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.
Mots clés : Mean field games, Hamilton–Jacobi equations, Sobolev regularity of the solutions
@article{AIHPC_2018__35_6_1557_0,
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://www.numdam.org/articles/10.1016/j.anihpc.2018.01.002/}
}
TY - JOUR AU - Jameson Graber, P. AU - Mészáros, Alpár R. TI - Sobolev regularity for first order mean field games JO - Annales de l'I.H.P. Analyse non linéaire PY - 2018 SP - 1557 EP - 1576 VL - 35 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2018.01.002/ DO - 10.1016/j.anihpc.2018.01.002 LA - en ID - AIHPC_2018__35_6_1557_0 ER -
%0 Journal Article %A Jameson Graber, P. %A Mészáros, Alpár R. %T Sobolev regularity for first order mean field games %J Annales de l'I.H.P. Analyse non linéaire %D 2018 %P 1557-1576 %V 35 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2018.01.002/ %R 10.1016/j.anihpc.2018.01.002 %G en %F AIHPC_2018__35_6_1557_0
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://www.numdam.org/articles/10.1016/j.anihpc.2018.01.002/
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