October  2021, 8(4): 445-465. doi: 10.3934/jdg.2021006

On some singular mean-field games

1. 

Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121, Padova, Italy

2. 

King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia

3. 

Department of Mathematics, Pontifícia Universidade Católica do Rio de Janeiro, 22451-900, Rio de Janeiro, Brazil

4. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cd. México 04510, México

Received  May 2020 Published  October 2021 Early access  March 2021

Fund Project: M. Cirant is partially supported by the Fondazione CaRiPaRo Project "Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games" and the INdAM-GNAMPA project "Fenomeni di segregazione in sistemi stazionari di tipo Mean Field Games a più popolazioni".
D. Gomes was partially supported by KAUST baseline and start-up funds.
E. Pimentel was partially supported by FAPESP (Grant 2015/13011-6) and PUC-Rio baseline funds.

Here, we prove the existence of smooth solutions for mean-field games with a singular mean-field coupling; that is, a coupling in the Hamilton-Jacobi equation of the form $ g(m) = -m^{- \alpha} $ with $ \alpha>0 $. We consider stationary and time-dependent settings. The function $ g $ is monotone, but it is not bounded from below. With the exception of the logarithmic coupling, this is the first time that MFGs whose coupling is not bounded from below is examined in the literature. This coupling arises in models where agents have a strong preference for low-density regions. Paradoxically, this causes the agents move towards low-density regions and, thus, prevents the creation of those regions. To prove the existence of solutions, we consider an approximate problem for which the existence of smooth solutions is known. Then, we prove new a priori bounds for the solutions that show that $ \frac 1 m $ is bounded. Finally, using a limiting argument, we obtain the existence of solutions. The proof in the stationary case relies on a blow-up argument and in the time-dependent case on new bounds for $ m^{-1} $.

Citation: Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics and Games, 2021, 8 (4) : 445-465. doi: 10.3934/jdg.2021006
References:
[1]

N. AlmullaR. Ferreira and D. Gomes, Two numerical approaches to stationary mean-field games, Dyn. Games Appl., 7 (2017), 657-682.  doi: 10.1007/s13235-016-0203-5.

[2]

P. Cardaliaguet, Notes on Mean-Field Games, 2011.

[3]

P. CardaliaguetP. J. GarberA. Porretta and D. Tonon, Second order mean field games with degenerate diffusion and local coupling, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1287-1317.  doi: 10.1007/s00030-015-0323-4.

[4]

P. Cardaliaguet and P. J. Graber, Mean field games systems of first order, ESAIM Control Optim. Calc. Var., 21 (2015), 690-722.  doi: 10.1051/cocv/2014044.

[5]

P. CardaliaguetJ.-M. LasryP.-L. Lions and A. Porretta, Long time average of mean field games, Netw. Heterog. Media, 7 (2012), 279-301.  doi: 10.3934/nhm.2012.7.279.

[6]

P. CardaliaguetA. Mészáros and F. Santambrogio, First order mean field games with density constraints: Pressure equals price, SIAM J. Control Optim., 54 (2016), 2672-2709.  doi: 10.1137/15M1029849.

[7]

M. Cirant, Multi-population mean field games systems with Neumann boundary conditions, J. Math. Pures Appl. (9), 103 (2015), 1294-1315.  doi: 10.1016/j.matpur.2014.10.013.

[8]

M. Cirant, Stationary focusing mean-field games, Comm. Partial Differential Equations, 41 (2016), 1324-1346.  doi: 10.1080/03605302.2016.1192647.

[9]

M. Cirant and A. Goffi, Maximal ${L}^q$-regularity for parabolic Hamilton-Jacobi equations and applications to Mean Field Games, 2020, arXiv: 2007.14873.

[10]

D. EvangelistaR. FerreiraD. A. GomesL. Nurbekyan and V. Voskanyan, First-order, stationary mean-field games with congestion, Nonlinear Anal., 173 (2018), 37-74.  doi: 10.1016/j.na.2018.03.011.

[11]

D. Evangelista and D. A. Gomes, On the existence of solutions for stationary mean-field games with congestion, J. Dynam. Differential Equations, 30 (2018), 1365-1388.  doi: 10.1007/s10884-017-9615-1.

[12]

D. Evangelista, D. Gomes and L. Nurbekyan, Radially symmetric mean-field games with congestion, In 2017 IEEE 56th Annual Conference on Decision and Control (CDC), (2017), 3158-3163.

[13]

L. C. Evans, Adjoint and compensated compactness methods for {H}amilton-{J}acobi PDE, Arch. Ration. Mech. Anal., 197 (2010), 1053-1088.  doi: 10.1007/s00205-010-0307-9.

[14]

R. Ferreira and D. Gomes, Existence of weak solutions to stationary mean-field games through variational inequalities, SIAM J. Math. Anal., 50 (2018), 5969-6006.  doi: 10.1137/16M1106705.

[15]

R. Ferreira, D. Gomes and T. Tada, Existence of weak solutions to time-dependent mean-field games, arXiv preprint. arXiv: 2001.03928.

[16]

R. FerreiraD. Gomes and T. Tada, Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions, Proc. Amer. Math. Soc., 147 (2019), 4713-4731.  doi: 10.1090/proc/14475.

[17]

D. A. Gomes and H. Mitake, Existence for stationary mean-field games with congestion and quadratic {H}amiltonians, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1897-1910.  doi: 10.1007/s00030-015-0349-7.

[18]

D. A. Gomes, L. Nurbekyan and M. Prazere, Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion, 2016 IEEE 55th Conference on Decision and Control, CDC 2016, (2016), 4534-4539.

[19]

D. A. GomesL. Nurbekyan and M. Prazeres, One-dimensional stationary mean-field games with local coupling, Dyn. Games Appl., 8 (2018), 315-351.  doi: 10.1007/s13235-017-0223-9.

[20]

D. A. GomesS. Patrizi and V. Voskanyan, On the existence of classical solutions for stationary extended mean field games, Nonlinear Anal., 99 (2014), 49-79.  doi: 10.1016/j.na.2013.12.016.

[21]

D. A. Gomes and E. Pimentel, Time dependent mean-field games with logarithmic nonlinearities, SIAM J. Math. Anal., 47 (2015), 3798-3812.  doi: 10.1137/140984622.

[22]

D. A. GomesE. A. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the subquadratic case, Comm. Partial Differential Equations, 40 (2015), 40-76.  doi: 10.1080/03605302.2014.903574.

[23]

D. A. GomesE. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the superquadratic case, ESAIM Control Optim. Calc. Var., 22 (2016), 562-580.  doi: 10.1051/cocv/2015029.

[24]

D. A. Gomes, E. A. Pimentel and V. Voskanyan, Regularity Theory for Mean-Field Game Systems, SpringerBriefs in Mathematics. Springer, [Cham], 2016. doi: 10.1007/978-3-319-38934-9.

[25]

D. Gomes and H. Sánchez Morgado, A stochastic Evans-Aronsson problem, Trans. Amer. Math. Soc., 366 (2014), 903-929.  doi: 10.1090/S0002-9947-2013-05936-3.

[26]

D. A. Gomes and J. Saude, Monotone numerical methods for finite-state mean-field games, arXiv preprint. arXiv: 1705.00174, 2017.

[27]

D. A. Gomes and V. K. Voskanyan, Short-time existence of solutions for mean-field games with congestion, J. Lond. Math. Soc. (2), 92 (2015), 778-799.  doi: 10.1112/jlms/jdv052.

[28]

J. Graber, Weak solutions for mean field games with congestion, Preprint. arXiv: 1503.04733, 2015.

[29]

O. Guéant, A reference case for mean field games models, J. Math. Pures Appl. (9), 92 (2009), 276-294.  doi: 10.1016/j.matpur.2009.04.008.

[30]

O. Guéant, Mean field games equations with quadratic Hamiltonian: A specific approach, Math. Models Methods Appl. Sci., 22 (2012), 1250022, 37 pp.

[31]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, In Paris-Princeton Lectures on Mathematical Finance 2010, volume 2003 of Lecture Notes in Math., pages 205-266. Springer, Berlin, (2011). doi: 10.1007/978-3-642-14660-2_3.

[32]

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 (1989), 583-630.  doi: 10.1007/BF01442856.

[33]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.  doi: 10.1016/j.crma.2006.09.019.

[34]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.

[35]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.

[36]

P. L. Lions, Collège de France course on mean-field games, 2007-2011.

[37]

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. (9), 104 (2015), 1135-1159.  doi: 10.1016/j.matpur.2015.07.008.

[38]

E. A. Pimentel and V. Voskanyan, Regularity for second-order stationary mean-field games, Indiana Univ. Math. J., 66 (2017), 1-22.  doi: 10.1512/iumj.2017.66.5944.

[39]

A. Porretta, On the planning problem for the mean field games system, Dyn. Games Appl., 4 (2014), 231-256.  doi: 10.1007/s13235-013-0080-0.

[40]

A. Porretta, Weak solutions to Fokker-Planck equations and mean field games, Arch. Ration. Mech. Anal., 216 (2015), 1-62.  doi: 10.1007/s00205-014-0799-9.

[41]

F. Santambrogio, A modest proposal for MFG with density constraints, Netw. Heterog. Media, 7 (2012), 337-347.  doi: 10.3934/nhm.2012.7.337.

[42]

J. Serrin, A Harnack inequality for nonlinear equations, Bull. Amer. Math. Soc., 69 (1963), 481-486.  doi: 10.1090/S0002-9904-1963-10971-4.

[43]

H. A. Tran, Adjoint methods for static Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 41 (2011), 301-319.  doi: 10.1007/s00526-010-0363-x.

show all references

References:
[1]

N. AlmullaR. Ferreira and D. Gomes, Two numerical approaches to stationary mean-field games, Dyn. Games Appl., 7 (2017), 657-682.  doi: 10.1007/s13235-016-0203-5.

[2]

P. Cardaliaguet, Notes on Mean-Field Games, 2011.

[3]

P. CardaliaguetP. J. GarberA. Porretta and D. Tonon, Second order mean field games with degenerate diffusion and local coupling, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1287-1317.  doi: 10.1007/s00030-015-0323-4.

[4]

P. Cardaliaguet and P. J. Graber, Mean field games systems of first order, ESAIM Control Optim. Calc. Var., 21 (2015), 690-722.  doi: 10.1051/cocv/2014044.

[5]

P. CardaliaguetJ.-M. LasryP.-L. Lions and A. Porretta, Long time average of mean field games, Netw. Heterog. Media, 7 (2012), 279-301.  doi: 10.3934/nhm.2012.7.279.

[6]

P. CardaliaguetA. Mészáros and F. Santambrogio, First order mean field games with density constraints: Pressure equals price, SIAM J. Control Optim., 54 (2016), 2672-2709.  doi: 10.1137/15M1029849.

[7]

M. Cirant, Multi-population mean field games systems with Neumann boundary conditions, J. Math. Pures Appl. (9), 103 (2015), 1294-1315.  doi: 10.1016/j.matpur.2014.10.013.

[8]

M. Cirant, Stationary focusing mean-field games, Comm. Partial Differential Equations, 41 (2016), 1324-1346.  doi: 10.1080/03605302.2016.1192647.

[9]

M. Cirant and A. Goffi, Maximal ${L}^q$-regularity for parabolic Hamilton-Jacobi equations and applications to Mean Field Games, 2020, arXiv: 2007.14873.

[10]

D. EvangelistaR. FerreiraD. A. GomesL. Nurbekyan and V. Voskanyan, First-order, stationary mean-field games with congestion, Nonlinear Anal., 173 (2018), 37-74.  doi: 10.1016/j.na.2018.03.011.

[11]

D. Evangelista and D. A. Gomes, On the existence of solutions for stationary mean-field games with congestion, J. Dynam. Differential Equations, 30 (2018), 1365-1388.  doi: 10.1007/s10884-017-9615-1.

[12]

D. Evangelista, D. Gomes and L. Nurbekyan, Radially symmetric mean-field games with congestion, In 2017 IEEE 56th Annual Conference on Decision and Control (CDC), (2017), 3158-3163.

[13]

L. C. Evans, Adjoint and compensated compactness methods for {H}amilton-{J}acobi PDE, Arch. Ration. Mech. Anal., 197 (2010), 1053-1088.  doi: 10.1007/s00205-010-0307-9.

[14]

R. Ferreira and D. Gomes, Existence of weak solutions to stationary mean-field games through variational inequalities, SIAM J. Math. Anal., 50 (2018), 5969-6006.  doi: 10.1137/16M1106705.

[15]

R. Ferreira, D. Gomes and T. Tada, Existence of weak solutions to time-dependent mean-field games, arXiv preprint. arXiv: 2001.03928.

[16]

R. FerreiraD. Gomes and T. Tada, Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions, Proc. Amer. Math. Soc., 147 (2019), 4713-4731.  doi: 10.1090/proc/14475.

[17]

D. A. Gomes and H. Mitake, Existence for stationary mean-field games with congestion and quadratic {H}amiltonians, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1897-1910.  doi: 10.1007/s00030-015-0349-7.

[18]

D. A. Gomes, L. Nurbekyan and M. Prazere, Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion, 2016 IEEE 55th Conference on Decision and Control, CDC 2016, (2016), 4534-4539.

[19]

D. A. GomesL. Nurbekyan and M. Prazeres, One-dimensional stationary mean-field games with local coupling, Dyn. Games Appl., 8 (2018), 315-351.  doi: 10.1007/s13235-017-0223-9.

[20]

D. A. GomesS. Patrizi and V. Voskanyan, On the existence of classical solutions for stationary extended mean field games, Nonlinear Anal., 99 (2014), 49-79.  doi: 10.1016/j.na.2013.12.016.

[21]

D. A. Gomes and E. Pimentel, Time dependent mean-field games with logarithmic nonlinearities, SIAM J. Math. Anal., 47 (2015), 3798-3812.  doi: 10.1137/140984622.

[22]

D. A. GomesE. A. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the subquadratic case, Comm. Partial Differential Equations, 40 (2015), 40-76.  doi: 10.1080/03605302.2014.903574.

[23]

D. A. GomesE. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the superquadratic case, ESAIM Control Optim. Calc. Var., 22 (2016), 562-580.  doi: 10.1051/cocv/2015029.

[24]

D. A. Gomes, E. A. Pimentel and V. Voskanyan, Regularity Theory for Mean-Field Game Systems, SpringerBriefs in Mathematics. Springer, [Cham], 2016. doi: 10.1007/978-3-319-38934-9.

[25]

D. Gomes and H. Sánchez Morgado, A stochastic Evans-Aronsson problem, Trans. Amer. Math. Soc., 366 (2014), 903-929.  doi: 10.1090/S0002-9947-2013-05936-3.

[26]

D. A. Gomes and J. Saude, Monotone numerical methods for finite-state mean-field games, arXiv preprint. arXiv: 1705.00174, 2017.

[27]

D. A. Gomes and V. K. Voskanyan, Short-time existence of solutions for mean-field games with congestion, J. Lond. Math. Soc. (2), 92 (2015), 778-799.  doi: 10.1112/jlms/jdv052.

[28]

J. Graber, Weak solutions for mean field games with congestion, Preprint. arXiv: 1503.04733, 2015.

[29]

O. Guéant, A reference case for mean field games models, J. Math. Pures Appl. (9), 92 (2009), 276-294.  doi: 10.1016/j.matpur.2009.04.008.

[30]

O. Guéant, Mean field games equations with quadratic Hamiltonian: A specific approach, Math. Models Methods Appl. Sci., 22 (2012), 1250022, 37 pp.

[31]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, In Paris-Princeton Lectures on Mathematical Finance 2010, volume 2003 of Lecture Notes in Math., pages 205-266. Springer, Berlin, (2011). doi: 10.1007/978-3-642-14660-2_3.

[32]

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 (1989), 583-630.  doi: 10.1007/BF01442856.

[33]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.  doi: 10.1016/j.crma.2006.09.019.

[34]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.

[35]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.

[36]

P. L. Lions, Collège de France course on mean-field games, 2007-2011.

[37]

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. (9), 104 (2015), 1135-1159.  doi: 10.1016/j.matpur.2015.07.008.

[38]

E. A. Pimentel and V. Voskanyan, Regularity for second-order stationary mean-field games, Indiana Univ. Math. J., 66 (2017), 1-22.  doi: 10.1512/iumj.2017.66.5944.

[39]

A. Porretta, On the planning problem for the mean field games system, Dyn. Games Appl., 4 (2014), 231-256.  doi: 10.1007/s13235-013-0080-0.

[40]

A. Porretta, Weak solutions to Fokker-Planck equations and mean field games, Arch. Ration. Mech. Anal., 216 (2015), 1-62.  doi: 10.1007/s00205-014-0799-9.

[41]

F. Santambrogio, A modest proposal for MFG with density constraints, Netw. Heterog. Media, 7 (2012), 337-347.  doi: 10.3934/nhm.2012.7.337.

[42]

J. Serrin, A Harnack inequality for nonlinear equations, Bull. Amer. Math. Soc., 69 (1963), 481-486.  doi: 10.1090/S0002-9904-1963-10971-4.

[43]

H. A. Tran, Adjoint methods for static Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 41 (2011), 301-319.  doi: 10.1007/s00526-010-0363-x.

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