Time dependent first-order Mean Field Games with Neumann boundary conditions

¹ King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia
² Università LUISS - Guido Carli. Viale Romania 32, 00197 Roma, Italy

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 23 October 2024
Accepted: 1 December 2025

Abstract

The primary objective of this paper is to understand first-order, time-dependent mean-field games with Neumann boundary conditions, a question that remains under-explored in the literature. This matter is particularly relevant given the importance of boundary conditions in crowd models. In our model, the Neumann conditions result from players entering the domain Ω according to a prescribed current j, for instance, in a crowd entry scenario into an open-air concert or stadium. We formulate the model as a standard mean-field game coupling a Hamilton-Jacobi equation with a Fokker-Planck equation. Then, we introduce a relaxed variational problem and use Fenchel-Rockafellar duality to study the relation between these problems. Finally, we prove the existence and uniqueness of solutions for the system using variational methods.