First-Order Mean-Field Games with Entry-Exit Flow Constraints and Contact-Set Conditions
This dissertation develops and rigorously analyzes first-order mean-field game models incorporating novel mixed boundary conditions to realistically represent population dynamics in bounded domains by eliminating unrealistic entry phenomena that are typically induced by standard Dirichlet conditions.
Overview
This dissertation explores first-order mean-field game (MFG) models that arise in the study of large-population dynamics in a bounded domain under nonstandard mixed boundary conditions. The proposed boundary conditions are tailored to eliminate unrealistic entry phenomena that are typically induced by standard Dirichlet conditions. This is achieved by imposing a no-entry condition on a predefined part of the boundary and a relaxed Dirichlet condition on the exit cost, specified via an inequality. An auxiliary contact-set condition is also imposed to ensure the well-posedness of the MFG model. The remaining part of the boundary is equipped with a Neumann-type condition that prescribes an inflow of agents. The interior dynamics of the game remain governed by a classical MFG system, which consists of a Hamilton–Jacobi equation coupled with a transport equation.
The first part of the dissertation introduces the proposed MFG model in detail and rigorously establishes its well-posedness. The proof of well-posedness is based on the analysis of the corresponding variational formulation and employs a modification of the direct method of the calculus of variations. We also provide some examples that illustrate the limitations of standard Dirichlet boundary conditions and the boundary behavior under the proposed boundary conditions. We also demonstrate the presence of regions of vanishing density via a two-dimensional example.
The second part of the dissertation establishes interior regularity results for stationary local MFG systems by connecting the system to quasilinear elliptic equations in divergence form. The main result of this part is the local Hölder continuity of the value function under minimal structural assumptions on the Hamiltonian in the non-separable case.
The third part of the dissertation investigates the monotone differential operator arising from the MFG system and the associated variational inequality in the regime of mixed boundary conditions. It also establishes the existence of solutions via the Browder–Minty theorem and addresses the analytical challenges posed by the lack of coercivity due to boundary terms and the possible vanishing of the density.
The last part of the dissertation discusses some observations concerning the geometric properties of agent flows at the free boundary formed by the interface between empty and populated regions.
Presenters
Brief Biography
AbdulRahman Mohammad Alharbi is a Ph.D. candidate in Applied Mathematics at King Abdullah University of Science and Technology (KAUST), working under the supervision of Professor Diogo Gomes. His primary area of research focuses on first-order mean-field games on bounded domains, with an emphasis on entry-exit flow constraints, contact-set conditions, and the analytical challenges posed by nonstandard boundary behavior.
AbdulRahman received his M.S. degree in Applied Mathematics from KAUST in 2019, with a thesis exploring the Lp-integrability of Green’s functions for elliptic differential operators. Prior to that, he received his BA in Mathematics from Rutgers University with a minor in Physics in 2017.
In addition to his research, AbdulRahman has held teaching roles at KAUST and the Islamic University of Madinah, assisting in courses on calculus, linear algebra, and partial differential equations at both undergraduate and graduate levels. He has also participated in and presented his research at international and local conferences, including the American Institute of Mathematical Sciences (AIMS ) Conference on Dynamical Systems, the conference of the S ITE center hosted by NYU Abu Dhabi, and the CMSA conference hosted by the Saudi Association of Mathematics (SAM).
His most recent work examines the well-posedness of MFG systems with nonstandard mixed boundary conditions, variational formulations of MFGs, the local regularity of weak solutions, and the monotone operator structure underlying such systems. These topics lie at the interface of partial differential equations, variational analysis, and optimal control.
