On the variational formulation of some stationary second order mean field games systems

2017/04/07 by Alpár R. Mészáros, Mészáros, Alpár Richárd, Francisco J. Silva +1 · 3 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Stochastic processes and statistical mechanics

paper · doi:10.48550/arxiv.1704.02125

openalex publication_date 2017/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the variational approach to prove the existence of solutions of second order stationary Mean Field Games on a bounded domain Ω⊆ ℝ^d, with Neumann boundary conditions, and with and without density constraints. We consider Hamiltonians which growth as |⋅|^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 \citeMesSil. Furthermore, our approach can be used to obtain solutions of systems with multiple populations.

Cited by

• The local Turnpike Property in Mean Field Control and Games with quadratic Hamiltonian
• Parallel-in-Time Preconditioning for Time-Dependent Variational Mean Field Games
• Solving Mean-Field Games with Monotonicity Methods in Banach Spaces

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