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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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