December  2020, 15(4): 681-710. doi: 10.3934/nhm.2020019

The selection problem for some first-order stationary Mean-field games

1. 

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

2. 

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

Received  August 2019 Revised  June 2020 Published  August 2020

Fund Project: D. Gomes was partially supported by King Abdullah University of Science and Technology (KAUST) baseline funds and KAUST OSR-CRG2017-3452. H. Mitake was partially supported by the JSPS grants: KAKENHI #19K03580, #19H00639, #17KK0093, #20H01816. K. Terai was supported by King Abdullah University of Science and Technology (KAUST) through the Visiting Student Research Program (VSRP) and by the JSPS grants: KAKENHI #20J10824

Here, we study the existence and the convergence of solutions for the vanishing discount MFG problem with a quadratic Hamiltonian. We give conditions under which the discounted problem has a unique classical solution and prove convergence of the vanishing-discount limit to a unique solution up to constants. Then, we establish refined asymptotics for the limit. When those conditions do not hold, the limit problem may not have a unique solution and its solutions may not be smooth, as we illustrate in an elementary example. Finally, we investigate the stability of regular weak solutions and address the selection problem. Using ideas from Aubry-Mather theory, we establish a selection criterion for the limit.

Citation: Diogo A. Gomes, Hiroyoshi Mitake, Kengo Terai. The selection problem for some first-order stationary Mean-field games. Networks & Heterogeneous Media, 2020, 15 (4) : 681-710. doi: 10.3934/nhm.2020019
References:
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E. Al-AidarousE. AlzahraniH. Ishii and A. Younas, A convergence result for the ergodic problem for Hamilton-Jacobi equations with Neumann-type boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 225-242.  doi: 10.1017/S0308210515000517.  Google Scholar

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F. CamilliI. Capuzzo-Dolcetta and D. Gomes, Error estimates for the approximation of the effective Hamiltonian, Appl. Math. Optim., 57 (2008), 30-57.  doi: 10.1007/s00245-007-9006-9.  Google Scholar

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A. DaviniA. FathiR. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted Hamilton-Jacobi equation: Convergence of the discounted solutions, Invent. Math., 206 (2016), 29-55.  doi: 10.1007/s00222-016-0648-6.  Google Scholar

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

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D. Evangelista and D. Gomes, On the existence of solutions for stationary mean-field games with congestion, J. Dyn. Diff. Equ., (2016), 1–24. doi: 10.1007/s10884-017-9615-1.  Google Scholar

[9]

L. C. Evans, Some new PDE methods for weak KAM theory, Calculus of Variations and Partial Differential Equations, 17 (2003), 159-177.  doi: 10.1007/s00526-002-0164-y.  Google Scholar

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L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. I, Arch. Ration. Mech. Anal., 157 (2001), 1-33.  doi: 10.1007/PL00004236.  Google Scholar

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L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics II, Arch. Ration. Mech. Anal., 161 (2002), 271-305.  doi: 10.1007/s002050100181.  Google Scholar

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A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649-652.  doi: 10.1016/S0764-4442(97)84777-5.  Google Scholar

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A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046.  doi: 10.1016/S0764-4442(97)87883-4.  Google Scholar

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A. Fathi, Orbite hétéroclines et ensemble de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1213-1216.  doi: 10.1016/S0764-4442(98)80230-9.  Google Scholar

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A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.  doi: 10.1016/S0764-4442(98)80144-4.  Google Scholar

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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

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R. Ferreira, D. Gomes and T. Tada, Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions, To appear in Proc. Amer. Math. Society, 2018. doi: 10.1090/proc/14475.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

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D. Gomes, Generalized Mather problem and selection principles for viscosity solutions and Mather measures, Adv. Calc. Var., 1 (2008), 291-307.  doi: 10.1515/ACV.2008.012.  Google Scholar

[20]

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

[21]

D. GomesH. Mitake and H. Tran, The selection problem for discounted Hamilton-Jacobi equations: Some non-convex cases, J. Math. Soc. Japan, 70 (2018), 345-364.  doi: 10.2969/jmsj/07017534.  Google Scholar

[22]

D. Gomes, L. Nurbekyan and M. Prazeres, 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. doi: 10.1007/s13235-017-0223-9.  Google Scholar

[23]

D. Gomes, L. Nurbekyan and M. Prazeres, One-dimensional stationary mean-field games with local coupling, Dyn. Games and Applications, (2017). doi: 10.1007/s13235-017-0223-9.  Google Scholar

[24]

D. 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

[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]

H. IshiiH. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus, J. Math. Pures Appl. (9), 108 (2017), 125-149.  doi: 10.1016/j.matpur.2016.10.013.  Google Scholar

[27]

H. IshiiH. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems, J. Math. Pures Appl. (9), 108 (2017), 261-305.  doi: 10.1016/j.matpur.2016.11.002.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

P.-L. Lions, G. Papanicolao and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations, Preliminary Version, (1988). Google Scholar

[32]

R. Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638.  doi: 10.1088/0951-7715/5/3/001.  Google Scholar

[33]

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

[34]

H. Mitake and H. Tran, Selection problems for a discount degenerate viscous Hamilton-Jacobi equation, Adv. Math., 306 (2017), 684-703.  doi: 10.1016/j.aim.2016.10.032.  Google Scholar

[35]

E. 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

show all references

References:
[1]

E. Al-AidarousE. AlzahraniH. Ishii and A. Younas, A convergence result for the ergodic problem for Hamilton-Jacobi equations with Neumann-type boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 225-242.  doi: 10.1017/S0308210515000517.  Google Scholar

[2]

F. CamilliI. Capuzzo-Dolcetta and D. Gomes, Error estimates for the approximation of the effective Hamiltonian, Appl. Math. Optim., 57 (2008), 30-57.  doi: 10.1007/s00245-007-9006-9.  Google Scholar

[3]

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

[4]

P. Cardaliaguet and A. Porretta, Long time behavior of the master equation in mean field game theory, Anal. PDE, 12 (2019), 1397-1453.  doi: 10.2140/apde.2019.12.1397.  Google Scholar

[5]

A. DaviniA. FathiR. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted Hamilton-Jacobi equation: Convergence of the discounted solutions, Invent. Math., 206 (2016), 29-55.  doi: 10.1007/s00222-016-0648-6.  Google Scholar

[6] J. Dieudonné, Foundations of Modern Analysis, Enlarged and Corrected Printing, Pure and Applied Mathematics, 10-I, Academic Press, New York-London, 1969.   Google Scholar
[7]

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

[8]

D. Evangelista and D. Gomes, On the existence of solutions for stationary mean-field games with congestion, J. Dyn. Diff. Equ., (2016), 1–24. doi: 10.1007/s10884-017-9615-1.  Google Scholar

[9]

L. C. Evans, Some new PDE methods for weak KAM theory, Calculus of Variations and Partial Differential Equations, 17 (2003), 159-177.  doi: 10.1007/s00526-002-0164-y.  Google Scholar

[10]

L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. I, Arch. Ration. Mech. Anal., 157 (2001), 1-33.  doi: 10.1007/PL00004236.  Google Scholar

[11]

L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics II, Arch. Ration. Mech. Anal., 161 (2002), 271-305.  doi: 10.1007/s002050100181.  Google Scholar

[12]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649-652.  doi: 10.1016/S0764-4442(97)84777-5.  Google Scholar

[13]

A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046.  doi: 10.1016/S0764-4442(97)87883-4.  Google Scholar

[14]

A. Fathi, Orbite hétéroclines et ensemble de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1213-1216.  doi: 10.1016/S0764-4442(98)80230-9.  Google Scholar

[15]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.  doi: 10.1016/S0764-4442(98)80144-4.  Google Scholar

[16]

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

[17]

R. Ferreira, D. Gomes and T. Tada, Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions, To appear in Proc. Amer. Math. Society, 2018. doi: 10.1090/proc/14475.  Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[19]

D. Gomes, Generalized Mather problem and selection principles for viscosity solutions and Mather measures, Adv. Calc. Var., 1 (2008), 291-307.  doi: 10.1515/ACV.2008.012.  Google Scholar

[20]

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

[21]

D. GomesH. Mitake and H. Tran, The selection problem for discounted Hamilton-Jacobi equations: Some non-convex cases, J. Math. Soc. Japan, 70 (2018), 345-364.  doi: 10.2969/jmsj/07017534.  Google Scholar

[22]

D. Gomes, L. Nurbekyan and M. Prazeres, 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. doi: 10.1007/s13235-017-0223-9.  Google Scholar

[23]

D. Gomes, L. Nurbekyan and M. Prazeres, One-dimensional stationary mean-field games with local coupling, Dyn. Games and Applications, (2017). doi: 10.1007/s13235-017-0223-9.  Google Scholar

[24]

D. 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

[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]

H. IshiiH. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus, J. Math. Pures Appl. (9), 108 (2017), 125-149.  doi: 10.1016/j.matpur.2016.10.013.  Google Scholar

[27]

H. IshiiH. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems, J. Math. Pures Appl. (9), 108 (2017), 261-305.  doi: 10.1016/j.matpur.2016.11.002.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

P.-L. Lions, G. Papanicolao and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations, Preliminary Version, (1988). Google Scholar

[32]

R. Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638.  doi: 10.1088/0951-7715/5/3/001.  Google Scholar

[33]

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

[34]

H. Mitake and H. Tran, Selection problems for a discount degenerate viscous Hamilton-Jacobi equation, Adv. Math., 306 (2017), 684-703.  doi: 10.1016/j.aim.2016.10.032.  Google Scholar

[35]

E. 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

Figure 1.  Density $ m $ for (2.2) which exhibits areas with no agents
Figure 2.  Two distinct solutions, $ \hat u $ and $ \tilde u $, of the Hamilton-Jacobi equation in (2.2). Their gradients differ only when $ m $ vanishes
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