| Issue |
ESAIM: COCV
Volume 29, 2023
|
|
|---|---|---|
| Article Number | 14 | |
| Number of page(s) | 21 | |
| DOI | https://doi.org/10.1051/cocv/2023004 | |
| Published online | 15 February 2023 | |
Classical solutions to local first-order extended mean field games*
Department of Mathematics, University of Chicago,
Chicago,
IL
60637,
USA
** Corresponding author: sbstn@math.uchicago.edu
Received:
29
April
2022
Accepted:
6
January
2023
We study the existence of classical solutions to a broad class of local, first order, forward-backward extended mean field games systems, that includes standard mean field games, mean field games with congestion, and mean field type control problems. We work with a strictly monotone cost that may be fully coupled with the Hamiltonian, which is assumed to have superlinear growth. Following previous work on the standard first order mean field games system, we prove the existence of smooth solutions under a coercivity condition that ensures a positive density of players, assuming a strict form of the uniqueness condition for the system. Our work relies on transforming the problem into a partial differential equation with oblique boundary conditions, which is elliptic precisely under the uniqueness condition.
Mathematics Subject Classification: 35Q89 / 35B65 / 35J66 / 35J70
Key words: Quasilinear elliptic equations / oblique derivative problems / Bernstein method / non-linear method of continuity / Hamilton-Jacobi equations
The author would like to thank P.E. Souganidis for valuable discussions, comments, and suggestions. He also thanks the anonymous referees for their invaluable help in improving and clarifying the manuscript. The author was partially supported by P.E. Souganidis’ National Science Foundation grant DMS-1900599, the Office for Naval Research grant N000141712095 and the Air Force Office for Scientific Research grant FA9550-18-1-0494.
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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