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Minimax Theory and its Applications 05 (2020), No. 2, 221--250
Copyright Heldermann Verlag 2020

Mild and Weak Solutions of Mean Field Game Problems for Linear Control Systems

Piermarco Cannarsa
Dip. di Matematica, UniversitÓ di Roma "Tor Vergata", 00133 Roma, Italy

Cristian Mendico
GSSI-Gran Sasso Science Institute, 67100 L'Aquila, Italy;
and: CEREMADE, UniversitÚ Paris-Dauphine, 75775 Paris, France

The aim of this paper is to study first order Mean field games subject to a linear controlled dynamics on Rd. For this kind of problems, we define Nash equilibria (called Mean Field Games equilibria), as Borel probability measures on the space of admissible trajectories, and mild solutions as solutions associated with such equilibria. Moreover, we prove the existence and uniqueness of mild solutions and we study their regularity: we prove H÷lder regularity of Mean Field Games equilibria and fractional semiconcavity for the value function of the underlying optimal control problem. Finally, we address the PDEs system associated with the Mean Field Games problem and we prove that the class of mild solutions coincides with a suitable class of weak solutions.

Keywords: Mean field games, mean field games equilibrium, semiconcave estimates, control systems.

MSC: 35A01, 35A02, 49J30, 49J53, 49N90.

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