Abstract
In this last chapter, we address the existence problem for local mean-field games. First, we illustrate the bootstrapping technique and put together the previous estimates. Thanks to this technique, we show that solutions of stationary MFGs are bounded a priori in all Sobolev spaces. This is an essential step for the two existence methods developed next. The first method is a regularization procedure in which we perturb the original local MFG into a non-local problem. By the results of the preceding chapter, this non-local problem admits a solution. Then, a limiting procedure gives the existence of a solution. The second method we consider is the continuation method. The key idea is to deform the original MFG into a problem that can be solved explicitly. Then, a topological argument shows that it is possible to deform the solution of the latter MFG into the former. This argument uses both the earlier bounds and an infinite dimensional version of the implicit function theorem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J. Dieudonné, Foundations of Modern Analysis (Academic, New York, 1969). Enlarged and corrected printing, Pure and Applied Mathematics, Vol. 10-I
L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics (American Mathematical Society, Providence, 1998)
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics (Springer, Berlin, 2001). Reprint of the 1998 edition
D. Gomes, H. Mitake, Existence for stationary mean-field games with congestion and quadratic Hamiltonians. Nonlinear Differ. Equ. Appl. 22 (6), 1897–1910 (2015)
D. Gomes, E. Pimentel, Time dependent mean-field games with logarithmic nonlinearities. SIAM J. Math. Anal. 47 (5), 3798–3812 (2015)
D. Gomes, V. Voskanyan, Short-time existence of solutions for mean-field games with congestion. J. Lond. Math. Soc. (2) 92 (3), 778–799 (2015)
D. Gomes, S. Patrizi, V. Voskanyan, On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal. 99, 49–79 (2014)
D. Gomes, H. Sánchez Morgado, A stochastic Evans-Aronsson problem. Trans. Am. Math. Soc. 366 (2), 903–929 (2014)
D. Gomes, E. Pimentel, H. Sánchez-Morgado, Time dependent mean-field games in the superquadratic case. ESAIM Control Optim. Calc. Var. 22 (2), 562–580 (2016)
D. Gomes, E. Pimentel, H. Sánchez-Morgado, Time-dependent mean-field games in the subquadratic case. Commun. Partial Differ. Equ. 40 (1), 40–76 (2015)
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires (Dunod; Gauthier-Villars, Paris, 1969
A. Porretta, On the planning problem for the mean field games system. Dyn. Games Appl. 4 (2), 231–256 (2014)
A. Porretta, Weak solutions to Fokker-Planck equations and mean field games. Arch. Ration. Mech. Anal. 216 (1), 1–62 (2015)
T. Tao, Nonlinear Dispersive Equations. CBMS Regional Conference Series in Mathematics, vol. 106 Published for the Conference Board of the Mathematical Sciences, Washington, DC; by (American Mathematical Society, Providence, 2006). Local and global analysis
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Gomes, D.A., Pimentel, E.A., Voskanyan, V. (2016). Local Mean-Field Games: Existence. In: Regularity Theory for Mean-Field Game Systems. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-38934-9_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-38934-9_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-38932-5
Online ISBN: 978-3-319-38934-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)