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Weak Solutions for First Order Mean Field Games with Local Coupling

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Analysis and Geometry in Control Theory and its Applications

Part of the book series: Springer INdAM Series ((SINDAMS,volume 11))

Abstract

Existence and uniqueness of a weak solution for first order mean field game systems with local coupling are obtained by variational methods. This solution can be used to devise ε−Nash equilibria for deterministic differential games with a finite (but large) number of players. For smooth data, the first component of the weak solution of the MFG system is proved to satisfy (in a viscosity sense) a time-space degenerate elliptic differential equation.

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Acknowledgements

The author wishes to thank the anonymous referee for his useful comments and remarks. This work has been partially supported by the Commission of the European Communities under the 7-th Framework Programme Marie Curie Initial Training Networks Project SADCO, FP7-PEOPLE-2010-ITN, No 264735, and by the French National Research Agency ANR-10-BLAN 0112 and ANR-12-BS01-0008-01.

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Authors and Affiliations

  1. Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Pierre Cardaliaguet

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  1. Pierre Cardaliaguet
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Correspondence to Pierre Cardaliaguet .

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Editors and Affiliations

  1. Department of Mathematics, Université de Bretagne Occidentale, Brest, France

    Piernicola Bettiol

  2. Department of Mathematics, Università di Roma "Tor Vergata", Roma, Italy

    Piermarco Cannarsa

  3. Department of Mathematics, Università di Padova, Padova, Italy

    Giovanni Colombo

  4. Department of Mathematics, Università di Padova, Padova, Italy

    Monica Motta

  5. Department of Mathematics, Università di Padova, Padova, Italy

    Franco Rampazzo

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Cardaliaguet, P. (2015). Weak Solutions for First Order Mean Field Games with Local Coupling. In: Bettiol, P., Cannarsa, P., Colombo, G., Motta, M., Rampazzo, F. (eds) Analysis and Geometry in Control Theory and its Applications. Springer INdAM Series, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-06917-3_5

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