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

TL;DR

This work develops a rigorous framework for a first-order, stationary Mean-Field Game on a bounded domain with mixed boundary conditions, separating entry via Neumann inflow on Γ_N from exit via a relaxed Dirichlet condition on Γ_D. It recasts the coupled HJ and transport equations into a variational problem with functional $\mathcal{I}[u]=\int_\Omega G(H(x,Du))\,dx - \int_{Γ_N} j u\,ds$, proving existence of minimizers and establishing a precise correspondence: any minimizer yields a weak MFG solution and vice versa, with the exit-constraint encoded by a contact-set condition on Γ_D. The analysis introduces a Neumann trace framework to give rigorous meaning to the flux on the boundary and proves partial uniqueness for the gradient where density is positive and full uniqueness for the density itself, aided by monotonicity of the MFG operator. Through one- and two-dimensional examples, the paper demonstrates realistic entry-exit dynamics, the possibility of empty regions, and a robust variational approach that can handle nonstandard boundary behavior, providing a solid mathematical foundation for crowd-dynamics and related applications.

Abstract

Entry-exit dynamics are crucial in modeling crowd movement. Here, we present a novel first-order, stationary mean-field game model on a bounded domain that accurately captures these dynamics. The interior dynamics of the system are governed by a standard first-order stationary MFG system consisting of a Hamilton-Jacobi equation coupled with a transport equation. The model incorporates nonstandard mixed boundary conditions corresponding to an entry region $Γ_N$, where a Neumann condition prescribes agent inflow, and an exit region $Γ_D$, where a no-entry condition prevents inward flow. Additionally, we impose an upper bound on the exit cost through $Γ_D$, combined with a complementary contact-set condition. The contact-set condition distinguishes boundary contact points, where the exit cost is attained and exit is permitted, from non-contact points, where a strict no-penetration condition is enforced. This mixed approach overcomes the limitations of classical Dirichlet conditions, which can artificially force boundary points to serve as both entry and exit locations. We analyze the system through a variational formulation, applying the direct method of the calculus of variations to establish the existence of solutions under minimal regularity assumptions. Furthermore, we prove a partial uniqueness result for the gradient of the value function (particularly in regions with positive agent density) and establish the uniqueness of the density function. Several examples, including one- and two-dimensional cases, illustrate the proper assignment of entry and exit roles and demonstrate that contact does not necessarily enforce exit. Additionally, they showcase first-order MFG phenomena, such as the formation of empty regions, where agent density vanishes. These results provide a rigorous mathematical foundation for modeling realistic entry-exit scenarios.

Figures (3)

• Figure 1: Comparison of the analytical (red dashed) and numerical (green) solutions for $m$ and two distinct $u_1$ and $u_2$, with $j_0 = 0$, and $V(x) = 0.5 \sin\bigl(3\pi (x+0.25)\bigr)$.
• Figure 2: Comparison of the analytical (red dashed) and numerical (green) solutions for $u$ and $m$, with $j_0=0.1$ and $V(x)=\gamma + 0.5 \sin\bigl(3\pi (x+0.25)\bigr)$.
• Figure 3: Illustration of MFG Behavior: Density $m$ and Flux $J$ Highlighting Empty Regions and Flow Patterns.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark
• Remark
• Remark
• Remark
• Definition 2.1
• Remark 2.2
• Lemma 2.3