Sobolev regularity for first order mean field games

  • P. Jameson Graber

    Department of Mathematics, Baylor University, One Bear Place, Waco, TX 97328, USA
  • Alpár R. Mészáros

    Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095, USA


In this paper we obtain Sobolev estimates for weak solutions of first order variational Mean Field Game systems with coupling terms that are local functions of the density variable. Under some coercivity conditions on the coupling, we obtain first order Sobolev estimates for the density variable, while under similar coercivity conditions on the Hamiltonian we obtain second order Sobolev estimates for the value function. These results are valid both for stationary and time-dependent problems. In the latter case the estimates are fully global in time, thus we resolve a question which was left open in [23]. Our methods apply to a large class of Hamiltonians and coupling functions.

Cite this article

P. Jameson Graber, Alpár R. Mészáros, Sobolev regularity for first order mean field games. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 6, pp. 1557–1576

DOI 10.1016/J.ANIHPC.2018.01.002