Uniqueness of Solutions in Mean Field Games with Several Populations and Neumann Conditions

Abstract

We study the uniqueness of solutions to systems of PDEs arising in Mean Field Games with several populations of agents and Neumann boundary conditions. The main assumption requires the smallness of some data, e.g., the length of the time horizon. This complements the existence results for MFG models of segregation phenomena introduced by the authors and Achdou. An application to robust Mean Field Games is also given.

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). They are partially supported by the research projects “Mean-Field Games and Nonlinear PDEs” of the University of Padova and “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games” of the Fondazione CaRiPaRo.

Appendix: A Comparison Principle

The next result is known but we give its elementary proof for lack of a precise reference.

Proposition 3.1

Assume $\Omega \subseteq {\mathbb R}^d$ is bounded with C ² boundary, $H : \overline \Omega \times {\mathbb R}^d$ is of class C ¹ with respect to p, and $u, v : [0,T]\times \overline \Omega \to {\mathbb R}$ are C ¹ in t and C ² in x and satisfy

$$\displaystyle \begin{aligned} \left\{ \begin{array}{ll} - \partial_t u - \Delta u + H(x,Du) \leq - \partial_t v - \Delta v + H(x,Dv) , & \mathit{in} \quad (0,T)\times\Omega , \\ \\ \partial_n u \leq \partial_n v , & \mathit{on}\quad (0,T)\times\partial \Omega , \\ \\ u(T,x) \leq v(T,x) & \mathit{in}\quad \Omega. \end{array} \right. \end{aligned}$$

Then u ≤ v in $[0,T]\times \overline \Omega$.

Proof

Let us assume first that

$$\displaystyle \begin{aligned} - \partial_t (u-v) - \Delta (u-v) + H(x,Du) - H(x, Dv) < 0 \quad \mbox{ in } [0,T)\times\Omega , \end{aligned}$$

∂ _n(u − v) < 0 on [0, T) × ∂ Ω, and (u − v)(T, x) ≤ δ. Then the maximum of u − v can be attained only at t = T, which implies u − v ≤ δ in $[0,T]\times \overline \Omega$.

Now take $g\in C^2(\overline \Omega )$ such that Dg(x) = n(x) for all x ∈ ∂ Ω and define

$$\displaystyle \begin{aligned} v_\varepsilon(t,x):= v(t,x)+\varepsilon(T-t)C+\varepsilon g(x). \end{aligned}$$

Then ∂ _n(u − v _ε) = ∂ _n(u − v) − ε < 0 and (u − v _ε)(T, x) ≤ ε∥g∥_∞. Moreover, by Taylor’s formula, for some q with |q|≤∥Dg∥_∞,

$$\displaystyle \begin{aligned}\displaystyle - \partial_t (u-v_\varepsilon) - \Delta (u-v_\varepsilon) + H(x,Du) - H(x, Dv_\varepsilon) = \\\displaystyle - \partial_t (u-v) - \Delta (u-v) + H(x,Du) - H(x, Dv) -\varepsilon (C -\Delta g+D_pH(x,q)\cdot Dg)\,{<}\,-\varepsilon \end{aligned} $$

if C is chosen large enough. Then

$$\displaystyle \begin{aligned} u\leq v_\varepsilon + \varepsilon\|g\|{}_\infty \leq v +\varepsilon(TC+ 2\|g\|{}_\infty) \end{aligned}$$

and we conclude by letting ε → 0. □

Remark 3.1

The result remains true if ∂ Ω is merely C ¹ and satisfies an interior sphere condition. This can be proved in a less direct way by linearizing the inequality for u − v and then using the parabolic Strong Maximum Principle and the parabolic version of Hopf’s Lemma for linear equations (see, e.g., [33]).