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On the existence of classical solutions for stationary extended mean field games. (English) Zbl 1284.49044

Summary: In this paper we consider extended stationary mean-field games, that is mean-field games which depend on the velocity field of the players. We prove various a-priori estimates which generalize the results for quasi-variational mean-field games in [D. A. Gomes et al., Netw. Heterog. Media 7, No. 2, 303–314 (2012; Zbl 1260.49071)]. In addition we use adjoint method techniques to obtain higher regularity bounds. Then we establish the existence of smooth solutions under fairly general conditions by applying the continuity method. When applied to standard stationary mean-field games as in [J.-M. Lasry and P.-L. Lions, C. R., Math., Acad. Sci. Paris 343, No. 9, 619–625 (2006; Zbl 1153.91009); C. R., Math., Acad. Sci. Paris 343, No. 10, 679–684 (2006; Zbl 1153.91010)], [D. A. Gomes and H. Sanchez-Morgado, “On the stochastic Evans-Aronsson problem”, Preprint (2011)] or [D. A. Gomes et al., loc. cit.], this paper yields various new estimates and regularity properties not available previously. We discuss additionally several examples where the existence of classical solutions can be proved.

MSC:

49N70 Differential games and control
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References:

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