The duality method for mean field games systems

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April 2022, 21(4): 1343-1360. doi: 10.3934/cpaa.2022021

The duality method for mean field games systems

Lucio Boccardo ^1, and Luigi Orsina ^2,,

1.	Istituto Lombardo - Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy
2.	Dipartimento di Matematica, Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy

^*Corresponding author

Received August 2021 Revised December 2021 Published April 2022 Early access January 2022

In this paper we prove, using a duality method, existence of solutions for the nonlinear elliptic mean field games type system

$ \left\{ \begin{array}{cl} -{\mathop{{{\rm{div}}}}}(M(x)\,{\nabla} u) + u - {\mathop{{{\rm{div}}}}}(u\,A(x)\,{\nabla}\psi) = f(x) & {\rm{in \; \Omega ,}}\\ -{\mathop{{{\rm{div}}}}}(M(x)\,{\nabla}\psi) + \psi + A(x)\,{\nabla}\psi \cdot {\nabla} \psi = u^{p-1} & {\rm{in \; \Omega ,}} \\ u = 0 = \psi & {\rm{on \; \partial\Omega ,}} \end{array} \right. $

under different assumptions on

$ p > 1 $

, and the function

$ f(x) $

in Lebesgue spaces.

Keywords: Quasilinear elliptic equations, elliptic systems, mean field games, maximum principle.

Mathematics Subject Classification: 35J47, 35J60, 35J62.

Citation: Lucio Boccardo, Luigi Orsina. The duality method for mean field games systems. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1343-1360. doi: 10.3934/cpaa.2022021

References:

[1]	P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), 241-273.
[2]	L. Boccardo, Some developments on Dirichlet problems with discontinuous coefficients, Boll. Unione Mat. Ital. (9), 2 (2009), 285-297.
[3]	L. Boccardo, Dirichlet problems with singular convection terms and applications, J. Differ. Equ., 258 (2015), 2290-2314. doi: 10.1016/j.jde.2014.12.009.
[4]	L. Boccardo, Weak maximum principle for Dirichlet problems with convection or drift terms, Math. Eng., 3 (2021), 1-9. doi: 10.3934/mine.2021026.
[5]	L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Commun. Partial Differ. Equ., 17 (1992), 641-655. doi: 10.1080/03605309208820857.
[6]	L. Boccardo and T. Gallouët, Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data, Nonlinear Anal., 19 (1992), 573-579. doi: 10.1016/0362-546X(92)90022-7.
[7]	L. Boccardo and T. Gallouët, $W_0^{1, 1}$ solutions in some borderline cases of Calderon-Zygmund theory, J. Differ. Equ., 253 (2012), 2698-2714. doi: 10.1016/j.jde.2012.07.003.
[8]	L. Boccardo and G. Croce, Elliptic partial differential equations, in De Gruyter Studies in Mathematics, De Gruyter, Berlin, 2014.
[9]	L. Boccardo, G. Croce and L. Orsina, Nonlinear degenerate elliptic problems with $W^{1, 1}_0(\Omega)$ solutions, Manuscripta Math., 137 (2012), 419-439. doi: 10.1007/s00229-011-0473-6.
[10]	L. Boccardo and L. Orsina, Strong maximum principle for some quasilinear Dirichlet problems having natural growth terms, Adv. Nonlinear Stud., 20 (2020), 503-510. doi: 10.1515/ans-2020-2088.
[11]	L. Boccardo, L. Orsina and A. Porretta, Strongly coupled elliptic equations related to mean-field games systems, J. Differ. Equ., 261 (2016), 1796-1834. doi: 10.1016/j.jde.2016.04.018.
[12]	H. Brezis and A.C. Ponce, Remarks on the strong maximum principle, Differ. Integral Equ., 16 (2003), 1-12.
[13]	M. Cirant, Stationary focusing mean-field games, Commun. Partial Differ. Equ., 41 (2016), 1324-1346. doi: 10.1080/03605302.2016.1192647.
[14]	M. Cirant and A. Goffi, Maximal $L^q$-regularity for parabolic Hamilton-Jacobi equations and applications to mean field games, Ann. Partial Differ Equ., 7 (2021), 40 pp. doi: 10.1007/s40818-021-00109-y.
[15]	D. A. Gomes, E. A. 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.
[16]	D. A. Gomes, E. A. Pimentel and V. Voskanyan, Regularity Theory for Mean-Field Game Systems, SpringerBriefs in Mathematics, Springer, 2016. doi: 10.1007/978-3-319-38934-9.
[17]	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.
[18]	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.
[19]	J. M. Lasry and P. L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.
[20]	L. Orsina and A.C. Ponce, Strong maximum principle for Schrödinger operators with singular potential, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 477-493. doi: 10.1016/j.anihpc.2014.11.004.
[21]	G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.

show all references

References:

[1]	P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), 241-273.
[2]	L. Boccardo, Some developments on Dirichlet problems with discontinuous coefficients, Boll. Unione Mat. Ital. (9), 2 (2009), 285-297.
[3]	L. Boccardo, Dirichlet problems with singular convection terms and applications, J. Differ. Equ., 258 (2015), 2290-2314. doi: 10.1016/j.jde.2014.12.009.
[4]	L. Boccardo, Weak maximum principle for Dirichlet problems with convection or drift terms, Math. Eng., 3 (2021), 1-9. doi: 10.3934/mine.2021026.
[5]	L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Commun. Partial Differ. Equ., 17 (1992), 641-655. doi: 10.1080/03605309208820857.
[6]	L. Boccardo and T. Gallouët, Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data, Nonlinear Anal., 19 (1992), 573-579. doi: 10.1016/0362-546X(92)90022-7.
[7]	L. Boccardo and T. Gallouët, $W_0^{1, 1}$ solutions in some borderline cases of Calderon-Zygmund theory, J. Differ. Equ., 253 (2012), 2698-2714. doi: 10.1016/j.jde.2012.07.003.
[8]	L. Boccardo and G. Croce, Elliptic partial differential equations, in De Gruyter Studies in Mathematics, De Gruyter, Berlin, 2014.
[9]	L. Boccardo, G. Croce and L. Orsina, Nonlinear degenerate elliptic problems with $W^{1, 1}_0(\Omega)$ solutions, Manuscripta Math., 137 (2012), 419-439. doi: 10.1007/s00229-011-0473-6.
[10]	L. Boccardo and L. Orsina, Strong maximum principle for some quasilinear Dirichlet problems having natural growth terms, Adv. Nonlinear Stud., 20 (2020), 503-510. doi: 10.1515/ans-2020-2088.
[11]	L. Boccardo, L. Orsina and A. Porretta, Strongly coupled elliptic equations related to mean-field games systems, J. Differ. Equ., 261 (2016), 1796-1834. doi: 10.1016/j.jde.2016.04.018.
[12]	H. Brezis and A.C. Ponce, Remarks on the strong maximum principle, Differ. Integral Equ., 16 (2003), 1-12.
[13]	M. Cirant, Stationary focusing mean-field games, Commun. Partial Differ. Equ., 41 (2016), 1324-1346. doi: 10.1080/03605302.2016.1192647.
[14]	M. Cirant and A. Goffi, Maximal $L^q$-regularity for parabolic Hamilton-Jacobi equations and applications to mean field games, Ann. Partial Differ Equ., 7 (2021), 40 pp. doi: 10.1007/s40818-021-00109-y.
[15]	D. A. Gomes, E. A. 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.
[16]	D. A. Gomes, E. A. Pimentel and V. Voskanyan, Regularity Theory for Mean-Field Game Systems, SpringerBriefs in Mathematics, Springer, 2016. doi: 10.1007/978-3-319-38934-9.
[17]	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.
[18]	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.
[19]	J. M. Lasry and P. L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.
[20]	L. Orsina and A.C. Ponce, Strong maximum principle for Schrödinger operators with singular potential, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 477-493. doi: 10.1016/j.anihpc.2014.11.004.
[21]	G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.

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