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

The duality method for mean field games systems

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.
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. BénilanL. BoccardoT. GallouëtR. GariepyM. 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. BoccardoG. 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. BoccardoL. 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. GomesE. 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. BénilanL. BoccardoT. GallouëtR. GariepyM. 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. BoccardoG. 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. BoccardoL. 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. GomesE. 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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