Mean Field Games of Controls with Boundary Conditions & Invariance Constraints

TL;DR

This paper studies mean field games of controls with boundary conditions and invariance constraints, establishing well-posedness, existence, uniqueness, and regularity of solutions under various conditions.

Contribution

It introduces new well-posedness results for mean field games of controls with boundary conditions and invariance constraints, including existence, uniqueness, and regularity of solutions.

Findings

Well-posedness under smallness or monotonicity conditions

Existence and uniqueness of weak solutions with invariance constraints

Higher regularity of solutions under additional assumptions

Abstract

In a mean field game of controls, a large population of identical players seek to minimize a cost that depends on the joint distribution of the states of the players and their controls. We first consider the classes of mean field games of controls in which the value function and the distribution of player states satisfy either Dirichlet or Neumann boundary conditions. We prove that such systems are well-posed either with sufficient smallness conditions or in the case of monotone couplings. Next, we consider mean field games of controls under invariance constraints imposed on the state space. We prove the existence and uniqueness of weak solutions to our mean field game system, and then we prove higher regularity of solutions under some additional assumptions.

Equations

$$\{\begin{array}{ll}-u_t-{\sum }_{i,j}{a}_{ij}(x){\partial }_{ij}^{2}u+H(t,x,D_{x}u)=f(t,x,m),& (t,x)\in [0,T]\times \overline{\Omega }\\ m_t-{\sum }_{i,j}{\partial }_{ij}^2(a_{ij}(x)m)-\nabla \cdot (m{D}_{p}H(t,x,D_{x}u))=0,& (t,x)\in [0,T]\times \overline{\Omega }\\ u(T,x)=g(x,m(T)),m(0,x)=m_0(x),& x\in \overline{\Omega }\end{array}$$

$${\Gamma }_{\epsilon }:=\{x\in {\mathbb{R}}^n:d(x,\Omega )\le \epsilon \}{\Omega }_{\epsilon }:-\Omega \setminus {\Gamma }_{\epsilon }.$$

$$d^{\ast }(\mu ,\nu )=W_1(\mu ,\nu )=\inf \left\{\int \left|x-y\right|d\pi (x,y):\pi \in \Pi (\mu ,\nu )\right\}$$

$$W_1(\mu ,\nu )=\sup \left\{\int \phi d(\mu -\nu ):\Vert \phi {\Vert }_{Lip}\le 1\right\}.$$

$$d^{\ast }(\mu ,\nu )=\sup \left\{\int \phi d(\mu -\nu ):\Vert \phi {\Vert }_{Lip}\le 1,\Vert \phi {\Vert }_{\infty }\le 1\right\}.$$

$${\Lambda }_q(\mu ):={\left({\int }_{\Omega \times {\mathbb{R}}^n}|\alpha {|}^{q}d\mu (x,\alpha )\right)}^{\frac{1}{q}}$$

$${\Lambda }_{\infty }(\mu ):=\sup \{|\alpha |:(x,\alpha )\in \text{supp}\mu \}$$

$$\{\begin{array}{ll}-u_t-\nu \Delta u+H(t,x,D_{x}u,\mu )=f(t,x,m),& (t,x)\in Q\\ m_t-\nu \Delta m-\nabla \cdot (m{D}_{p}H(t,x,D_{x}u,\mu ))=0,& (t,x)\in Q\\ \mu =(I,-D_{p}H(t,\cdot ,D_{x}u,\mu ))\#m,& t\in [0,T]\\ u(T,x)=g(x,m(T)),m(0,x)=m_0(x),& x\in \overline{\Omega }\end{array}$$

$$u=m=0,(t,x)\in \Sigma$$

$$\frac{\partial u}{\partial \vec{n}}=\nu \frac{\partial m}{\partial \vec{n}}+m{D}_{p}H(t,x,D_{x}u,\mu )\cdot \vec{n}=0,(t,x)\in \Sigma$$

$${\int }_{\Omega }(m_t\phi +(\nu D_{x}m+m{D}_{p}H)\cdot D_x\phi )dx=0$$

$$-u_t-\nu \Delta u+H(t,x,D_{x}u,\mu )=f(t,x,m)$$

$$X(t)=x_0-{\int }_0^{t\wedge \tau }{D}_{p}H(s,X(s),D_{x}u(s,X(s)),\mu (s))ds+\sqrt{2\nu }{\int }_0^{t\wedge \tau }dB(s),$$

$$\tau :=\inf \{t\in [0,T]:X(t)\in \partial \Omega \text{ or }t=T\}$$

$$\{\begin{array}{l}dX(t)=-D_{p}H(t,X(t),D_{x}u(t,X(t)),\mu (t))dt+\sqrt{2\nu }dB(t)-{\int }_{\partial \Omega }\vec{n}(X(t))dL(t)\\ X(0)=x_0\end{array}$$

$$\{\begin{array}{ll}-u_t-{\sum }_{i,j}{a}_{ij}(x){\partial }_{ij}^{2}u+H(t,x,D_{x}u,\mu )=f(t,x,m),& (t,x)\in Q\\ m_t-{\sum }_{i,j}{\partial }_{ij}^2(a_{ij}(x)m)-\nabla \cdot (m{D}_{p}H(t,x,D_{x}u,\mu ))=0,& (t,x)\in Q\\ \mu =(I,-D_{p}H(t,\cdot ,D_{x}u,\mu ))\#m,& t\in [0,T]\\ u(T,x)=g(x,m(T)),m(0,x)=m_0(x),& x\in \overline{\Omega }\end{array}$$

$$tr(a(x)D^{2}d(x))-D_{p}H(t,x,p,\mu )\cdot Dd(x)\ge \frac{a(x)Dd(x)\cdot Dd(x)}{d(x)}-Cd(x)$$

$$\{\begin{array}{ll}-u_t-{\sum }_{i,j}{a}_{ij}(x){\partial }_{ij}^{2}u+H(t,x,D_{x}u)=0,& (t,x)\in Q\\ u(T,x)=g(x),& x\in \Omega \end{array}$$

$${\int }_0^T{\int }_{\Omega }\left[u{\phi }_t+a{D}_{x}u\cdot D_x\phi +(\tilde{b}{D}_{x}u+H)\phi \right]dxdt={\int }_{\Omega }g\phi (T)dx$$

$$\{\begin{array}{ll}{m}_t-{\sum }_{i,j}{\partial }_{ij}^2(a_{ij}(x)m)-\nabla \cdot (mb)=0,& (t,x)\in (0,T)\times \Omega \\ m(0,x)=m_0(x),& x\in \Omega \end{array}$$

$$\{\begin{array}{l}-{\phi }_t-{\sum }_{i,j}{a}_{ij}{\partial }_{ij}^2\phi +b\cdot D_x\phi \in L^{\infty }(Q)\\ \phi (T)=0\end{array}$$

$${\int }_0^T{\int }_{\Omega }m\left(-{\phi }_t-\sum _{i,j}{a}_{ij}{\partial }_{ij}^2\phi +b\cdot D_x\phi \right)dxdt={\int }_{\Omega }{m}_0\phi (0)dx.$$

$$\{\begin{array}{ll}-u_t-\nu \Delta u+\lambda H(t,x,D_{x}u,\mu )=0,& (t,x)\in Q\\ m_t-\nu \Delta m-\lambda \nabla \cdot (m{D}_{p}H(t,x,D_{x}u,\mu ))=0,& (t,x)\in Q\\ \mu =(I,-\lambda D_{p}H(t,\cdot ,D_{x}u,\mu ))\#m,& t\in [0,T]\\ u(T,x)=\lambda g(x,m(T)),m(0,x)=\lambda m_0(x),& x\in \overline{\Omega }\end{array}$$

$$u=m=0,(t,x)\in \Sigma$$

$$\frac{\partial u}{\partial \vec{n}}=\nu \frac{\partial m}{\partial \vec{n}}+\lambda m{D}_{p}H(t,x,D_{x}u,\mu )\cdot \vec{n}=0,(t,x)\in \Sigma$$

$$\underset{|p|\to \infty }{\lim }\frac{H(t,x,p,\mu )}{|p|}=+\infty ,$$

$$|D_{p}H(t,x,p,{\mu }_1)-D_{p}H(t,x,p,{\mu }_2)|\le L_1\Vert {\alpha }^{{\mu }_1}-{\alpha }^{{\mu }_2}{\Vert }_{L^{q_0}(\rho )}$$

$$|D_{p}H(t,x,p,{\mu }_1)-D_{p}H(t,x,p,{\mu }_2)|\le d^{\ast }({\rho }_1,{\rho }_2{)}^{\beta }{\xi }_1(\left|p\right|,\Vert \alpha {\Vert }_{\infty })$$

$$|D_{p}H(t,x,p_1,\mu )-D_{p}H(s,y,p_2,\mu )|$$

$$\le (|p_1-p_2{|}^{\beta }+|x-y{|}^{\beta }+|t-s{|}^{\beta /2}){\xi }_2(\left|p_1\right|,\left|p_2\right|,{\Lambda }_{\infty }(\mu ))$$

$$|H(t,x,p,{\mu }_1)-H(s,x,p,{\mu }_2)|\le (|t-s{|}^{\beta /2}+d^{\ast }({\mu }_1,{\mu }_2{)}^{\beta }){\xi }_2(\left|p\right|,{\Lambda }_{\infty }({\mu }_1),{\Lambda }_{\infty }({\mu }_2)).$$

Full text

The authors are grateful to be supported by National Science Foundation through NSF Grant DMS-2045027.

1. Introduction

A mean field game (MFG) is a type of differential game in which a large population of identical rational players seek to minimize a cost (or maximize a utility) that depends on the distribution of player states. The theory of mean field games was introduced independently by Lasry and Lions in [lasry2007mean] and by Caines, Huang, and Malhamé in [huang2006large]. A typical mean field game can be characterized by a forward-backward system of PDE in which the optimal cost $u$ for a generic agent at time $t$ and state $x$ satisfies a backward-in-time Hamilton-Jacobi (HJ) equation and the distribution $m$ of player states satisfies a forward-in-time Fokker-Planck (FP) equation:

[TABLE]

In System (1), the coupling is only through the distribution of players’ states, i.e. through $m$.

By contrast, in mean field games of controls (MFGCs), each player’s cost depends on the joint distribution $\mu$ of states and controls (see System (6)). This type of game is elsewhere referred to as an extended mean field game (see [gomes2016extended, gomes2014existence]), but the terminology “mean field game of controls” now appears to be standard, cf. [cardaliaguet2018mean]. Compared to System (1), MFGCs have been much less studied in the literature. Perhaps the most comprehensive results on the existence of solutions to MFGCs are found in Kobeissi’s 2022 papers [kobeissi2022mean, kobeissi2022classical]. To derive these results, it is necessary to make a detailed study of how the Hamiltonian $H$ depends on the distribution of states and especially controls, in comparison with its dependence on the momentum variable $D_{x}u$, since this comparison will determine which a priori estimates are possible. A certain class of “potential” MFGCs has been studied in [bonnans2019schauder, graber2021weak], where the potential provides an alternative way to establish a priori estimates using the calculus of variations. The existence of solutions is now well-established on the torus $\mathbb{T}^{n}$ (i.e. with periodic boundary conditions) or on $\mathbb{R}^{n}$.

The focus of the first sections of the present article is on MFGCs with Dirichlet and Neumann boundary conditions on a smooth bounded domain. Such problems naturally arise in applications such as economics [achdou2014partial, gueant2011mean]. Dirichlet boundary conditions model players who must leave the game on reaching a certain threshold. This is the case, for example, in the exhaustible resource production models found in [chan2015bertrand, chan2017fracking], which have inspired a number of mathematical results for particular classes of MFGCs with Dirichlet and Neumann boundary conditions [graber2018existence, graber2018variational, graber2020mean, graber2023master, camilli2025learning]. Our purpose is to present a more general theory of MFGCs with boundary conditions.

A recent paper by Bongini and Salvarani has addressed the existence of solutions to mean field games of controls under Dirichlet boundary conditions [bongini2024mean]. The authors assumed that the set of controls is compact, and therefore the Hamiltonian is linearly bounded. By contrast, we wish to examine the case where the Hamiltonian is coercive with respect to the momentum variable. As for Neumann boundary conditions, some probabilistic results are given by [bo2025mean] for mean field games of controls with state reflections. Beyond this, we are unaware of any previous works on MFGC with Neumann boundary conditions. Finally, some have examined invariance conditions for MFGs (see [porretta2020mean]) as well as the master equation (see [zitridis2022master]). However, to our knowledge, there has been no investigation of MFGCs under invariance conditions for the state space.

The purpose of the first part of this article is to prove the well-posedness of mean field games of controls with both Dirichlet and Neumann boundary conditions. We prove that our results for Dirichlet and Neumann boundary conditions hold under two different sets of assumptions on the Hamiltonian and/or Lagrangian. In the first set, we carefully parametrize the growth of the Hamiltonian with respect to the distribution of controls and relate it to the growth with respect to the momentum variable, in the spirit of [kobeissi2022classical]. Uniqueness is then proved under a smallness assumption on the data. In the second set, we impose the well-known Lasry-Lions monotonicity condition, which in the case of MFGCs is most conveniently imposed on the Lagrangian $L$ rather than the Hamiltonian $H$, under the standard assumption that $H$ is the Legendre transform of $L$ with respect to the velocity variable; cf. [kobeissi2022mean, cardaliaguet2018mean]. This structure allows an alternative way to prove many of the a priori estimates leading to the existence of solutions, and in addition it guarantees uniqueness of solutions without any smallness assumptions.

In the last section of the article, we consider the class of MFGCs under invariance constraints, in which we impose conditions on the drift-diffusion terms such that the domain $\Omega$ is an invariant set for the controlled dynamics of the players, regardless of their controls. In the control community, this property is sometimes referred to as the viability of the state space. In the spirit of [porretta2020mean], most of our investigation will be done by taking solutions to an approximating MFGC on a sequence of subdomains. This will rely on the well-posedness of MFGCs under Neumann boundary conditions, and we will focus our attention in this section on the case of monotone coupling.

This type of invariance constraint is a special case of state constraints, under which the dynamical state is forced to remain inside the domain (with probability $1$), often by putting restrictions on the class of admissible controls. In [cannarsa2018existence, cannarsa2021mean], the authors examine the first-order mean field game system under state constraints. More recent papers (see [porretta2023ergodic, sardarli2021ergodic]) have considered the second-order ergodic MFG system with state constraints (including infinite Dirichlet boundary conditions), which relied on the aysmptotic behavior of $D_{x}u$ near the boundary, established in [alessio2006asymptotic]. In [lasry1989nonlinear], Lasry and Lions establish properties of solutions to elliptic equations with state constraints in both the ergotic and non-ergotic cases.

Our main contribution is as follows. We provide a systematic, comprehensive set of results on existence and uniqueness to MFGCs with both Dirichlet and Neumann boundary conditions. We synthesize many of the ideas found in Kobeissi’s papers [kobeissi2022classical, kobeissi2022mean] so as to provide one self-contained treatment of several classes of coupling, both monotone and non-monotone. More importantly, we provide a priori estimates on classical solutions that apply even when the boundary conditions create additional difficulties. For example, to prove an a priori bound on the gradient of the value function is considerably more technical, since to apply a Bernstein type argument one has to analyze its behavior near the bounday (see Section 2.4). As for the distribution of states and controls, Dirichlet boundary conditions create the added difficulty of mass absorption, so that the standard Wasserstein metric is no longer the appropriate tool to analyze the behavior of the distribution of states. We handle this differently from [bongini2024mean], preferring to use a metric very much akin to the Wasserstein metric (cf. [graber2023master]). Additionally, we combine these results with the methods used in [porretta2020mean] to give a number of results on the existence, uniqueness, and regularity solutions to MFGCs under invariance constraints on the state space.

The remainder of this manuscript is organized as follows. In the rest of this introduction, we introduce some notation and useful preliminaries, followed by a presentation of the PDE systems that we study. In Section 2, we introduce the first set of assumptions and prove existence of solutions to (6) under Dirichlet and Neumann boundary conditions with non-monotone coupling, under carefully parametrized smallness assumptions. Then in Section 3, we introduce the second set of assumptions, in particular the Lasry-Lions monotonicity condition, and prove the existence of solutions to this system under such couplings. In Section 4, we prove the uniqueness of solutions under both monotone and non-monotone coupling. Finally, in Section 5, we introduce the assumptions that we will use in our analysis of MFGCs under invariance constraints; we prove the existence and uniqueness of solutions, as well as increased regularity of solutions under additional assumptions.

1.1. Notation and Preliminaries

We begin by recalling a few definitions from [porretta2020mean].

Definition 1.1**.**

We say $K\subseteq\mathbb{R}^{n}$ is a compact domain of class $C^{2}$ if $K$ is a compact connected set and there exists $M\in\mathbb{N}$ such that for $1\leq i\leq M$, there exist $x_{i}\in\partial K$, $r_{i}>0$, $\phi_{i}\mathrel{\mathop{\ordinarycolon}}B_{r_{i}}(x_{i})\rightarrow\mathbb{R}$ so that

1. (1)

$\partial K\subseteq\bigcup_{i=1}^{M}B_{r_{i}}(x_{i})$** 2. (2)

$\partial K\cap B_{r_{i}}(x_{i})=\{x\in B_{r_{i}}(x_{i})\mathrel{\mathop{\ordinarycolon}}\phi_{i}(x)=0\}$** 3. (3)

$\phi_{i}\in C^{2}(B_{r_{i}}(x_{i}))$* with $D_{xx}^{2}\phi_{i}$ bounded.*

Definition 1.2**.**

For every $\varepsilon>0$, we will denote by $\Gamma_{\varepsilon}$ and $\Omega_{\varepsilon}$ the sets

[TABLE]

Furthermore, we will denote by $d_{\Omega}$ a $C^{2}(\overline{\Omega})$ function such that there exists $\varepsilon_{0}>0$ with $d_{\Omega}(x)=d(x,\Omega)$ in $\Gamma_{\varepsilon_{0}}$ (see [cannarsa2010invariant, delfour1994shape]). When there is no ambiguity, we will merely write $d$ for $d_{\Omega}$.

Next, we define the following spaces of measures and the metrics we impose on them, which will vary depending on the boundary conditions. Given $R>0$ and sets $A,B\subseteq\mathbb{R}^{n}$, define $\mathfrak{M}(A)$ to be the set of Borel measures $\rho$ on $A$ with $\rho(A)\leq 1$ and define $\mathfrak{M}_{\infty,R}(A\times B)$ to be the set of Borel measures $\rho\in\mathfrak{M}(A\times B)$ with $\text{supp}\rho\subseteq\{(a,b)\in A\times B\mathrel{\mathop{\ordinarycolon}}|b|\leq R\}$. We endow these spaces with a metric, depending on the boundary conditions. In the Neumann case, as the “total mass” $\int_{\Omega}m(t)dx$ remains constant, we can restrict to the space $\mathcal{P}\subset\mathfrak{M}$ consisting of probability measures, which is endowed with the Wasserstein metric

[TABLE]

where $\Pi(\mu,\nu)$ denotes the set of all couplings between $\mu$ and $\nu$. By Kantorovitch duality, we also have the characterization

[TABLE]

See [villani2021topics] for more details. We will also denote by $\mathcal{P}_{\infty,R}(A\times B)$ the set of all probability measures in $\mathfrak{M}_{\infty,R}(A\times B)$.

To deal with the “mass escape” that occurs in the Dirichlet case and the variation of mass in the sequence of solutions to approximating problems found in Section 5, we will endow $\mathfrak{M}$ with the metric

[TABLE]

We will also define the quantities

[TABLE]

for $1\leq q<\infty$ and

[TABLE]

as in [kobeissi2022classical, kobeissi2022mean]. These will allow us to quantify the dependence of $H$ on $\mu$.

As for function spaces, for non-negative integers $j,k$ we denote by $C^{j,k}$ the space of all functions $u(t,x)$ on $Q$ that are $j$ times continuously differentiable with respect to $t$ and $k$ times continuously differentiable with respect to $x$. For a fraction $\alpha\in(0,1)$ we denote by $C^{1+\alpha/2,2+\alpha}$ the usual parabolic Hölder space as introduced for instance in [ladyzhenskaia1968linear]. As for Sobolev spaces, we use similar notation as in [ladyzhenskaia1968linear], in particular $W^{1,2}_{p}$ denotes the space of all functions with weak derivatives up to order 1 in time and 2 in space, whose weak derivatives are all $L^{p}$ summable.

Finally, we recall the well-known Leray-Schauder fixed point theorem by which we will prove existence of solutions.

Theorem 1.3** (Leray-Schauder).**

Let $X$ be a Banach space and let $T\mathrel{\mathop{\ordinarycolon}}X\times[0,1]\rightarrow X$ be a continuous and compact mapping. Assume there exist $x_{0}\in X$ and $C>0$ so that $T(x,0)=x_{0}$ for all $x\in X$ and $\|x\|_{X}<C$ for all $(x,\tau)\in X\times[0,1]$ such that $T(x,\tau)=x$. Then there exists $x\in X$ such that $T(x,1)=x$.

Aside from these preliminaries, we specify that the constant $C$ appearing in many results denotes a generic constant that depends only on the data, specifically on the constants found in the Assumptions (that is, in Section 2.1 or 3.1).

1.2. The Systems of PDE

Let $\Omega\subseteq\mathbb{R}^{n}$ be a bounded, convex (we need not assume convexity in the Dirichlet case), $C^{2+\beta}$ domain for some $\beta\in(0,1)$. In Sections 2 and 3, we will consider the system

[TABLE]

paired with either Dirichlet

[TABLE]

or Neumann

[TABLE]

boundary conditions, where $Q\coloneqq[0,T]\times\overline{\Omega}$ and $\Sigma\coloneqq[0,T]\times{\partial\Omega}$ for some $T>0$.

For the purpose of variation, we will investigate classical solutions in Section 2 and strong solutions in Section 3. We define these respectively as follows.

Definition 1.4**.**

We will say $(u,m,\mu)$ is a classical solution to (6)-(6d) (resp., (6)-(6n)) if

1. (1)

$u\in C^{1,2}(Q)$* is a classical solution to the Hamilton-Jacobi equation;* 2. (2)

$m\in\{\rho\in L^{2}(0,T;H_{0}^{1}(\Omega))\mathrel{\mathop{\ordinarycolon}}\rho_{t}\in L^{2}(0,T;H^{-1}(\Omega))\}$* (resp. $m\in\{\rho\in L^{2}(0,T;H^{1}(\Omega))\mathrel{\mathop{\ordinarycolon}}\rho_{t}\in L^{2}(0,T;H^{1}(\Omega)^{*})\}$) satisfies*

[TABLE]

for all $\varphi\in H^{1}_{0}(\Omega)$ (resp., $\varphi\in H^{1}(\Omega)$) and a.e. $0\leq t\leq T$; 3. (3)

$\mu\in C^{0}(0,T;\mathfrak{M}(\overline{\Omega}\times\mathbb{R}^{n}))$* (resp. $\mu\in C^{0}(0,T;\mathcal{P}(\overline{\Omega}\times\mathbb{R}^{n}))$) satisfies the fixed-point relation $\mu=(I,-D_{p}H(t,\cdot,D_{x}u,\mu))\#m$ at every $t\in[0,T]$;* 4. (4)

$(u,m)$* satisfies $u(T,\cdot)=g(\cdot,m(T))$, $m(0,\cdot)=m_{0}$, and $u|_{\Sigma}=0$ (resp., $\frac{\partial u}{\partial\vec{n}}|_{\Sigma}=0$) pointwise.*

Definition 1.5**.**

We will say $(u,m,\mu)$ is a strong solution to (6)-(6d) (resp., (6)-(6n)) if

1. (1)

$u\in C^{0,1}(Q)\cap W^{1,2}_{2}(Q)$* is a strong solution to the HJ equation, i.e.,*

[TABLE]

for a.e. $(t,x)\in Q$; 2. (2)

$m\in\{\rho\in L^{2}(0,T;H_{0}^{1}(\Omega))\mathrel{\mathop{\ordinarycolon}}\rho_{t}\in L^{2}(0,T;H^{-1}(\Omega))\}$* (resp. $m\in\{\rho\in L^{2}(0,T;H^{1}(\Omega))\mathrel{\mathop{\ordinarycolon}}\rho_{t}\in L^{2}(0,T;H^{1}(\Omega)^{*})\}$) satisfies (7) for all $\varphi\in H^{1}_{0}(\Omega)$ (resp., $\varphi\in H^{1}(\Omega)$) and a.e. $0\leq t\leq T$;* 3. (3)

$\mu\in C^{0}(0,T;\mathfrak{M}(\overline{\Omega}\times\mathbb{R}^{n}))$* satisfies $\mu=(I,-D_{p}H(t,\cdot,D_{x}u,\mu))\#m$ at every $t\in[0,T]$;* 4. (4)

$(u,m)$* satisfies $u(T,\cdot)=g(\cdot,m(T))$, $m(0,\cdot)=m_{0}$, and $u|_{\Sigma}=0$ (resp., $\frac{\partial u}{\partial\vec{n}}|_{\Sigma}=0$) pointwise.*

It will be useful in later sections to notice the following stochastic interpretations of our boundary value problems.

Remark 1.6**.**

We note that $X(t)\sim m(t)$ and $(X(t),-D_{p}H(t,X(t),D_{x}u(t,X(t)),\mu(t)))\sim\mu(t)$, where

1. (1)

In the Dirichlet case,

[TABLE]

where $B(t)$ denotes $n$-dimensional Brownian motion and $\tau$ is the stopping time

[TABLE]

(See [bongini2024mean]). 2. (2)

In the Neumann case,

[TABLE]

where $L(t)$ is the local time of $X(t)$ and $\vec{n}(x)$ is the unit outward normal vector. For convex $\Omega$, [tanaka1979stochastic] shows that this equation can be solved. See [bo2025mean] for an analysis of reflection boundary conditions in the case of MFGCs specifically.

In Section 5, we will show that the system

[TABLE]

is well-posed in a weak sense without need for boundary conditions, given the assumption that the diffusion coefficient $a$ and the Hamiltonian $H$ satisfy the following invariance constraint:

[TABLE]

for all $t\in[0,T]$ and $p\in\mathbb{R}^{n}$, for some $C>0$, for $x$ in some neighborhood of ${\partial\Omega}$. To this end, we define weak solutions as in [porretta2020mean].

Definition 1.7**.**

We will say $u$ is a weak solution of

[TABLE]

provided

1. (1)

$u\in L^{\infty}(Q)$; 2. (2)

$u\in L^{2}(0,T;W^{1,2}(K))$* for each $K\subset\subset\Omega$* 3. (3)

For every $\varphi\in C^{\infty}((0,T]\times\Omega)$, $u$ satisfies

[TABLE]

where $\tilde{b}_{j}(x)=\sum_{i=1}^{n}\partial_{i}a_{ij}(x)$ for $j=1,\dots,n$.

Definition 1.8**.**

Given a locally bounded vector field $b\mathrel{\mathop{\ordinarycolon}}[0,T]\times\Omega\rightarrow\mathbb{R}^{n}$, we will say $m\in L^{1}([0,T]\times\Omega)$ is a weak solution of

[TABLE]

provided

1. (1)

$m\in C^{0}(0,T;L^{1}(\Omega))$* with $m\geq 0$;* 2. (2)

For every $\varphi\in C(0,T;L^{1}(\Omega))\cap L^{\infty}(Q)$ satisfying

[TABLE]

in the sense of distributions, we have

[TABLE]

Definition 1.9**.**

Given $R>0$, we will consider $(u,m,\mu)\in C^{0,1}([0,T]\times\Omega)\times C^{0}(0,T;L^{1}(\Omega))_{+}\times C^{0}(0,T;\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n}))$ a weak solution to (8) if

1. (1)

$u$* is a weak solution to the first line of (8) in sense of Definition 1.7;* 2. (2)

$m$* is a weak solution to the second line of (8) in sense of Definition 1.8;* 3. (3)

$\mu$* satisfies the third equation of (8) at every $t\in[0,T]$;* 4. (4)

$(u,m)$* satisfy the last line of (8) pointwise.*

2. Dirichlet & Neumann Boundary Conditions with Non-Monotone Coupling

For the first problem, we will make the majority of our assumptions on the Hamiltonian, and we will prove existence of classical solutions using an approach similar to that of [kobeissi2022classical].

With the aim of applying the Leray-Schauder fixed point theorem, we will consider the following parametrized system:

[TABLE]

paired with either

[TABLE]

or

[TABLE]

where $\lambda\in[0,1]$ is given.

2.1. Assumptions

In the case of non-monotone couplings, we will make the following assumptions. The constants $C_{0},\lambda_{0},\lambda_{1},\lambda_{2},q,q_{0}$ and the functions $\xi_{1},\xi_{2},\xi_{3}$ listed below are fixed independent of the data.

A 1**.**

For every $R>0$, the Hamiltonian $H\mathrel{\mathop{\ordinarycolon}}[0,T]\times\overline{\Omega}\times\mathbb{R}^{n}\times\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})\rightarrow\mathbb{R}$ is differentiable with respect to $(x,p)$, strictly convex with respect to $p$, satisfies the coercivity condition

[TABLE]

and $H$ and its derivatives are continuous with respect to $(t,x,p,\mu)$. Moreover, the Lagrangian $L\mathrel{\mathop{\ordinarycolon}}[0,T]\times\overline{\Omega}\times\mathbb{R}^{n}\times\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})\rightarrow\mathbb{R}$ is $C^{1}$ and strictly convex with respect to $\alpha$.

A 2**.**

We take $m_{0}\in C^{\beta}(\overline{\Omega})_{+}$ with $\int_{\Omega}m_{0}dx=1$. For all $m\in\mathfrak{M}(\overline{\Omega})$, we have $g(\cdot,m)\in C^{3+\beta}(\overline{\Omega})$ with $g|_{\partial\Omega}=0$ (resp., $\frac{\partial g}{\partial\vec{n}}|_{\partial\Omega}=0$) and $\|g(\cdot,m)\|_{C^{3+\beta}(\overline{\Omega})}\leq C_{0}$ for some constant $C_{0}$. Furthermore, $m\mapsto g(\cdot,m)$ is continuous from $L^{2}(\Omega)$ into $L^{2}(\Omega)$.

A 3**.**

In this problem, we will assume $f\equiv 0$. (If $f$ is sufficiently smooth such that $f$ and its derivatives are bounded, we need only replace $H$ with $H-f$. Then there is no loss of generality in assuming $f\equiv 0$.)

A 4**.**

$|D_{p}H(t,x,p,\mu)|\leq C_{0}(1+|p|^{q-1})+\lambda_{0}\Lambda_{q_{0}}(\mu)$* for some constants $q\in(1,\infty)$, $q_{0}\leq q^{\prime}\coloneqq q/(q-1)$, and $\lambda_{0}\in(0,1)$; in the Dirichlet case we require $q\leq 2$.*

A 5**.**

$D_{p}H(t,x,p,\mu)\cdot p-H(t,x,p,\mu)\geq C_{0}^{-1}(|p|^{q}-\lambda_{1}\Lambda_{q_{0}}(\mu)^{q^{\prime}})-C_{0}$.

A 6**.**

$|H(t,x,0,\mu)|\leq\lambda_{2}\Lambda_{q_{0}}(\mu)^{q^{\prime}}+C_{0}$, where $\lambda_{1}$ and $\lambda_{2}$ satisfy $0\leq\lambda_{1}<\frac{(1-\lambda_{0})^{q^{\prime}}}{C_{0}^{q^{\prime}}}-C_{0}\lambda_{2}$.

A 7**.**

$|D_{x}H(t,x,p,\mu)|\leq C_{0}(1+|p|^{q}+\Lambda_{q_{0}}(\mu)^{q^{\prime}})$.

A 8**.**

For every $R>0$, $p\in\mathbb{R}^{n}$, and $\mu_{1},\mu_{2}\in\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$ with $\mu_{i}=(I,\alpha^{\mu_{i}})\#\rho$, we have

[TABLE]

for some $L_{1}\in(0,1)$.

A 9**.**

For every $R>0$, $p\in\mathbb{R}^{n}$, and $\mu_{1},\mu_{2}\in\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$ with $\mu_{i}=(I,\alpha)\#\rho_{i}$, we have

[TABLE]

for some continuous function $\xi_{1}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{2}\rightarrow[0,\infty)$.

A 10**.**

For every $R>0$, $p\in\mathbb{R}^{n}$, and $\mu\in\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$, we have

[TABLE]

for some continuous function $\xi_{2}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{3}\rightarrow[0,\infty)$.

A 11**.**

For every $R>0$, $p\in\mathbb{R}^{n}$, and $\mu_{1},\mu_{2}\in\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$, we have

[TABLE]

A 12**.**

For every $R>0$, $p_{1},p_{2}\in\mathbb{R}^{n}$, and $\mu_{1},\mu_{2}\in\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$, we have

[TABLE]

for some continuous function $\xi_{3}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{4}\rightarrow[0,\infty)$.

For motivating examples, one could pair the examples in [kobeissi2022classical, Section 6] with relevant boundary conditions. For instance, one could consider the linear-demand exhaustible resource model with non-positively correlated resources paired with Dirichlet boundary conditions, which would model a situation in which players must leave the game when their production capacities reach the boundary of a specified region. Alternatively, one consider models of crowd dynamics or flocking birds paired with Neumann boundary conditions, which would model a situation in which members are reflected off of the boundary and must therefore remain inside the enclosed region. It is straightforward to check that our assumptions are satisfied by these examples.

2.2. Fixed-Point Relation in $\mu$

In this section, we address the well-posedness of the third equation in System (6), which we regard as a fixed-point relation for the measure $\mu$.

Lemma 2.1**.**

Assume A1, A4, and A8 hold. Given $t\in[0,T]$, $\lambda\in[0,1]$, $p\in C^{0}(\Omega;\mathbb{R}^{n})$, and $m\in\mathfrak{M}(\overline{\Omega})$, we have the following:

There exists a unique $\mu\in\mathfrak{M}(\overline{\Omega}\times\mathbb{R}^{n})$ such that

[TABLE] 2. 2)

For any $\tilde{q}\in[1,\infty]$, we have

[TABLE]

Combining (13) and A4 gives $R>0$ (depending on $p,m$) such that $\mu\in\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$. Moreover, if $m\in\mathcal{P}(\overline{\Omega})$ then $\mu\in\mathcal{P}(\overline{\Omega}\times\mathbb{R}^{n})$.

Remark 2.2**.**

Note that since we have $\int_{\Omega}mdx\leq 1$, Jensen’s inequality gives that

[TABLE]

for $1\leq q_{1}\leq q_{2}\leq\infty$, where $\mu_{\alpha}=(I,\alpha)\#m$ for some $\alpha$.

Proof of Lemma 2.1.

1. Define $\Phi_{(p,m)}^{t}\mathrel{\mathop{\ordinarycolon}}C^{0}(\Omega;\mathbb{R}^{n})\rightarrow C^{0}(\Omega;\mathbb{R}^{n})$ given by

[TABLE]

Then

[TABLE]

By the Banach fixed point theorem, since $L_{1}<1$, $\Phi_{(p,m)}^{t}$ has a unique fixed point $\alpha$. In particular, $\mu\coloneqq(I,\alpha)\#m$ uniquely solves (12).

2. If $\tilde{q}\geq q_{0}$,

[TABLE]

and so

[TABLE]

If $\tilde{q}\leq q_{0}$, we use a similar argument to get

[TABLE]

∎

Lemma 2.3**.**

Assume A1-8 hold. Let $(t_{k},\lambda_{k},p_{k},m_{k})_{n\in\mathbb{N}}$ be a sequence in $[0,T]\times[0,1]\times C^{0}(\overline{\Omega},\mathbb{R}^{n})\times L^{1}(\Omega)$ such that

• •

$t_{k}\rightarrow t$* in $[0,T]$ and $\lambda_{k}\rightarrow\lambda$ in $[0,1]$;*

• •

$p_{k}\rightarrow p$* in $C^{0}(\overline{\Omega};\mathbb{R}^{n})$;*

• •

$\|m_{k}\|_{L^{1}(\Omega)}\leq 1$* and $m_{k}\rightarrow m$ in $L^{1}(\Omega)$.*

Then $\mu^{k}\rightarrow\mu$ in $\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$, where $\mu^{k}$ and $\mu$ are the fixed points associated to $(t_{k},\lambda_{k},p_{k},m_{k})$ and $(t,\lambda,p,m)$, respectively.

Proof.

Since $(p_{k})_{k\in\mathbb{N}}$ is bounded in $C^{0}(\overline{\Omega};\mathbb{R}^{n})$, by (13),

[TABLE]

and so $\text{supp}\mu^{k}\subseteq\{(x,\alpha)\mathrel{\mathop{\ordinarycolon}}|\alpha|\leq C\}$. Thus, the $\mu^{k}$ are uniformly compactly supported. Hence, $(\mu^{k})_{k\in\mathbb{N}}$ is compact in $\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$. Take a subsequence $(\mu^{k_{j}})_{j\in\mathbb{N}}$ converging to some $\tilde{\mu}$. In the spirit of [villani2008optimal, Special case 6.16], we have

[TABLE]

Therefore, $\tilde{\mu}=(I,-\lambda D_{p}H(t,\cdot,p(\cdot),\tilde{\mu}))\#m$. By uniqueness of the fixed point, this shows that $\mu=\tilde{\mu}$. Thus, the map $(t,\lambda,p,m)\mapsto\mu$ is continuous. ∎

Remark 2.4**.**

This shows that for any $(\lambda,u,m)\in[0,1]\times C^{0,1}(Q)\times C^{0}(0,T;L^{1}(\Omega))$, the map $t\mapsto\mu(t)=(I,-\lambda D_{p}H(t,\cdot,D_{x}u(t,\cdot),\mu(t))\#m(t)$ is continuous. Furthermore, if $(\lambda_{k},p_{k},m_{k})_{k\in\mathbb{N}}$ converges to $(\lambda,D_{x}u,m)$ in $[0,1]\times C^{0}(Q;\mathbb{R}^{n})\times C^{0}(0,T;L^{1}(\Omega))$, then $\mu_{k}(t)$ defined by

[TABLE]

converges to $\mu$ uniformly on $[0,T]$.

2.3. Estimates for $u$

The main purpose of this section is to establish a priori bounds on $u$ for solutions of System (6). This will require some a priori estimates on $\mu$ as well, which follow from so-called “energy estimates” that are now standard in mean field games.

Proposition 2.5**.**

Assume A1-3 hold. Fix $\lambda\in[0,1]$ and let $(u,m,\mu)$ be a classical solution of (11). Then

[TABLE]

Proof.

Note that for $w(t,x)=u(t,x)-\lambda\int_{t}^{T}\|H(t,\cdot,0,\mu(s))\|_{\infty}ds$, we have

[TABLE]

By Gronwall’s inequality,

[TABLE]

Choosing $k=\underset{x}{\max}\,w(T,x)_{+}=\underset{x}{\max}\,u(T,x)_{+}$ gives $u(t,x)\leq\underset{x}{\max}\,u(T,x)_{+}+\lambda\int_{t}^{T}\|H(t,x,0,\mu(s))\|_{\infty}ds$. By an analogous argument, $u(t,x)\geq-\underset{x}{\max}\,u(T,x)_{-}-\lambda\int_{t}^{T}\|H(t,x,0,\mu(s))\|_{\infty}ds$. ∎

Lemma 2.6**.**

Assume A1-4, A6 and A8 hold. Then for any $\theta\in(0,1)$, we have

[TABLE]

As a corollary,

[TABLE]

Proof.

By Proposition 2.5, we have

[TABLE]

By Lemma 2.1, we get

[TABLE]

Using $q_{0}\leq q^{\prime}$ and recalling $m(t,\cdot)$ is a probability density, we have $\||D_{x}u|^{q-1}\|_{L^{q_{0}}(m)}^{q^{\prime}}\leq\|D_{x}u\|_{L^{q}(m)}^{q}\leq\|D_{x}u\|_{\infty}^{q}$, which concludes the proof. ∎

Lemma 2.7**.**

Assume A1-5 hold. Then

[TABLE]

for all $\theta\in(0,1)$ with $\frac{\lambda_{1}\lambda^{q^{\prime}}C_{0}^{q^{\prime}}}{(1-\lambda\lambda_{0})^{q^{\prime}}}(1-\theta)^{1-q^{\prime}}<1$.

Proof.

Multiply the HJ equation in (11) by $m$ and integrate by parts to get the standard “energy identity” for mean field games:

[TABLE]

By A5,

[TABLE]

Thus for $\theta\in(0,1)$ with $\frac{\lambda_{1}\lambda^{q^{\prime}}C_{0}^{q^{\prime}}}{(1-\lambda\lambda_{0})^{q^{\prime}}(1-\theta)^{q^{\prime}-1}}<1$, we have

[TABLE]

∎

Proposition 2.8**.**

Assume A1-6 and A8 hold. Then $\|u\|_{\infty}\leq C$, where $C$ depends only on $\|u(T,\cdot)\|_{\infty}$ and the constants in the assumptions.

Proof.

Choose $\theta\in(0,1)$ such that $\lambda_{1}\lambda^{q^{\prime}}+C_{0}\lambda_{2}\lambda^{q^{\prime}+1}<\frac{(1-\theta)^{q^{\prime}-1}(1-\lambda\lambda_{0})^{q^{\prime}}}{C_{0}^{q^{\prime}}}$. By lemmas 2.6 and 2.7, we have

[TABLE]

Since $C_{0}\lambda_{2}\lambda^{q^{\prime}+1}\left(\frac{(1-\lambda\lambda_{0})^{q^{\prime}}(1-\theta)^{q^{\prime}-1}}{C_{0}^{q^{\prime}}}-\lambda_{1}\lambda^{q^{\prime}}\right)^{-1}<1$, this concludes the proof. ∎

2.4. Gradient Estimate

In this section we prove a priori bounds on $|D_{x}u|$ for solutions of System (11). We use what is commonly known as a Bernstein method. For mean field games of controls, we follow an outline similar to that which is found in [kobeissi2022classical], where the boundary conditions play no role. Here, we will need to adapt the argument to the case of Neumann or Dirichlet type boundary conditions. We will start with the Neumann case.

Theorem 2.9**.**

Assume A1-8 hold. Let $(u,m,\mu)$ be a classical solution of (11)-(11n) with $u\in C^{1,3}([0,T]\times\Omega)$. Then

[TABLE]

Proof.

By vector calculus,

[TABLE]

Define the following functions:

[TABLE]

[TABLE]

where $a>1$ and $b>0$ are constants that will be defined below. Note that

[TABLE]

[TABLE]

and hence

[TABLE]

Furthermore,

[TABLE]

We now bound the right-hand side from above. To begin, notice that $\varphi^{\prime\prime}\varphi-2(\varphi^{\prime})^{2}\geq 0$ provided $a\geq b$. By A7,

[TABLE]

Since $\varphi^{\prime}<0$, A5 gives

[TABLE]

By Lemma 2.1, we get

[TABLE]

for $\theta\in(0,1)$ defined below. Also note that

[TABLE]

Combining these inequalities, we get

[TABLE]

where $\tilde{C}=C_{0}^{-1}b\|u\|_{\infty}^{-1}e^{-a-b}e^{-\frac{q}{2}e^{-a+b}}$. Since $\|w(0,\cdot)\|_{\infty}\leq eC_{0}^{2}$, we get that the constant function

[TABLE]

is a super-solution to (14) with $\|w(0,\cdot)\|_{\infty}\leq\tilde{v}$ (where $C=C+(eC_{0}^{2})^{1+\frac{q}{2}}$). Note that $v=w-\tilde{v}$ satisfies

[TABLE]

Recall that we are assuming in the case of Neumann boundary conditions that $\Omega$ is convex. Thus, for all $z\in C^{2}(\overline{\Omega})$ with $\frac{\partial z}{\partial\vec{n}}=0$ on ${\partial\Omega}$, we have $\frac{\partial}{\partial\vec{n}}|D_{x}z|^{2}\leq 0$ on ${\partial\Omega}$ (See [lions1980resolution, lions1985quelques]). Hence, we get $\frac{\partial}{\partial\vec{n}}w\leq 0$. Since

[TABLE]

this implies

[TABLE]

By Gronwall’s inequality, it follows that $w\leq\tilde{v}$. This implies

[TABLE]

By A5, we can choose $a,b,\theta$ such that $\frac{\lambda_{1}C_{0}^{q^{\prime}}e^{2b}e^{\frac{q}{2}e^{-a+b}}}{(1-\theta)^{q^{\prime}-1}(1-\lambda_{0})^{q^{\prime}}}<1$ and so

[TABLE]

If $\|w\|_{\infty}^{\frac{1}{2}}\leq 2\lambda^{\frac{1}{2}}C^{\prime}(1+\|u\|_{\infty})$, we are done. So suppose $\|w\|_{\infty}^{\frac{1}{2}}>2\lambda^{\frac{1}{2}}C^{\prime}(1+\|u\|_{\infty})$. Then we have $\|w\|_{\infty}^{1+\frac{q}{2}}\leq\frac{1}{2}\lambda^{\frac{1}{2}}\|w\|_{\infty}^{\frac{1}{2}}\left(1+\|w\|_{\infty}^{\frac{1+q}{2}}\right)$, which implies $\|w\|_{\infty}\leq\lambda^{\frac{1}{2}}$. Since $\varphi\geq 1$, this completes the proof. ∎

To handle the Dirichlet case, we first need to prove an a priori bound on $\mathinner{\!\left\lvert D_{x}u\right\rvert}$ on the boundary. Arguments of a similar spirit can be found in [ladyzhenskaia1968linear, Chapter V]. Here it is necessary to have $q\leq 2$ (see Assumption 4). Indeed, for superquadratic Hamiltonians, the phenomenon of “gradient blow-up” is well-known for Hamilton-Jacobi equations with Dirichlet boundary conditions. See e.g. [attouchi2020gradient] and references therein.

Lemma 2.10**.**

Assume A1-8 hold. Let $(u,m,\mu)$ be a classical solution of (11)-(11d). Then

[TABLE]

on $\Sigma$.

Proof.

Let $S$ be a section of ${\partial\Omega}$ such that, under a $C^{2}$ smooth change of coordinates $y=y(x)$, the image of $S$ is contained in $\mathbb{R}^{n-1}\times\{0\}$ and the image of $\overline{\Omega}$ is contained in $\mathbb{R}^{n}_{+}=\{y\in\mathbb{R}^{n}\mathrel{\mathop{\ordinarycolon}}y_{n}\geq 0\}$. We will denote by $\tilde{S}$ and $\tilde{\Omega}$ the images of $S$ and $\Omega$, respectively. Define $\tilde{u}(t,y(x))=u(t,x)$. Note that

[TABLE]

where we follow the convention of summing over repeated indices, and we define

[TABLE]

Let $A_{kl}=\sum_{i=1}^{n}A_{iikl}$, $B_{k}=\sum_{i=1}^{n}B_{iik}$, and $\tilde{H}(t,y,D_{y}\tilde{u}(t,y)\mu)=\lambda H(t,x,Du(t,x),\mu)-\nu B_{k}\partial_{k}\tilde{u}(t,y)$. Note that all coefficients are bounded, there exists $a_{0}>0$ with

[TABLE]

and

[TABLE]

where $M$ depends on the change of coordinates. (Here we have used $q\leq 2$.) Now note that $\tilde{u}$ satisfies

[TABLE]

Define $v=e^{\gamma\tilde{u}}-1$ with $\gamma>0$ to be chosen below. Then

[TABLE]

and so

[TABLE]

Choosing $\gamma=\frac{M}{a_{0}\nu}$ gives

[TABLE]

using the bound on $\tilde{u}$ coming from the maximum principle.

Define $w(t,y)=v(t,y)+\eta e^{-y_{n}}$, where $\eta$ is a constant to be chosen below. Note that

[TABLE]

where $d$ is the diameter of $\tilde{\Omega}$. For $\eta$ large enough, we get

[TABLE]

By the maximum principle,

[TABLE]

Since $v(t,y)=0$ for all $y\in\partial\tilde{\Omega}$ and since $\tilde{\Omega}\subseteq\mathbb{R}^{n}_{+}$, it follows that

[TABLE]

Moreover, for $\eta$ large enough, we have

[TABLE]

and hence

[TABLE]

Since $w$ attains its maximum value of $\eta$ on $[0,T]\times\tilde{S}$ provided $\eta$ is large enough, we get that for $y\in\tilde{S}$, $\partial_{n}w\leq 0$ and hence $\partial_{n}\tilde{u}\leq\frac{1}{\gamma}\eta e^{\gamma\|u\|_{\infty}}$. Since $D_{x}u\cdot\vec{n}=C\partial_{n}\tilde{u}$, this gives an upper bound on the normal derivative of $u$ on the boundary. A similar argument gives a lower bound on the normal derivative. Since $u=0$ on $\Sigma$, the tangential derivatives are all [math]. ∎

Theorem 2.11**.**

Assume A1-8 hold. Let $(u,m,\mu)$ be a classical solution of (11)-(11d) with $u\in C^{1,3}([0,T]\times\Omega)$. Then

[TABLE]

Proof.

The argument is almost identical to the proof of theorem 2.9, taking into account Lemma 2.10 to estimate the gradient on the boundary. ∎

2.5. Bootstrapping

Let $X\coloneqq C^{\beta/2,1+\beta}(Q)\times C^{0}(0,T;L^{2}(\Omega))$. With the aim of applying the Leray-Schauder fixed point theorem, we will define map the $\Psi\mathrel{\mathop{\ordinarycolon}}X\times[0,1]\rightarrow X$ as follows: Given $(\tilde{u},\tilde{m})\in X$ and $\lambda\in[0,1]$, define

[TABLE]

and let $(u,m)=\Psi(\tilde{u},\tilde{m},\lambda)$ be the classical solution to

[TABLE]

paired with either

[TABLE]

or

[TABLE]

Remark 2.12**.**

By A4, A6, A7, we have uniform bounds for $H$ and each of its first-order derivatives (depending only on $\|D_{x}\tilde{u}\|_{\infty}$ and the constants in the assumptions). By Lemma 2.1 and the arguments from [evans2022partial, Section 7.1.2], we get that $\tilde{\mu}$, $m$, and $\overline{\mu}$ are well-defined.

In order to prove the continuity and compactness required to apply Leray-Schauder, we will need prove higher regularity and a priori estimates for $(u,m)$, possibly depending on $\|\tilde{u}\|_{C^{\beta/2,1+\beta}(Q)}$ and $\|\tilde{m}\|_{C^{0}(0,T;L^{2}(\Omega)}$.

Lemma 2.13**.**

Assume A1-8 hold. Then there exists $C>0$ (depending only on $\|D_{x}\tilde{u}\|_{\infty}$, $\|m_{0}\|_{\infty}$, $\nu$, $n$, $T$, and $|\Omega|$) such that

[TABLE]

Proof.

In the Dirichlet case, this follows directly from [ladyzhenskaia1968linear, Theorem V.2.1]. For the convenience of the reader, we sketch a proof that holds in both Dirichlet and Neumann problems. We use Moser iteration. First note that for $p\geq 2$, we have

[TABLE]

The inequality is established formally by multiplying the FP equation by $m^{p-1}$, which is a valid test function when $m$ is bounded; to establish the inequality for all weak solutions, one can multiply by $\phi(m)$ where $\phi(s)$ is a smooth bounded function chosen to approximate $s^{p-1}$ for $s\geq 0$ (we omit the details). Now, by Sobolev’s inequality,

[TABLE]

and so

[TABLE]

By interpolation,

[TABLE]

Thus,

[TABLE]

Let $\gamma=1+\frac{2}{n}$ and define a sequence $(p_{k})_{k=0}^{\infty}$ by $p_{k}=2\gamma^{k}$. Now define the sequence $(a_{k})_{k=0}^{\infty}$ by $a_{k}=\max\{\|m_{0}\|_{L^{\infty}(\Omega)},\|m\|_{L^{p_{k}}(Q)}\}$. By induction,

[TABLE]

As the series

[TABLE]

converges, it follows that

[TABLE]

also converges, thus completing the proof. ∎

Lemma 2.14**.**

Assume A1-8 hold. Then there exist $\alpha\in(0,\beta)$ and $C>0$ so that $m\in C^{\alpha/2,\alpha}(Q)$ with

[TABLE]

where $C$ depends on $(\tilde{u},\tilde{m})$.

Proof.

In the Dirichlet case, this follows from [ladyzhenskaia1968linear, Theorem V.1.1] or [dibenedetto2012degenerate, Theorem II.1.2]. In the Neumann case, this follows by arguments similar to those used to prove [dibenedetto2012degenerate, Theorem II.1.3]. ∎

Lemma 2.15**.**

Assume A1-9 hold. Then $h(t,x)\coloneqq D_{p}H(t,x,D_{x}\tilde{u}(t,x),\overline{\mu}(t))\in C^{\alpha\beta/2,\alpha\beta}(Q)$ with

[TABLE]

Proof.

Note that for $x,y\in\overline{\Omega}$ and $t\in[0,T]$,

[TABLE]

Furthermore, for all $t,s\in[0,T]$,

[TABLE]

and

[TABLE]

As $L_{1}\in(0,1)$, this completes the proof. ∎

Remark 2.16**.**

This gives us estimates for $\overline{\mu}$ in $C^{\frac{\alpha\beta}{2}}(0,T;\mathfrak{M}_{\infty,R}(\overline{\Omega}\times\mathbb{R}^{n}))$ (resp. in $C^{\frac{\alpha\beta}{2}}(0,T;\mathcal{P}_{\infty,R}(\overline{\Omega}\times\mathbb{R}^{n}))$). By A11 and A12, this gives us estimates for $H$ in $C^{\alpha\beta^{2}/2,\alpha\beta^{2}}(Q)$ and $D_{x}H$ in $C^{\alpha\beta^{2}/2,\alpha\beta^{2}}(Q;\mathbb{R}^{n})$. By [ladyzhenskaia1968linear, Theorem IV.5.2] (resp., [ladyzhenskaia1968linear, Theorem IV.5.3]), this shows that $\Psi$ is well-defined. Furthermore, we get bounds for $u$ in $W^{1,2}_{p}(Q)$ for arbitrarily large $p$ (see [ladyzhenskaia1968linear, Section IV.9]). By [ladyzhenskaia1968linear, Lemma II.3.3], we get estimates for $u$ and $D_{x}u$ in $C^{\beta/2,\beta}(Q)$ and $C^{\beta/2,\beta}(Q;\mathbb{R}^{n})$, respectively.

Lemma 2.17**.**

Assume A1-11 hold. Then $u\in C^{1+\alpha\beta^{2}/2,2+\alpha\beta^{2}}(Q)$ with

[TABLE]

Proof.

This follows from Theorem 5.2 (resp., Theorem 5.3) from chapter IV of [ladyzhenskaia1968linear]. ∎

Finally, in order to use the results from Section 2.4 to get a priori estimates for fixed-points, we will need to prove even higher of $u$.

Lemma 2.18**.**

Assume A1-12 hold and fix $\psi\in C^{2+\alpha\beta^{2}}(\overline{\Omega})$ such that $\psi|_{\partial\Omega}=0$. Then for $j=1,\dots,n$, we get $\|\psi\partial_{j}u\|_{C^{1+\alpha\beta/2,2+\alpha\beta}(Q)}\leq C_{\psi}$. In particular, $u\in C^{1,3}([0,T]\times\Omega)$.

Proof.

Note that for $G\coloneqq\left(2\nu D_{x}\psi\cdot D_{x}\partial_{j}u+\nu\partial_{j}u\Delta\psi+\lambda\psi\left(\partial_{j}H+D_{p}H\cdot D_{x}\partial_{j}\tilde{u}\right)\right)$, we have $G\in C^{\alpha\beta^{2}/2,\alpha\beta^{2}}(Q)$. By Theorem IV.5.2 from [ladyzhenskaia1968linear], there exists a solution $w\in C^{1+\alpha\beta^{2}/2,2+\alpha\beta^{2}}(Q)$ to

[TABLE]

and it satisfies

[TABLE]

By uniqueness of solutions in the sense of distributions, it follows that $w=\psi\partial_{j}u$. ∎

2.6. Existence

With the results from Section 2.5, we are ready to prove our first existence result.

Theorem 2.19**.**

Under assumptions A1-12, there exists a classical solution to (6)-(6d) (resp., (6)-(6n)).

Proof.

Define $X$ and $\Psi\mathrel{\mathop{\ordinarycolon}}X\times[0,1]\rightarrow X$ as in section 2.5.

$\Psi(\cdot,0)$* is constant:* First, note that for every $(\tilde{u},\tilde{m})\in X$, $\Psi(\tilde{u},\tilde{m},0)=(0,0)$.

Bound for fixed-points: By the results from the previous sections, we get

[TABLE]

for all $(u,m,\lambda)\in X\times[0,1]$ with $\Psi(u,m,\lambda)=(u,m)$.

Continuity: That $\Psi$ is continuous in $\lambda$ is clear. Now fix $\lambda\in[0,1]$. Take $(\tilde{u}_{k},\tilde{m}_{k})\rightarrow(\tilde{u},\tilde{m})$ in $X$. Define $(u_{k},m_{k})=\Psi(\tilde{u}_{k},\tilde{m}_{k},\lambda)$ and $(u,m)=\Psi(\tilde{u},\tilde{m},\lambda)$. By Remark 2.4, it follows that $\tilde{\mu}^{k}=(I,-\lambda D_{p}H(t,\cdot,D_{x}\tilde{u}_{k},\tilde{\mu}^{k}))\#\overline{m}_{k}$ converges to $\tilde{\mu}=(I,-\lambda D_{p}H(t,\cdot,D_{x}\tilde{u},\tilde{\mu}))\#\overline{m}$ uniformly in $[0,T]$. Since $\tilde{u}_{k}\rightarrow\tilde{u}$ in $C^{0,1}(Q)$, we get uniform bounds for $D_{x}\tilde{u}_{k}$, and hence for $H(t,x,D_{x}\tilde{u}_{k},\tilde{\mu}^{k})$ and each of its first-order derivatives. Since

[TABLE]

Gronwall’s inequality gives

[TABLE]

and so $m_{k}\rightarrow m$ in $C^{0}(0,T;L^{2}(\Omega))$. Again, by Lemma 2.3, we get $\overline{\mu}^{k}\rightarrow\overline{\mu}$. Furthermore, by the results in Section 2.5, see in particular Lemma 2.17, we get uniform bounds for $\|u_{k}\|_{C^{1+\alpha\beta^{2}/2,2+\alpha\beta^{2}}(Q)}$. By the Arzela-Ascoli theorem, there is a subsequence $(u_{k_{j}})_{j=1}^{\infty}$ converging to some $v$ in $C^{\alpha/2,1+\alpha}(Q)$. Since

[TABLE]

it follows from Gronwall’s inequality that

[TABLE]

and so $u_{k_{j}}\rightarrow u$ in $L^{2}(Q)$. By the uniqueness of the limit in $L^{2}(Q)$, this gives $v=u$ and so $(u_{k})_{k=1}^{\infty}$ converges to $u$ in $C^{\alpha/2,1+\alpha}(Q)$.

Compactness: Take $(\tilde{u}_{k},\tilde{m}_{k})_{k\in\mathbb{N}}$ bounded in $X$ and let $(u_{k},m_{k})=\Psi(\tilde{u}_{k},\tilde{m}_{k},\lambda)$ and $(u,m)=\Psi(\tilde{u},\tilde{m},\lambda)$. By similar arguments to those above, there is a subsequence $(u_{k_{j}})_{j=1}^{\infty}$ converging to some $u$ in $C^{0,1}(Q)$. Likewise, we get uniform bounds for $m_{k_{j}}$ in $C^{\alpha/2,\alpha}(Q)$, and so $m_{k_{j}}$ converges in $C^{0}(Q)$ (and hence $C^{0}(0,T;L^{2}(\Omega))$), passing to a subsequence if necessary.

By the Leray-Schauder fixed point theorem, it follows that there exists some $(u,m)\in X$ with $\Psi(u,m,1)=(u,m)$. Letting

[TABLE]

we get that $(u,m,\mu)$ is a classical solution to (6)-(6d) (resp., (6)-(6n)). ∎

3. Dirichlet & Neumann Boundary Conditions with Monotone Coupling

For this problem, we will make most of our assumptions on the Lagrangian, and we will prove the existence of strong solutions to (6) using an approach similar to that of [kobeissi2022mean], which leverages Lasry-Lions monotonicity to obtain a priori estimates. To this end, for $\theta\in(0,1]$, define $L^{\theta}(t,x,\alpha,\mu)=\theta L(t,x,\frac{\alpha}{\theta},\Theta(\mu))$ where $\Theta\mathrel{\mathop{\ordinarycolon}}\mathfrak{M}(\overline{\Omega}\times\mathbb{R}^{n})\rightarrow\mathfrak{M}(\overline{\Omega}\times\mathbb{R}^{n})$ is given by $\Theta(\mu)=(I\otimes\frac{1}{\theta}I)\#\mu$. Note that the associated Hamiltonian is given by $H^{\theta}(t,x,p,\mu)=\theta H(t,x,p,\Theta(\mu))$. Extending to $\theta=0$ gives $H^{0}=0$ and

[TABLE]

With the aim of applying the Leray-Schauder fixed point theorem, we will consider the following parametrized system:

[TABLE]

paired with either

[TABLE]

or

[TABLE]

3.1. Assumptions

In the case of monotone couplings, we will still assume A10-12. However, we will replace A1-9 with the following assumptions:

A 13**.**

For all $R>0$ and $\mu_{1},\mu_{2}\in\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$, we have

[TABLE]

A 14**.**

The Lagrangian $L\mathrel{\mathop{\ordinarycolon}}[0,T]\times\overline{\Omega}\times\mathbb{R}^{n}\times\mathfrak{M}(\overline{\Omega}\times\mathbb{R}^{n})\rightarrow\mathbb{R}$ is differentiable with respect to $(x,\alpha)$, and $L$ and its derivatives are continuous on $[0,T]\times\overline{\Omega}\times\mathbb{R}^{n}\times\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$ for any $R>0$.

A 15**.**

For every $t\in[0,T]$, $x\in\overline{\Omega}$, and $p\in\mathbb{R}^{n}$, the maximum in

[TABLE]

is achieved at a unique $\alpha\in\mathbb{R}^{n}$.

A 16**.**

$L(t,x,\alpha,\mu)\geq\frac{1}{C_{0}}|\alpha|^{q^{\prime}}-C_{0}\left(1+\Lambda_{q^{\prime}}(\mu)^{q^{\prime}}\right)$* for some $q\in(1,\infty)$, $q^{\prime}=\frac{q}{q-1}$, and $C_{0}>0$. In the Dirichlet case, we require $q\leq 2$.*

A 17**.**

$|L(t,x,\alpha,\mu)|+|D_{x}L(t,x,\alpha,\mu)|\leq C_{0}\left(1+|\alpha|^{q^{\prime}}+\Lambda_{q^{\prime}}(\mu)^{q^{\prime}}\right)$.

A 18**.**

We take $m_{0}\in C^{\beta}(\overline{\Omega})_{+}$ with $\int_{\Omega}m_{0}dx=1$. For every $m\in\mathfrak{M}(\overline{\Omega})$, we have $g|_{\Sigma}=0$ (resp., $\frac{\partial g}{\partial\vec{n}}|_{\partial\Omega}=0$) and $\|g(\cdot,m)\|_{C^{3+\beta}(\overline{\Omega})}+\|f(\cdot,\cdot,m)\|_{C^{\beta/2,1+\beta}(Q)}\leq C_{0}$. Furthermore, $m\mapsto g(\cdot,m)$ is continuous from $L^{2}(\Omega)$ into $L^{2}(\Omega)$, and $m\mapsto f(\cdot,\cdot,m)$ is continuous from $L^{2}(Q)$ into $L^{2}(Q)$.

As an example, one could consider a variation of the exhaustible resource model discussed in [kobeissi2022mean] paired with Dirichlet boundary conditions. This would correspond to a situation in which players are forced to leave the game when their production capacities reach the boundary.

Remark 3.1**.**

Note that for $\theta\in(0,1]$, A16-17 give

[TABLE]

[TABLE]

Remark 3.2**.**

If we assume further regularity of $D_{p}H$ with respect to $\mu$ (e.g. A8-9) such that we have Hölder estimates for $\mu$, we get classical solutions as in the previous problem. However, due to the nature of $\mu$ as a fixed-point, such estimates seem to require some kind of “smallness condition”. In this problem, we will instead show that we get strong (not classical) solutions under relatively modest assumptions.

3.2. Estimates on the Hamiltonian and Lagrangian

We start by recalling results from [kobeissi2022mean] for the Lagrangian and Hamiltonian, which rely heavily on properties of convex functions (see [rockafellar1997convex]).

Lemma 3.3**.**

Fix $\theta\in(0,1]$. If $L^{\theta}$ is coercive and differentiable with respect to $\alpha$, then $L^{\theta}$ being strictly convex is equivalent to A15.

Lemma 3.4**.**

Fix $\theta\in(0,1]$. Under assumptions A14-17, the Hamiltonian $H^{\theta}(t,x,p,\mu)$ is differentiable with respect to $x$ and $p$, and $H$ and its derivatives are continuous on $[0,T]\times\overline{\Omega}\times\mathbb{R}^{n}\times\mathfrak{M}_{\infty,R}(\overline{\Omega}\times\mathbb{R}^{n})$ for all $R>0$. Furthermore, there exists $\tilde{C}_{0}>0$ (depending only on $C_{0}$) so that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for all $t\in[0,T]$, $x\in\overline{\Omega}$, $p\in\mathbb{R}^{n}$, and $\mu\in\mathfrak{M}(\overline{\Omega}\times\mathbb{R}^{n})$. Without loss of generality, we will assume $\tilde{C}_{0}=C_{0}$.

3.3. Fixed-Point Relation in $\mu$

As was done in [kobeissi2022mean], we will use our monotonicity assumption A13 to get a priori estimates for $\Lambda_{\infty}(\mu)$ and will prove the existence and uniqueness of the fixed-point $\mu$ using, respectively, the Leray-Schauder fixed point theorem and the monotonicity approach found in [cardaliaguet2018mean, Lemma 5.2].

Lemma 3.5**.**

Assume A13-17 hold. Given $t\in[0,T]$, $\theta\in(0,1]$, $p\in C^{0}(\Omega;\mathbb{R}^{n})$, and $m\in\mathfrak{M}(\overline{\Omega})$, we have the following:

If $\mu$ satisfies

[TABLE]

then we have

[TABLE]

[TABLE] 2. 2)

There exists a unique $\mu\in\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$ satisfying (22), where $R\geq\|D_{p}H^{\theta}\|_{\infty}$.

Proof.

1. Applying A13 to $\mu$ and $m\otimes\delta_{0}$ gives

[TABLE]

A17 gives

[TABLE]

Thus, by A16 and Lemma 3.3, we get

[TABLE]

where $\mu=(I,\alpha^{\mu})\#m$. Since $p(x)=-D_{\alpha}L^{\theta}(t,x,\alpha^{\mu},\mu)$ and $yz\leq\frac{y^{q^{\prime}}}{c^{q^{\prime}}q^{\prime}}+\frac{c^{q}z^{q}}{q}$ for $z,y\geq 0$ and $c>0$, we get

[TABLE]

Since $\frac{q}{q^{\prime}}+1=q$ and $(q^{\prime}-1)\frac{q}{q^{\prime}}=1$, this gives (23). Combining this with (18) gives (24).

2. Take $\overline{p}\in C^{0}(\Omega;\mathbb{R}^{n})$. For $\lambda\in[0,1]$, define

[TABLE]

and denote by $H^{\overline{p},\lambda}$ the associated Legendre transform. As $|\alpha\cdot\overline{p}(x)|\leq\frac{|\alpha|^{q^{\prime}}}{2q^{\prime}}+\frac{2^{q-1}}{q}\|\overline{p}\|_{\infty}$, we can assume (up to a change of $C_{0}$) that $L^{\overline{p},\lambda}$ satisfies A13-17. Thus, $(\lambda,t,x,p,\mu)\mapsto D_{p}H^{\overline{p},\lambda}(t,x,p,\mu)$ is continuous on $[0,1]\times[0,T]\times\overline{\Omega}\times\mathbb{R}^{n}\times\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$ for all $R>0$.

Define $\Phi\mathrel{\mathop{\ordinarycolon}}C^{0}(\overline{\Omega};\mathbb{R}^{n})\times[0,1]\rightarrow C^{0}(\overline{\Omega};\mathbb{R}^{n})$ by $\Phi(\alpha,\lambda)=-D_{p}H^{\overline{p},\lambda}(t,x,\overline{p},(I,\alpha)\#m)$. Note that $\Phi(\cdot,0)$ is constant, and that $\Phi$ is continuous by the continuity of the map $(\lambda,t,x,p,\mu)\mapsto D_{p}H^{\overline{p},\lambda}(t,x,p,\mu)$. For every $R>0$, $A_{R}\coloneqq[0,1]\times[0,T]\times\overline{\Omega}\times B_{R}(0)\times\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$ is compact, and hence $(\lambda,t,x,p,\mu)\mapsto D_{p}H^{\overline{p},\lambda}(t,x,p,\mu)$ is uniformly continuous on $A_{R}$ (by the Heine-Cantor theorem). Furthermore, $\overline{p}$ is uniformly continuous on $\overline{\Omega}$. Thus, the Arzela-Ascoli theorem gives that $\Phi$ is compact. Finally, if $\Phi(\alpha,\lambda)=\alpha$ for some $\lambda\in[0,1]$, then (23),(24) give a uniform bound for $\|\alpha\|_{\infty}$. By the Leray-Schauder fixed point theorem, there exists some $\alpha_{0}$ such that $\Phi(\alpha_{0},1)=\alpha_{0}$ and hence $\mu=(I,\alpha_{0})\#m$ satisfies (22).

To prove uniqueness, suppose $\mu_{1},\mu_{2}$ both satisfy (22). Letting $\alpha_{i}=-D_{p}H^{\theta}(t,\cdot,p,\mu_{i})$, A13 gives

[TABLE]

As $D_{\alpha}L^{\theta}(t,x,\alpha_{i},\mu_{i})=p$, this shows that $\alpha_{1}=\alpha_{2}$ $m$-a.e. ∎

We conclude this section by introducing two continuity results, the first will be useful in this section and the second will be useful in Section 5. The proofs are omitted as they are nearly identical to that of Lemma 2.3.

Lemma 3.6**.**

Assume A13-17 hold. Fix $\theta\in(0,1]$ and let $(t_{k},p_{k},m_{k})_{n\in\mathbb{N}}$ be a sequence in $[0,T]\times C^{0}(\overline{\Omega},\mathbb{R}^{n})\times L^{1}(\Omega)$ such that

• •

$t_{k}\rightarrow t$* in $[0,T]$;*

• •

$p_{k}\rightarrow p$* in $C^{0}(\overline{\Omega};\mathbb{R}^{n})$;*

• •

$\|m_{k}\|_{L^{1}(\Omega)}\leq 1$* and $m_{k}\rightarrow m$ in $L^{1}(\Omega)$.*

Then $\mu^{k}\rightarrow\mu$ in $\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$, where $\mu^{k}$ and $\mu$ are the fixed points associated to $(t_{k},p_{k},m_{k})$ and $(t,p,m)$, respectively.

Lemma 3.7**.**

Assume A13-17 hold. Fix $\theta\in(0,1]$ and let $(t_{k},p_{k},m_{k})_{k\in\mathbb{N}}$ be a sequence in $[0,T]\times C^{0}(\Omega,\mathbb{R}^{n})\times L^{1}(\Omega)$ such that

• •

$t_{k}\rightarrow t$* in $[0,T]$;*

• •

$\mathinner{\!\left\lvert p_{k}\right\rvert}\leq C$* and $p_{k}\rightarrow p$ pointwise in $\Omega$;*

• •

$\|m_{k}\|_{L^{1}(\Omega)}\leq C$* and $m_{k}\rightarrow m$ in $L^{1}(\Omega)$.*

Then $\mu^{k}\rightarrow\mu$ in $\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$, where $\mu^{k}$ and $\mu$ are the fixed points associated to $(t_{k},p_{k},m_{k})$ and $(t,p,m)$, respectively.

Remark 3.8**.**

This shows that for any $\theta\in(0,1]$, $u\in C^{0,1}([0,T]\times\Omega)$, and $m\in C^{0}(0,T;L^{1}(\Omega))$, the map $t\mapsto\mu(t)=(I,-D_{p}H^{\theta}(t,\cdot,D_{x}u(t,\cdot),\mu(t))\#m(t)$ is continuous. Furthermore, if $(p_{k},m_{k})_{k\in\mathbb{N}}$ converges to $(D_{x}u,m)$ in $C^{0}(Q;\mathbb{R}^{n})\times C^{0}(0,T;L^{1}(\Omega))$, then $\mu_{k}(t)$ defined by

[TABLE]

converges to $\mu$ uniformly on $[0,T]$.

3.4. A Priori Estimates

In this section, we prove a priori bounds on $|u|$, $|D_{x}u|$, and $\Lambda_{\infty}(\mu)$ using an approach similar to that found in [kobeissi2022mean], which we adapt to account for the boundary conditions. This approach leverages our Lasry-Lions monotonicity assumption, allowing us to eliminate some of the “smallness conditions” that we required in the non-monotone case. As before, we start with the Neumann case.

Theorem 3.9**.**

Assume A13-18 hold and suppose $(u,m,\mu)$ is a strong solution to (16)-(16n). Then there exists some $C>0$ (depending only on the constants in the assumptions) such that $\|u\|_{\infty}\leq C\theta$. Furthermore, if $u\in C^{1,3}([0,T]\times\Omega)$, then up to a change of constants, $\|D_{x}u\|_{\infty}\leq C\theta^{\frac{1}{2}}$ and $\underset{t\in[0,T]}{\sup}\Lambda_{\infty}(\mu(t))\leq C\theta$.

Proof.

Estimating $\int_{0}^{T}\Lambda_{q^{\prime}}(\mu)^{q^{\prime}}dt$:

Define $(X,\alpha)$ by

[TABLE]

where $\{B_{t}\}_{t\in[0,T]}$ is a Brownian motion independent of $\xi$ and $K_{t}$ enforces the reflection boundary conditions. Note that for $t\in[0,T]$ and $s\in[t,T]$,

[TABLE]

where

[TABLE]

$\{B^{\prime}_{t}\}_{t\in[0,T]}$ is a Brownian motion independent of $\xi^{\prime}$, and $K_{t}^{\alpha^{\prime}}$ enforces the reflection boundary conditions. Define $\tilde{X}$ by

[TABLE]

and define the measures $\tilde{m}(t)=\mathcal{L}(\tilde{X}_{t})$ and $\tilde{\mu}(t)=\mathcal{L}(\tilde{X}_{t})\otimes\delta_{0}$, where $\mathcal{L}(\tilde{X}_{t})$ denotes the law of $\tilde{X}_{t}$.

Taking $\alpha^{\prime}=0$ gives

[TABLE]

By A18,

[TABLE]

Furthermore, by A13 and A17, we get

[TABLE]

and

[TABLE]

By A16, these estimates give

[TABLE]

*Estimating $\|u\|_{\infty}$: * For $w(t,x)=u(t,x)+2C_{0}\theta(t-T)-C_{0}\theta^{1-q^{\prime}}\int_{t}^{T}\Lambda_{q^{\prime}}(\mu)dt$, we have

[TABLE]

By Gronwall’s inequality,

[TABLE]

Choosing $k=\underset{x}{\max}\,w(T,x)_{+}=\underset{x}{\max}\,u(T,x)_{+}$ gives $u(t,x)\leq\underset{x}{\max}\,u(T,x)_{+}+2C_{0}\theta(T-t)+C_{0}\theta^{1-q^{\prime}}\int_{t}^{T}\Lambda_{q^{\prime}}(\mu)dt$. Similarly, $u(t,x)\geq-\underset{x}{\max}\,u(T,x)_{-}+2C_{0}\theta(t-T)-C_{0}\theta^{1-q^{\prime}}\int_{t}^{T}\Lambda_{q^{\prime}}(\mu)dt$. In particular,

[TABLE]

*Estimating $\|D_{x}u\|_{\infty}$: * Note that if $u\in C^{1,3}([0,T]\times\Omega)$, then

[TABLE]

If $\|u\|_{\infty}=0$, then $u\equiv 0$. So suppose $\|u\|_{\infty}>0$. Define the following functions:

[TABLE]

[TABLE]

Recall that

[TABLE]

Furthermore, by similar arguments to those in the proof of Theorem 2.9,

[TABLE]

We now bound the right-hand side from above. To begin, notice that $\varphi^{\prime\prime}\varphi-2(\varphi^{\prime})^{2}\geq 0$ for $a\geq b$. By (21),

[TABLE]

Since $\varphi^{\prime}<0$, (20) gives

[TABLE]

Combining these inequalities, we get

[TABLE]

for all $\varepsilon>0$. Note that $v=w-\tilde{C}_{\varepsilon,q}\left(\theta^{1-q^{\prime}}\int_{0}^{t}\Lambda_{q^{\prime}}(\mu(s))^{q^{\prime}}ds+\theta t\right)$ satisfies

[TABLE]

Recall that for all $z\in C^{2}(\overline{\Omega})$ with $\frac{\partial z}{\partial\vec{n}}=0$ on ${\partial\Omega}$, we have $\frac{\partial}{\partial\vec{n}}|D_{x}z|^{2}\leq 0$ on ${\partial\Omega}$. Hence, $\frac{\partial}{\partial\vec{n}}w\leq 0$. Since

[TABLE]

we have

[TABLE]

By Gronwall’s inequality, it follows that $v\leq\|v(0,\cdot)\|_{\infty}$ and so

[TABLE]

Since $\varphi\geq 1$, this shows $\|D_{x}u\|_{\infty}\leq\theta^{\frac{1}{2}}C$.

*Estimating $\Lambda_{\infty}(\mu)$: * Combining our gradient estimate with (23),(24) gives $\Lambda_{q^{\prime}}(\mu)+\Lambda_{\infty}(\mu)\leq C\theta$ for some constant $C$ depending only on the constants in the assumptions. ∎

Theorem 3.10**.**

Assume A13-18 hold and suppose $(u,m,\mu)$ is a strong solution to (16)-(16d). Then there exists some $C>0$ (depending only on the constants in the assumptions) such that $\|u\|_{\infty}\leq C\theta$. Furthermore, if $u\in C^{1,3}([0,T]\times\Omega)$, then $\|D_{x}u\|_{\infty}\leq C$ and $\underset{t\in[0,T]}{\sup}\Lambda_{\infty}(\mu(t))\leq C\theta$.

Proof.

The proof is nearly identical to the proof of Theorem 3.9, with an estimate for the gradient on the boundary given by an argument similar to the proof of Lemma 2.10. ∎

3.5. Bootstrapping

Let $X\coloneqq C^{0,1}(Q)\times C^{0}(0,T;L^{2}(\Omega))$. With the aim of applying the Leray-Schauder fixed point theorem, we will define the map $\Gamma\mathrel{\mathop{\ordinarycolon}}X\times[0,1]\rightarrow X$ so that $(u,m)=\Gamma(\tilde{u},\tilde{m},\theta)$ is the unique strong solution to

[TABLE]

paired with either

[TABLE]

or

[TABLE]

(where $\overline{m}$ is defined as in section 2.5).

Remark 3.11**.**

As in section 2.5, $m$ and $\mu$ are well-defined. Moreover, [ladyzhenskaia1968linear, Theorem IV.9.1] (resp., the discussion at the end of [ladyzhenskaia1968linear, Section IV.9]) gives that there is a unique strong solution $u$ in $W^{1,2}_{\tilde{q}}(Q)$ for arbitrarily large $\tilde{q}$. Hence, $u\in C^{\frac{\beta}{2},1+\beta}(Q)$ with bounds depending only on $\|D_{x}\tilde{u}\|_{\infty}$. Thus, $\Gamma$ is well-defined.

Lemma 3.12**.**

Assume A13-18 hold. Then there exist $\alpha\in(0,\beta)$ and $C>0$ so that $m\in C^{\alpha/2,\alpha}(Q)$ with

[TABLE]

where $C$ depends only on $\|D_{x}\tilde{u}\|_{\infty}$ and the constants in the assumptions.

Theorem 3.13**.**

Assume A10-18 hold. If $(u,m,\mu)$ is a strong solution to (16)-(16d) (resp., (16)-(16n)), then we have

[TABLE]

where $C$ depends only on the constants in the assumptions.

Proof.

Let $\alpha_{k}$ be a sequence in $C^{\beta_{k}/2,\beta_{k}}(Q;\mathbb{R}^{n})$ converging to $-D_{p}H^{\theta}(t,x,D_{x}u,\mu)$ with $\|\alpha_{k}\|_{\infty}\leq C$. Letting $\theta\in[0,1]$ and $\mu^{k}=(I,\alpha_{k})\#m$, arguments similar to those in section 2 give $C^{1,3}([0,T]\times\Omega)$ classical solutions to the HJ equation

[TABLE]

paired with $u_{k}|_{\Sigma}=0$ (resp., $\frac{\partial u_{k}}{\partial\vec{n}}|_{\Sigma}=0$). By nearly identical arguments to those used to prove our a priori estimates, this gives uniform bounds for $\|D_{x}u_{k}\|_{\infty}$ and hence $\|u_{k}\|_{C^{\beta/2,1+\beta}(Q)}$. By the Arzela-Ascoli theorem, we get that there is a subsequence $u_{k_{j}}$ converging to some $v\in C^{0,1}(Q)$.

Since

[TABLE]

for a.e. $0\leq t\leq T$, Gronwall’s inequality gives that

[TABLE]

for a.e. $0\leq t\leq T$. By uniqueness, it follows that $u=v$ and so $\|u\|_{C^{\beta/2,1+\beta}(Q)}\leq C$ (by Remark 3.11). ∎

3.6. Existence

Theorem 3.14**.**

Under assumptions A10-18, there is a strong solution to (6)-(6d) (resp., (6)-(6n)).

Proof.

Define $X$ and $\Gamma\mathrel{\mathop{\ordinarycolon}}X\times[0,1]\rightarrow X$ as in section 3.5.

$\Gamma(\cdot,0)$* is constant:* First, note that for every $(\tilde{u},\tilde{m})\in X$, $\Gamma(\tilde{u},\tilde{m},0)=(0,\rho)$, where $\rho$ is the weak solution to the heat equation $\rho_{t}=\nu\Delta\rho$ with $\rho(0,x)=m_{0}$ and $\rho|_{\Sigma}=0$ (resp., $\frac{\partial\rho}{\partial\vec{n}}|_{\Sigma}=0$).

Bound for fixed-points: By the results from the previous sections, we get

[TABLE]

for all $(u,m,\theta)\in X\times[0,1]$ with $\Gamma(u,m,\theta)=(u,m)$.

Continuity: Fix $\theta\in[0,1]$ and take $(\tilde{u}_{k},\tilde{m}_{k})\rightarrow(\tilde{u},\tilde{m})$ in $X$. Define $(u_{k},m_{k})=\Gamma(\tilde{u}_{k},\tilde{m}_{k},\theta)$ and $(u,m)=\Gamma(\tilde{u},\tilde{m},\theta)$. By Remark 3.8, it follows that $\mu^{k}=(I,-D_{p}H^{\theta}(t,\cdot,D_{x}\tilde{u}_{k},\mu^{k}))\#\overline{m}_{k}$ converges to $\mu=(I,-D_{p}H^{\theta}(t,\cdot,D_{x}\tilde{u},\mu))\#\overline{m}$ uniformly in $[0,T]$. Since $D_{x}\tilde{u}_{k}$ is uniformly bounded, so are $H^{\theta}(t,x,D_{x}\tilde{u}_{k},\mu^{k})$ and each of its first-order derivatives.

By similar arguments to those in the proof of Theorem 2.19, we get $m_{k}\rightarrow m$ in $C^{0}(0,T;L^{2}(\Omega))$. Furthermore, by the results in Section 3.5, see in particular Lemma 3.13, we get uniform bounds for $\|u_{k}\|_{C^{\beta/2,1+\beta}(Q)}$. By the Arzela-Ascoli theorem, there is a subsequence $(u_{k_{j}})_{j=1}^{\infty}$ converging to some $v$ in $C^{0,1}(Q)$. By similar arguments to those in the proof of Theorem 2.19, $v=u$ and so $(u_{k})_{k=1}^{\infty}$ converges to $u$ in $C^{0,1}(Q)$.

Compactness: Take $(\tilde{u}_{k},\tilde{m}_{k})_{k\in\mathbb{N}}$ bounded in $X$ and let $(u_{k},m_{k})=\Gamma(\tilde{u}_{k},\tilde{m}_{k},\theta)$ and $(u,m)=\Gamma(\tilde{u},\tilde{m},\theta)$. By similar arguments to those above, there is a subsequence $(u_{k_{j}})_{j=1}^{\infty}$ converging to some $u$ in $C^{0,1}(Q)$. Likewise, we get uniform bounds for $m_{k_{j}}$ in $C^{\alpha/2,\alpha}(Q)$, and so $m_{k_{j}}$ converges in $C^{0}(Q)$ (and hence $C^{0}(0,T;L^{2}(\Omega))$), passing to a subsequence if necessary.

By the Leray-Schauder fixed point theorem, it follows that there exists some $(u,m)\in X$ with $\Gamma(u,m,1)=(u,m)$. Letting

[TABLE]

we get that $(u,m,\mu)$ is a strong solution to (6)-(6d) (resp., (6)-(6n)). ∎

4. Uniqueness

In this section, we prove uniqueness of solutions to our boundary-value problems under two different types of assumptions. For the first case, we use the approach found in [kobeissi2022mean], which relies on the monotonicity assumptions A13 and

A 19**.**

For all $t\in[0,T]$ and $m_{1},m_{2}\in\mathfrak{M}(\overline{\Omega})$, we have

[TABLE]

and

[TABLE]

Theorem 4.1**.**

Assume A13,A19 and either A1 or A14-15 hold. Then there is at most one strong solution $(u,m,\mu)$ to (6)-(6d) (resp., (6)-(6n)).

Proof.

Let $(u_{1},m_{1},\mu_{1})$ and $(u_{2},m_{2},\mu_{2})$ be strong solutions. By A19, we get

[TABLE]

Note that for $i=1,2$,

[TABLE]

and

[TABLE]

Thus,

[TABLE]

Since $L$ is strictly convex,

[TABLE]

with equality holding if and only if $\alpha_{1}=\alpha_{2}$. Hence,

[TABLE]

By A13, this gives

[TABLE]

By the condition for equality for (28), we get $|\{(t,x)\in Q\mathrel{\mathop{\ordinarycolon}}\alpha^{\mu_{1}}\neq\alpha^{\mu_{2}},m_{i}\neq 0\}|=0$ for $i=1,2$. Therefore, $\alpha^{\mu_{1}}=\alpha^{\mu_{2}}$. By the uniqueness of solutions to the Fokker-Planck equation, $m_{1}=m_{2}$. Therefore, $\mu_{1}=(I,\alpha^{\mu_{i}})\#m_{i}=\mu_{2}$. By the uniqueness of solutions to the Hamilton-Jacobi equation, $u_{1}=u_{2}$. ∎

For the second case, we refrain from assuming Lasry-Lions monotonicity and instead use an approach adapted from [kobeissi2022classical] to prove the uniqueness of classical solutions. This requires a short time horizon but may be more realistic for some models (e.g. models of crowd dynamics).

Theorem 4.2**.**

Suppose that $D_{p}H$ is Lipschitz in $p$ and that we have uniform bounds on $\|u\|_{C^{1,2}(Q)}$ and $\|m\|_{C^{\alpha,\alpha/2}(Q)}$ for solutions $(u,m,\mu)$ to (6)-(6d) (resp., (6)-(6n)) (as is the case if A1-12 hold). Then there exists $T_{0}>0$ such that if $T\leq T_{0}$ and

1. (1)

For all $m_{1},m_{2}\in\mathfrak{M}(\overline{\Omega})$, we have

[TABLE] 2. (2)

For every $p\in\mathbb{R}^{n}$, $\alpha_{1},\alpha_{2}\in C^{0}(\overline{\Omega};\mathbb{R}^{n})$, and $m_{1},m_{2}\in\mathfrak{M}(\overline{\Omega})$, we have

[TABLE]

[TABLE]

[TABLE]

for some $L_{1}\in(0,1)$;

then there is at most one classical solution to (6)-(6d) (resp., (6)-(6n)).

Proof.

Let $(u_{1},m_{1},\mu_{1})$ and $(u_{2},m_{2},\mu_{2})$ be solutions. Then there exist $\alpha\in(0,1)$ and $C>0$ (depending only on the constants in the assumptions) so that

[TABLE]

Define $u\coloneqq u_{1}-u_{2}$, $m\coloneqq m_{1}-m_{2}$, $\alpha_{i}(t,x)\coloneqq-D_{p}H(t,x,D_{x}u_{i}(t,x),\mu_{i})$, and $\alpha\coloneqq\alpha_{1}-\alpha_{2}$. Then

[TABLE]

and

[TABLE]

and so $\|\alpha\|_{\infty}\leq C\|D_{x}u\|_{\infty}$ provided $T$ is sufficiently small. Now note that $u$ solves

[TABLE]

(with either Dirichlet or Neumann boundary conditions). Note that

[TABLE]

Furthermore, by Theorem 6.48 (resp., Theorem 6.49) in [lieberman1996second], since $u(T,\cdot)\in C^{1+\beta}(\overline{\Omega})$, we get that $u\in C^{\frac{1+\beta}{2},1+\beta}(Q)$ with

[TABLE]

Combining this with our assumptions gives

[TABLE]

Thus, choosing $T$ sufficiently small gives $D_{x}u=0$ and hence $m=0$. This implies that $m_{1}=m_{2}$, $\mu_{1}=\mu_{2}$, and finally $u_{1}=u_{2}$. ∎

5. Invariance Constraints

In this final section, we shift our attention to MFGCs under invariance constraints imposed on the state space.

5.1. Assumptions

In the case of invariance constraints, we will still assume A10-17, and A19 hold. However, we will replace the rest of our assumptions with the following:

A 20**.**

There exists a sequence of non-negative numbers $\lambda_{k}$ such that

1. (1)

$a$* is uniformly bounded in $C^{1+\beta}(\overline{\Omega})^{n\times n}$;* 2. (2)

$\sum_{i,j}(a(x))_{ij}\xi_{i}\xi_{j}\geq\lambda_{k}|\xi|^{2}$* for every $\xi\in\mathbb{R}^{n}$ and $x\in\Omega_{1/k}$.*

A 21**.**

There exists a matrix $\sigma\in W^{1,\infty}(\Omega)$ such that $a(x)=\sigma(x)\sigma^{*}(x)$.

A 22**.**

$m_{0}\in C^{\beta/2,\beta}(\overline{\Omega})$* with $m_{0}\geq 0$ and $\int_{\Omega}m_{0}dx=1$. For every $m\in C^{0}(0,T;\mathfrak{M}(\Omega))$, we have $\|f(\cdot,\cdot,m)\|_{C^{\beta/2,1+\beta}(Q)}\leq C_{0}$. Furthermore, the map $m\mapsto f(\cdot,\cdot,m)$ is continuous from $L^{2}(Q)$ into $L^{2}(Q)$.*

A 23**.**

The map $g\mathrel{\mathop{\ordinarycolon}}L^{1}(\Omega)\rightarrow C^{0}(\Omega)$ given by $m\mapsto g(\cdot,m)$ is such that $\frac{\partial g}{\partial\vec{n}}|_{\partial\Omega}=0$ and the map $m\mapsto g(\cdot,m)$ is continuous from $L^{2}(\Omega)$ into $L^{2}(\Omega)$. Furthermore, there exist a sequence $\{\Omega^{\varepsilon}\}$ of convex, $C^{2+\alpha}$ subdomains such that $\Omega_{2\varepsilon}\subseteq\Omega^{\varepsilon}\subseteq\Omega_{\varepsilon}$ and a sequence of functions $g_{\varepsilon}(x,m)$ satisfying

1. (1)

For every $\varepsilon>0$ and $m\in\mathfrak{M}(\Omega^{\varepsilon})$, we have $\frac{\partial g_{\varepsilon}}{\partial\vec{n}_{\varepsilon}}|_{\partial\Omega^{\varepsilon}}=0$, $\|g_{\varepsilon}(\cdot,m)\|_{C^{3+\beta}(\overline{\Omega^{\varepsilon}})}\leq C(\varepsilon)$, and $\|g_{\varepsilon}(\cdot,m)\|_{C^{1+\beta}(\overline{\Omega^{\varepsilon}})}\leq C_{0}$; 2. (2)

For every $\varepsilon>0$, $m\mapsto g_{\varepsilon}(\cdot,m)$ is continuous from $L^{2}(\Omega^{\varepsilon})$ into $L^{2}(\Omega^{\varepsilon})$; 3. (3)

If $m_{\varepsilon}\rightarrow m$ in $L^{1}(\Omega)$, then (up to a subsequence) $g_{\varepsilon}(\cdot,m_{\varepsilon})\rightarrow g(\cdot,m)$ a.e.;

For an example satisfying Assumption 23, let ${\partial\Omega}$ be $C^{4+\beta}$-smooth, $m\mapsto g(\cdot,m)$ be continuous from $L^{1}(\Omega)$ into $L^{2}(\Omega)$, and $\|g(\cdot,m)\|_{C^{3+\beta}(\overline{\Omega})}\leq C$ for all $m$: Then there exists a collar region $\{x\mathrel{\mathop{\ordinarycolon}}\mathinner{\!\left\lvert d(x)\right\rvert}<r\}$ on which $d(x)$ is $C^{4+\beta}$. Let $\varepsilon<r/2$. Pick a $C^{\infty}$ smooth non-increasing function $\eta_{\varepsilon}\mathrel{\mathop{\ordinarycolon}}\mathinner{\left[0,\infty\right)}\to\mathinner{\left[0,\infty\right)}$ such that $\eta_{\varepsilon}(s)=\varepsilon$ for $s\in[0,\varepsilon]$ and $\eta_{\varepsilon}$ has compact support in $\mathinner{\left[0,2\varepsilon\right)}$. Define $\xi=\xi(t,x)$ to be the flow map given by solving the ODE

[TABLE]

As the vector field $V(\xi)=\eta_{\varepsilon}\mathinner{\left(d(\xi)\right)}Dd(\xi)$ is $C^{3+\beta}$ smooth, the map $x\mapsto\xi(t,x)$ is a $C^{3+\beta}$ diffeomorphism for every $t\geq 0$. Since $\mathinner{\!\left\lvert Dd\right\rvert}=1$, we have

[TABLE]

From the structure of $\eta_{\varepsilon}$ we can deduce that $d\mathinner{\left(\xi(t,x)\right)}\geq\varepsilon$ for all $t\geq 1$ and all $x\in\overline{\Omega}$, and on the other hand $\xi(t,x)=x$ whenever $d(x)\geq 2\varepsilon$. We now set $\psi_{\varepsilon}(x)=\xi(1,x)$. Then $\psi_{\varepsilon}$ is a $C^{3+\beta}$ diffeomorphism. Setting $\Omega^{\varepsilon}\coloneqq\psi_{\varepsilon}(\Omega)$, we have $\Omega_{2\varepsilon}\subseteq\Omega^{\varepsilon}\subseteq\Omega_{\varepsilon}$. If we set $g_{\varepsilon}(\cdot,m)\coloneqq g(\cdot,m)\circ\psi_{\varepsilon}^{-1}$, we have the desired properties.

A 24**.**

There exist some $\delta>0$ and $C>0$ so that $H$ satisfies (8*) for all $p\in\mathbb{R}^{n}$ and all $(t,x)\in[0,T]\times\Gamma_{\delta}$. Without loss of generality, we may assume that $\delta<\varepsilon_{0}$, where $\varepsilon_{0}$ is the constant given in Definition 1.2.

Remark 5.1**.**

In the previous sections, we proved the well-posedness of MFGCs with Neumann boundary conditions in the case where $a=\nu I$. However, the arguments can be generalized easily enough to the case of variable coefficients $a_{ij}\in C^{1+\beta}(\overline{\Omega})$, provided they satisfy the uniform ellipticity constraint $a_{ij}(x)\xi_{i}\xi_{j}\geq\nu|\xi|^{2}$ for some constant $\nu>0$. On the other hand, in Assumption A20 we take the ellipticity constraint to degenerate as $x$ approaches the boundary of $\Omega$. The reason for this is as follows. Consider the approximating system without variable coefficients in the diffusion:

[TABLE]

By careful review of the proofs in the previous sections, we get a solution $(u_{\varepsilon},m_{\varepsilon},\mu_{\varepsilon})$ with uniform bounds on $D_{x}u_{\varepsilon}$ and hence on $D_{p}H(t,x,D_{x}u_{\varepsilon},\mu_{\varepsilon})$. As $\varepsilon\to 0$ this is incompatible with the invariance constraint (8*) (see [porretta2020mean, Remark 4.4]).

5.2. Example (Cournot)

To motivate the application of invariance constraints to MFGCs, we will consider a modified version of the Cournot mean field game system (see [camilli2025learning]) without a discount term, which we adjust to allow for degeneracy of the diffusion coefficient. Let $\Omega=(0,L)$ for some $L>0$. Now let $a(x)=\sigma(x)^{2}$ for some $\sigma\in C^{2}([0,L])$ such that

1. (1)

$\sigma(0)=\sigma(L)=0$ and $\sigma(x)\geq\nu(\varepsilon)>0$ in $(\varepsilon,L-\varepsilon)$ for every $\varepsilon>0$. 2. (2)

There exist $C_{\sigma},\delta>0$ so that $\sigma(x)\leq C_{\sigma}x$ in $(0,\delta)$ and $\sigma(x)\leq C_{\sigma}(L-x)$ in $(L-\delta,L)$.

Let $c\in C^{2}([0,L])$ be a function such that $c(x)>0$ for $x>0$ and $c(x)\leq x$ in $[0,\varepsilon]$ for some $\varepsilon>0$.

Consider the system

[TABLE]

where $P(t,\cdot)$ is a decreasing function for all $t\in[0,T]$.

Note that

[TABLE]

satisfies A13 for $\mu_{1},\mu_{2}\in\mathfrak{M}([0,L]\times[0,\infty))$, as

[TABLE]

where $\mu_{i}=(I,\alpha_{i})\#\rho_{i}$. Furthermore, the Hamiltonian

[TABLE]

satisfies

[TABLE]

and

[TABLE]

a.e. In Lemma 5.3 we will establish bounds on $D_{x}u$ and $\mu$ that are independent of the diffusion coefficients. Therefore, to establish the invariance condition (8*), it is enough to show that there is some constant $C>0$ such that $\sigma(x)^{2}\leq-xD_{p}H(t,x,p,\mu)+Cx^{2}$ in $(0,\delta)$ and $\sigma(x)^{2}\leq(L-x)D_{p}H(t,x,p,\mu)+C(L-x)^{2}$ in $(L-\delta,L)$ for all $p,\mu$ satisfying $|p|,|P|\leq M$, where $M$ is defined a priori and $P$ is defined in terms of $\mu$ as above. Indeed, on $(L-\delta,L)$ we just need $C\geq C_{\sigma}^{2}$, while in $(0,\delta)$ we observe that $-xD_{p}H(t,x,p,\mu)=-xc(x)\left(\frac{(p+P+\gamma)}{2}\right)_{+}\geq-x^{2}(M+\gamma/2)$, so it suffices to take $C=C_{\sigma}^{2}+M+\gamma/2$.

We note that the assumptions A10-14 and A24 are not technically satisfied (e.g. we need to restrict ourselves to non-negative controls and $p,\mu$ satisfying given a priori estimates). However, the well-posedness of this system follows by nearly identical arguments to those found in the following sections.

5.3. Existence of Solutions

In this section, we will prove the following existence result:

Theorem 5.2**.**

Assume A10-17 and A19-24 hold. Then there is a weak solution to (8).

Throughout this section, we will assume A10-17 and A19-24 hold. For $\varepsilon>0$, define $(u_{\varepsilon},m_{\varepsilon},\mu_{\varepsilon})$ to be the unique strong solution to

[TABLE]

We extend to $Q$ by setting $u_{\varepsilon}=m_{\varepsilon}=0$ on $Q\setminus[0,T]\times\Omega^{\varepsilon}$. Note that $m_{\varepsilon}\geq 0$ with $\int_{\Omega^{\varepsilon}}m_{\varepsilon}dx=\int_{\Omega^{\varepsilon}}m_{0}dx\leq 1$ for all $t\in[0,T]$. By A22 and A23, $f(t,x,m_{\varepsilon})$ and $g_{\varepsilon}(x,m_{\varepsilon})$ are uniformly bounded.

By arguments nearly identical to those used to prove Theorem 3.9, we get

[TABLE]

and

[TABLE]

Furthermore, adapting the argument in [porretta2020mean] for Lipschitz regularity, we can actually bound $C_{\varepsilon}$ uniformly in $\varepsilon$.

Lemma 5.3**.**

Assume A10-17 and A19-24 hold. Then there exists some constant $C>0$ independent of $\varepsilon$ such that

[TABLE]

Proof.

We will assume that $u_{\varepsilon}$ is smooth (otherwise, we can approximate $u_{\varepsilon}$ as in Lemma 3.13). As was done in [porretta2020mean], we will define $w_{\varepsilon}=|D_{x}u_{\varepsilon}|^{2}e^{d^{\gamma}}$ for some $\gamma\in(0,1)$. Then we have

[TABLE]

Assumption A22 and (34) give

[TABLE]

and assumption A21 gives

[TABLE]

Thus,

[TABLE]

where

[TABLE]

By A24, we get

[TABLE]

in $\overline{\Omega}\setminus\Omega_{\delta}$. Combining this with (34), we get that

[TABLE]

By [leonori2011gradient, Lemma 4], the maximum of $w_{\varepsilon}$ must not be attained on $[0,T]\times{\partial\Omega}_{\varepsilon}$. Furthermore, applying the maximum principle to (35), we get that the maximum must be attained on $\{T\}\times\Omega^{\varepsilon}$. Thus, A23 gives the desired estimate. ∎

With this, we are ready to prove our existence result.

Proof of Theorem 5.2.

By Lemma 5.3, we get uniform bounds for $H$ and each of its first-order derivatives, which in turn gives uniform bounds for $u_{\varepsilon}$ in $C^{\beta/2,1+\beta}([0,T]\times K)$ for each $K\subset\subset\Omega$. By the Arzela-Ascoli theorem and a diagonal argument, there is some $u\in C^{0,1}([0,T]\times\Omega)$ such that $u_{\varepsilon}\rightarrow u$ in $C^{0,1}([0,T]\times K)$ for every $K\subset\subset\Omega$.

Similarly, for every $\varepsilon_{0}>0$ and every $\varepsilon\in(0,\varepsilon_{0})$, $m_{\varepsilon}$ is uniformly bounded in $C^{\alpha/2,1+\alpha}([0,T]\times\overline{\Omega}_{\varepsilon_{0}})$, and hence we have a subsequence converging to some $m$ uniformly on $[0,T]\times\overline{\Omega}_{\varepsilon_{0}}$. By a diagonal argument, there is a subsequence (which we will still denote by $m_{\varepsilon}$) converging to some $m$ in $C^{0,1}([0,T]\times K)$ for $K\subset\subset\Omega$. Note that $m\geq 0$ and $\int_{\Omega}m(t,\cdot)dx\leq 1$ for all $t\in[0,T]$.

By nearly identical arguments to those used to prove [porretta2020mean, Proposition 4.3], we get that for each $t\in[0,T]$,

[TABLE]

and hence $m_{\varepsilon}(t,\cdot)\rightarrow m(t,\cdot)$ in $L^{1}(\Omega)$ (by Scheffé’s lemma). Thus, applying Lemma 3.7 gives that $\mu_{\varepsilon}(t)$ converges to

[TABLE]

for all $t\in[0,T]$. By Lemma 3.4, this gives that $D_{p}H(t,x,D_{x}u_{\varepsilon},\mu_{\varepsilon})\rightarrow D_{p}H(t,x,D_{x}u,\mu)$ pointwise. By [porretta2020mean, Proposition 4.3], $m_{\varepsilon}\rightarrow m$ in $C^{0}(0,T;L^{1}(\Omega))$ and $m$ is a weak solution to the FP equation. By A22 and A23, $f(t,x,m_{\varepsilon})\rightarrow f(t,x,m)$ a.e. and $g(x,m_{\varepsilon}(T))\rightarrow g(x,m(T))$ in $L^{p}(\Omega)$ for $p<\infty$. By [porretta2020mean, Proposition 3.4], $u$ is a weak solution to the HJ equation, thus completing the proof. ∎

Notice that as an immediate consequence of Lemma 5.3, we have a gradient estimate for the solutions constructed in the proof of Theorem 5.2. Moreover, by the uniqueness of solutions (see Section 5.4), this gives an a priori bound for the gradient of solutions.

5.4. Uniqueness of Solutions

Now that we have proven the existence of solutions, we shift our focus to uniqueness. The first half of our uniqueness proof will follow the argument used in [porretta2020mean]. However, for the second half, we apply the strategy used in [kobeissi2022mean] to adapt the argument to the MFGC setting.

Theorem 5.4**.**

Assume A10-17 and A19-24 hold. Then there is at most one solution to (8).

Proof.

Let $(u,m,\mu)$ and $(v,\rho,\nu)$ be solutions to (8). Defining $\Omega^{\varepsilon}$ as before, let $(u_{\varepsilon},m_{\varepsilon},\mu_{\varepsilon})$ and $(v_{\varepsilon},\rho_{\varepsilon},\nu_{\varepsilon})$ be solutions to the approximating systems

[TABLE]

and

[TABLE]

respectively. As in section 5.3, for each $K\subset\subset\Omega$, we have uniform estimates for $u_{\varepsilon}$ and $v_{\varepsilon}$ in $C^{\beta/2,1+\beta}([0,T]\times K)$, which implies that every subsequence has a subsequence converging pointwise in $C^{0,1}([0,T]\times\Omega)$. Applying Lemmas 3.7 and 3.4 give convergence of $H(t,x,D_{x}u_{\varphi(\varepsilon)},\mu_{\varphi(\varepsilon)})$, $H(t,x,D_{x}v_{\varphi(\varepsilon)},\nu_{\varphi(\varepsilon)})$, $D_{p}H(t,x,D_{x}u_{\varphi(\varepsilon)},\mu_{\varphi(\varepsilon)})$, and $D_{p}H(t,x,D_{x}v_{\varphi(\varepsilon)},\nu_{\varphi(\varepsilon)})$. By [porretta2020mean, Corollary 3.9], it follows that the limits of the subsequences $u_{\varphi(\varepsilon)},v_{\varphi(\varepsilon)}$ (and hence the sequences themselves) are $u$ and $v$, respectively. Similarly, since we have $D_{p}H(t,x,D_{x}u_{\varepsilon},\mu_{\varepsilon})\rightarrow D_{p}H(t,x,D_{x}u,\mu)$ and $D_{p}H(t,x,D_{x}v_{\varepsilon},\nu_{\varepsilon})\rightarrow D_{p}H(t,x,D_{x}v,\nu)$ pointwise, [porretta2020mean, Proposition 4.3] gives that $m_{\varepsilon}\rightarrow m$ and $\rho_{\varepsilon}\rightarrow\rho$ in $C^{0}(0,T;L^{1}(\Omega))$.

By arguments of the same spirit as those in [kobeissi2022mean], we get

[TABLE]

By A22-23 and the fact that $(m_{\varepsilon}-\rho_{\varepsilon})\rightarrow(m-\rho)$ in $C^{0}(0,T;L^{1}(\Omega))$, it follows that

[TABLE]

In particular, these integrals must vanish. Recall that since $L$ is strictly convex,

[TABLE]

if and only if $\alpha_{1}=\alpha_{2}$. Hence, $\alpha^{\mu}=\alpha^{\nu}$. By uniqueness of solutions the FP equation, $m=\rho$. By Lemma 3.5, this implies $\mu=\nu$. Finally, by uniqueness of solutions to the HJ equation, $u=v$. ∎

5.5. Regularity of Solutions

In this section, we investigate the regularity of the solutions to our system. First, we state results on the semiconcavity of $u$ and the boundedness of $m$ in $L^{\infty}(Q)$, which are proved using identical arguments to those used for the analogous results in [porretta2020mean].

Theorem 5.5**.**

Assume A10-17 and A19-24 hold. Now suppose $\sigma\in W^{2,\infty}(\Omega)$ and

[TABLE]

for all $t\in[0,T]$ and $m\in L^{1}(\Omega)$. Finally, assume that for $\nu\in\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$, there exist constants $C_{1},C_{2}$, depending only on $\Lambda_{q^{\prime}}(\nu)$, such that

[TABLE]

for any $x,y,z\in\Omega$, $p,q\in\mathbb{R}^{n}$, and $t\in(0,T)$. If $(u,m,\mu)$ is a solution to (8), then $u(t,\cdot)$ is semiconcave for all $t\in(0,T)$, with a semiconcavity constant bounded uniformly in $t$. In particular,

[TABLE]

for all $t\in(0,T)$.

Theorem 5.6**.**

Assume the conditions of Theorem 5.5 hold. Now suppose that there exists $\delta_{0}>0$ such that

[TABLE]

for all $x\in\Gamma_{\delta_{0}}$ and $p\in\mathbb{R}^{n}$. Then if $(u,m,\mu)$ is a solution to (8), we have $m\in L^{\infty}(Q)$ with

[TABLE]

Finally, we prove that under a few additional assumptions, the value function $u$ will satisfy the HJ equation in a classical sense on the interior of our domain.

Theorem 5.7**.**

Assume A10-17 and A19-24 hold. Now assume that for $\rho\in L^{1}(\Omega)$ with $\|\rho\|_{L^{1}(\Omega)}\leq 1$, we have $\|g_{\varepsilon}(\cdot,m)\|_{C^{2+\beta}(\Omega)}\leq C_{0}$. Finally, assume there exist $L_{1}\in(0,1)$ and $\xi_{1}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{2}\rightarrow[0,\infty)$ continuous such that for $R>0$, $p\in\mathbb{R}^{n}$, and $\mu_{1},\mu_{2}\in\mathfrak{M}_{\infty,R}(\Omega\times\mathbb{R}^{n})$ we have:

• •

If $\mu_{i}=(I,\alpha^{\mu_{i}})\#\rho$, then

[TABLE]

• •

If $\mu_{i}=(I,\alpha)\#\rho_{i}$, then

[TABLE]

Then the unique weak solution $(u,m,\mu)$ to (8) is actually a classical solution (in the sense that $u$ is a classical solution to the HJ equation in $[0,T]\times\Omega$).

Proof.

As in the proof of Theorem 5.2, let $(u_{\varepsilon},m_{\varepsilon},\mu_{\varepsilon})$ be a sequence of solutions to (32) converging to $(u,m,\mu)$. However, in this case, the $(u_{\varepsilon},m_{\varepsilon},\mu_{\varepsilon})$ are classical solutions on $[0,T]\times\overline{\Omega}_{\varepsilon}$. Combining the Lipschitz estimate from Lemma 5.3 with the arguments from Sections 2.5 and 3.5, for each $K\subset\subset\Omega$, we get bounds for $u_{\varepsilon}$ in $C^{1+\alpha/2,2+\alpha}([0,T]\times K)$ uniformly in $\varepsilon$. By the Arzela-Ascoli theorem and a diagonal argument, we get a subsequence that converges in $C^{1,2}([0,T]\times K)$ for all $K\subset\subset\Omega$, and hence $u_{\varepsilon}$ and its derivatives converge pointwise in $[0,T]\times\Omega$. By the uniqueness of the limit, we get that $u\in C^{1,2}([0,T]\times K)$ for all $K\subset\subset\Omega$. Furthermore, by A23 and Lemma 3.4, passing to the limit gives that $u$ satisfies the HJ equation on $[0,T]\times\Omega$. ∎