Sobolev regularity for first order mean field games

Jameson Graber, P.; Mészáros, Alpár R.

doi:10.1016/j.anihpc.2018.01.002

Jameson Graber, P. ; Mészáros, Alpár R.

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 6, pp. 1557-1576.

Résumé

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.

MR Zbl

DOI : 10.1016/j.anihpc.2018.01.002

Classification : 49K20, 35Q91, 49N60, 49N15, 49N70
Mots clés : Mean field games, Hamilton–Jacobi equations, Sobolev regularity of the solutions

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     author = {Jameson Graber, P. and M\'esz\'aros, Alp\'ar R.},
     title = {Sobolev regularity for first order mean field games},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1557--1576},
     publisher = {Elsevier},
     volume = {35},
     number = {6},
     year = {2018},
     doi = {10.1016/j.anihpc.2018.01.002},
     mrnumber = {3846236},
     zbl = {1397.49054},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2018.01.002/}
}

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Jameson Graber, P.; Mészáros, Alpár R. Sobolev regularity for first order mean field games. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 6, pp. 1557-1576. doi : 10.1016/j.anihpc.2018.01.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2018.01.002/

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