Second order mean field games with degenerate diffusion and local coupling

  • Pierre Cardaliaguet
  • P. Jameson Graber
  • Alessio Porretta
  • Daniela Tonon
Article

Abstract

We analyze a (possibly degenerate) second order mean field games system of partial differential equations. The distinguishing features of the model considered are (1) that it is not uniformly parabolic, including the first order case as a possibility, and (2) the coupling is a local operator on the density. As a result we look for weak, not smooth, solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as minimizers of two optimal control problems. We also show that such solutions are stable with respect to the data, so that in particular the degenerate case can be approximated by a uniformly parabolic (viscous) perturbation.

Mathematics Subject Classification

35K55 49N70 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Pierre Cardaliaguet
    • 1
  • P. Jameson Graber
    • 2
  • Alessio Porretta
    • 3
  • Daniela Tonon
    • 1
  1. 1.Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de TassignyParis cedex 16France
  2. 2.828, Boulevard des MaréchauxPalaiseau CedexFrance
  3. 3.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

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