On the Variational Formulation of Some Stationary Second-Order Mean Field Games Systems

Mészáros, Alpár Richárd; Silva, Francisco J.

Abstract

We consider the variational approach to prove the existence of solutions of second-order stationary Mean Field Games systems on a bounded domain $\Omega\subseteq {\mathbb R}^{d}$ with Neumann boundary conditions and with and without density constraints. We consider Hamiltonians which grow as $|\cdot|^{q'}$, where $q'=q/(q-1)$ and $q>d$. Despite this restriction, our approach allows us to prove the existence of solutions in the case of rather general coupling terms. When density constraints are taken into account, our results improve those in [A. R. Mészáros and F. J. Silva, J. Math. Pures Appl., 104 (2015), pp. 1135--1159]. Furthermore, our approach can be used to obtain solutions of systems with multiple populations.

Citation

Mészáros, A. R., & Silva, F. J. (2018). On the Variational Formulation of Some Stationary Second-Order Mean Field Games Systems. SIAM Journal on Mathematical Analysis, 50(1), 1255-1277. https://doi.org/10.1137/17m1125960

• Journal Article Type: Article
• Acceptance Date: Oct 30, 2017
• Online Publication Date: Feb 15, 2018
• Publication Date: 2018
• Deposit Date: Oct 1, 2019
• Publicly Available Date: Feb 28, 2020
• Journal: SIAM Journal on Mathematical Analysis
• Print ISSN: 0036-1410
• Electronic ISSN: 1095-7154
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 50
• Issue: 1
• Pages: 1255-1277
• DOI: https://doi.org/10.1137/17m1125960
• Related Public URLs: https://arxiv.org/abs/1704.02125