A First-Order Mean-Field Game on a Bounded Domain with Mixed Boundary Conditions

This paper presents a novel first-order stationary mean-field game model with a predefined decomposition of the boundary into entry and exit regions. On the entry portion, we prescribe the incoming flow of agents (Neumann-type boundary condition), while on the exit portion, we impose a constraint that prevents inward flow, complemented by an upper bound on the value function. The interior dynamics are governed by the standard first-order stationary mean-field game system, consisting of a stationary Hamilton-Jacobi equation and a stationary transport equation. We establish the existence of solutions via a rigorous variational formulation. Furthermore, we prove a partial uniqueness result for solutions via a Lasry-Lions monotone operator, which is key to the analysis of the model. In addition to the theoretical results, we present several examples that illustrate the behavior of the system, including the emergence of regions with vanishing density.