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We study the regularity and well-posedness of the local, first-order forward–backward mean field games system, assuming a polynomially growing cost function and a Hamiltonian of quadratic growth. We consider systems and terminal data that are strictly monotone in density and study two different regimes depending on whether there exists a lower bound for the running cost function. The work relies on a transformation due to P.-L. Lions, which gives rise to an elliptic partial differential equation with oblique boundary conditions, that is strictly elliptic when the coupling is unbounded from below. In this case, we prove that the solution is smooth. When the problem is degenerate elliptic, we obtain existence and uniqueness of weak solutions analogous to those obtained by P. Cardaliaguet and P. J. Graber for the case of a terminal condition that is independent of the density. The weak solutions are shown to arise as viscous limits of classical solutions to strictly elliptic problems.
@article{AIHPC_2022__39_1_1_0,
author = {Mu\~noz, Sebastian},
title = {Classical and weak solutions to local first-order mean field games through elliptic regularity},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1--39},
volume = {39},
number = {1},
year = {2022},
doi = {10.4171/aihpc/1},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/1/}
}
TY - JOUR AU - Muñoz, Sebastian TI - Classical and weak solutions to local first-order mean field games through elliptic regularity JO - Annales de l'I.H.P. Analyse non linéaire PY - 2022 SP - 1 EP - 39 VL - 39 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4171/aihpc/1/ DO - 10.4171/aihpc/1 LA - en ID - AIHPC_2022__39_1_1_0 ER -
%0 Journal Article %A Muñoz, Sebastian %T Classical and weak solutions to local first-order mean field games through elliptic regularity %J Annales de l'I.H.P. Analyse non linéaire %D 2022 %P 1-39 %V 39 %N 1 %U http://geodesic.mathdoc.fr/articles/10.4171/aihpc/1/ %R 10.4171/aihpc/1 %G en %F AIHPC_2022__39_1_1_0
Muñoz, Sebastian. Classical and weak solutions to local first-order mean field games through elliptic regularity. Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 1, pp. 1-39. doi: 10.4171/aihpc/1
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