Ergodic Mean Field Games: existence of local minimizers up to the Sobolev critical case

2023/01/27 by Marco Cirant, Cirant, Marco, Alessandro Cosenza +3 · 3 citations
Economics, Econometrics and Finance · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2301.11692

openalex publication_date 2023/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case.

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• The local Turnpike Property in Mean Field Control and Games with quadratic Hamiltonian
• Existence and multiplicity of normalized solutions to a large class of elliptic equations on bounded domains with general boundary conditions
• Energy local minimizers for the nonlinear Schrödinger equation on product spaces

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