

[20]

IEEE Trans. Automat. Control, 52 (2007), 1560-1571. doi: 10.1109/TAC.2007.904450.  Google Scholar

[21]

Commun. Inf. Syst., 6 (2006), 221-251.  Google Scholar

[22]

C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[23]

C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[24]

Jpn. J. Math., 2 (2007), 229-260.  Google Scholar

[25]

Cahiers de la Chaire Finance et Développement Durable, 2007.  Google Scholar

[26]

preprint, 2010. Google Scholar

[27]

in "Paris-Princeton Lectures on Mathematical Finance 2010," Lecture Notes in Math., 2003, Springer, Berlin, (2011), 205-266.  Google Scholar

[28]

Math. Z, 207 (1991), 169-207. doi: 10.1007/BF02571383.  Google Scholar

[29]

Nonlinearity, 5 (1992), 623-638.  Google Scholar

[30]

Commun. Pure Appl. Anal., 7 (2008), 1211-1223.  Google Scholar

show all references

References:
[1]

SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477.  Google Scholar

[2]

Math. Models Methods Appl. Sci., 20 (2010), 567-588. doi: 10.1142/S0218202510004349.  Google Scholar

[3]

SIAM J. Control Opt., 50 (2012), 77-109. 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]

Arch. Ration. Mech. Anal., 201 (2011), 87-113. doi: 10.1007/s00205-011-0399-x.  Google Scholar

[7]

Calc. Var. Partial Differential Equations, 17 (2003), 159-177. doi: 10.1007/s00526-002-0164-y.  Google Scholar

[8]

Calc. Var. Partial Differential Equations, 35 (2009), 435-462. doi: 10.1007/s00526-008-0214-1.  Google Scholar

[9]

Arch. Ration. Mech. Anal., 197 (2010), 1053-1088. doi: 10.1007/s00205-010-0307-9.  Google Scholar

[10]

C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649-652. doi: 10.1016/S0764-4442(97)84777-5.  Google Scholar

[11]

C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046.  Google Scholar

[12]

C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1213-1216.  Google Scholar

[13]

C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270. doi: 10.1016/S0764-4442(98)80144-4.  Google Scholar

[14]

Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308-328.  Google Scholar

[15]

preprint, 2011. Google Scholar

[16]

Nonlinearity, 15 (2002), 581-603. doi: 10.1088/0951-7715/15/3/304.  Google Scholar

[17]

preprint, 2011. Google Scholar

[18]

Ph.D. Thesis, Université Paris Dauphine, Paris, 2009. Google Scholar

[19]

J. Math. Pures Appl. (9), 92 (2009), 276-294.  Google Scholar

[20]

IEEE Trans. Automat. Control, 52 (2007), 1560-1571. doi: 10.1109/TAC.2007.904450.  Google Scholar

[21]

Commun. Inf. Syst., 6 (2006), 221-251.  Google Scholar

[22]

C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[23]

C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[24]

Jpn. J. Math., 2 (2007), 229-260.  Google Scholar

[25]

Cahiers de la Chaire Finance et Développement Durable, 2007.  Google Scholar

[26]

preprint, 2010. Google Scholar

[27]

in "Paris-Princeton Lectures on Mathematical Finance 2010," Lecture Notes in Math., 2003, Springer, Berlin, (2011), 205-266.  Google Scholar

[28]

Math. Z, 207 (1991), 169-207. doi: 10.1007/BF02571383.  Google Scholar

[29]

Nonlinearity, 5 (1992), 623-638.  Google Scholar

[30]

Commun. Pure Appl. Anal., 7 (2008), 1211-1223.  Google Scholar

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