[Paper Review] A variational approach to second order mean field games with density constraints: The stationary case

Alpár R. Mészáros, Francisco J. Silva|arXiv (Cornell University)|Feb 20, 2015

Nonlinear Partial Differential Equations45 references57 citations

TL;DR

This paper establishes the existence of weak solutions to second-order stationary Mean Field Game systems with density constraints (m ≤ 1 a.e.) on a bounded domain using a variational approach. By formulating the problem as a convex optimization with a Benamou-Brenier-type functional, it proves existence for power-type Hamiltonians with arbitrary growth order q′, leveraging Sobolev regularity, subdifferential calculus, and approximation techniques when standard qualification fails.

ABSTRACT

In this paper we study second order stationary Mean Field Game systems under density constraints on a bounded domain $\\Omega \\subset \\mathbb{R}^d$. We show the existence of weak solutions for power-like Hamiltonians with arbitrary order of growth. Our strategy is a variational one, i.e. we obtain the Mean Field Game system as the optimality condition of a convex optimization problem, which has a solution. When the Hamiltonian has a growth of order $q' \\in ]1, d/(d-1)[$, the solution of the optimization problem is continuous which implies that the problem constraints are qualified. Using this fact and the computation of the subdifferential of a convex functional introduced by Benamou-Brenier, we prove the existence of a solution of the MFG system. In the case where the Hamiltonian has a growth of order $q'\\geq d/(d-1)$, the previous arguments do not apply and we prove the existence by means of an approximation argument.

Motivation & Objective

• To rigorously establish existence of weak solutions for second-order stationary Mean Field Games under the constraint that the population density m ≤ 1 almost everywhere.
• To develop a variational framework that formulates the MFG system as the optimality condition of a convex minimization problem involving a Benamou-Brenier-type functional.
• To extend existing existence results to cases where the Hamiltonian has arbitrary growth order q′, including the critical case q′ ≥ d/(d−1) where standard qualification conditions fail.
• To handle the non-strictly convex and non-smooth nature of the optimization problem by introducing a regularization and approximation scheme.

Proposed method

• Formulate the MFG system as the optimality condition of a convex minimization problem involving the Lq-norm of the current w and an energy term F(x,m), subject to continuity equation and density constraints.
• Use the direct method of calculus of variations to prove existence of minimizers (m,w) in W1,q(Ω) × Lq(Ω)d for q > d.
• Leverage the embedding W1,q(Ω) ↪ C(Ω) for q > d to ensure continuity of m, which implies constraint qualification for the optimization problem.
• Apply subdifferential calculus to the convex functional introduced by Benamou-Brenier to derive the dual variables (u, λ, µ, p) and derive the MFG system in weak form.
• For the critical case q′ ≥ d/(d−1), use a regularization of the Hamiltonian and an approximation scheme with ε > 0, proving convergence of solutions via weak and a.e. convergence in Sobolev and Lebesgue spaces.
• Use compactness arguments (Poincaré inequality, Sobolev embedding) and weak convergence in W1,q′(Ω) and Lq(Ω)d to extract convergent subsequences and pass to the limit in the regularized system.

Experimental results

Research questions

• RQ1 Can a variational approach be used to prove existence of weak solutions to stationary second-order MFG systems with density constraints m ≤ 1 a.e.?
• RQ2 What happens to the existence theory when the Hamiltonian has growth order q′ ≥ d/(d−1), where standard qualification conditions fail?
• RQ3 How can the subdifferential of the Benamou-Brenier-type functional be computed in the presence of density constraints?
• RQ4 Can the Lagrange multipliers µ and p in the MFG system be interpreted as measures supported on {m=0} and {m=1}, respectively, even in the non-smooth case?

Key findings

• For power-type Hamiltonians with growth order q′ ∈ (1, d/(d−1)), the solution (m,w) of the convex optimization problem is continuous, which ensures constraint qualification and allows direct application of subdifferential calculus.
• The existence of a weak solution (u, m, λ, µ, p) to the MFG system (MFGq) is proven in the sense of distributions, with µ and p as non-negative measures supported on {m=0} and {m=1}, respectively.
• In the critical case q′ ≥ d/(d−1), the existence is established via an approximation scheme using a regularized Hamiltonian, with convergence of solutions (uε, mε, wε) to a limit (u, m, w) in W1,q′(Ω) × W1,q(Ω) × Lq(Ω)d.
• The limit system satisfies the MFG equations in weak form, with the first equation involving a measure-valued term ρ = γ − (1/q′)|∇u|q′dx, and the final inequality (5.3) implies the weak concentration property ⟨µ − p, m⟩ ≤ 0.
• The paper proves that the dual variable p acts as a Lagrange multiplier for the constraint m ≤ 1, and its support is contained in {m = 1}, while µ is supported on {m = 0}.
• The analysis confirms that the variational approach remains valid even when standard PDE estimates fail, by relying on convex duality and approximation techniques.