Regularity Theory for Mean-Field Game Systems

123

S P R I N G E R B R I E F S I N M AT H E M AT I C S

Diogo A. GomesEdgard A. PimentelVardan Voskanyan

Regularity Theory for Mean-Field Game Systems

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Diogo A. Gomes • Edgard A. PimentelVardan Voskanyan

Regularity Theoryfor Mean-Field GameSystems

123

Diogo A. GomesCEMSE DivisionKing Abdullah University

of Science and TechnologyThuwal, Saudi Arabia

Vardan VoskanyanCEMSE DivisionKing Abdullah University

of Science and TechnologyThuwal, Saudi Arabia

Edgard A. PimentelDepartment of MathematicsUniversidade Federal de Sao CarlosSão Carlos, Brazil

ISSN 2191-8198 ISSN 2191-8201 (electronic)SpringerBriefs in MathematicsISBN 978-3-319-38932-5 ISBN 978-3-319-38934-9 (eBook)DOI 10.1007/978-3-319-38934-9

Library of Congress Control Number: 2016943315

Mathematics Subject Classification (2010): 35J47, 35A01

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Keywords Mean-field game • Hamilton-Jacobi equation • Fokker-Planck equa-tion • Transport equation • Regularity theory • Mathematical methods for partialdifferential equations

Preface

This book brings together several recent developments on the regularity theoryfor mean-field game systems. We detail several classes of methods and present aconcise overview of the main techniques developed in the last few years. Most ofthe forthcoming material deals with simple and computation-friendly examples; thisis intended to unveil the main ideas behind the methods rather than focus on thetechnicalities of particular cases.

The choice of topics presented here reflects the authors’ perspective on this fast-growing field of research; it is by no means exhaustive or intended as a completeaccount of the theory. Rather—and in the best scenario—it serves as an introductionto the material available in scientific papers.

Book Outline

Mean-field games comprise a wide range of models with distinct properties.Accordingly, no single method addresses existence or regularity issues in all cases.In a restricted number of problems, existence questions on MFGs can be settledthrough explicit solutions or special transformations. Some of these explicit methodsare presented in Chap. 2. Explicit solutions are also essential for the continuationarguments in Chap. 11.

When explicit solutions cannot be found, fixed-point methods, regularizationtechniques, and continuation arguments provide systematic tools to study theexistence of solutions. Usually, a priori bounds are a key ingredient in existenceproofs. These bounds are estimates for the size of solutions that are derived beforethe solution is known to exist. Then, it is often possible to show the existence ofthe solution. Unless otherwise stated, we work with classical (i.e., C1 or at leastregular-enough solutions).

We begin our study of a priori bounds for MFGs in Chap. 3, where we examinethe Hamilton–Jacobi equation. There, some of the estimates rely only on the

vii

viii Preface

optimal control interpretation (see Sect. 3.2) or parabolic regularization effects (seeSect. 3.5). In contrast, other results (see Sect. 3.3 or 3.4) illustrate a subtle interplaybetween these two mechanisms.

In Chap. 4, we consider transport and Fokker–Planck equations. Both equationspreserve mass and positivity. However, the Fokker–Planck equation enjoys strongregularizing properties that we investigate in detail. The chapter ends with a briefdiscussion of relative entropy inequalities and weak solutions.

A recent development in the theory of solutions of Hamilton–Jacobi equationsis the nonlinear adjoint method introduced by L.C. Evans. This method relies oncoupling a Hamilton–Jacobi equation with a Fokker–Planck equation. This systemresembles (1.1) with F D 0. In Chap. 5, we develop the main techniques of thismethod. The nonlinear adjoint method gives bounds for Hamilton–Jacobi equationsthat go beyond maximum principle methods. These bounds are obtained by carefulintegration techniques. In addition to bounds relevant to MFGs, to illustrate themethod, we prove semiconcavity estimates and consider the vanishing viscosityproblem.

Next, in Chap. 6, we develop techniques that are specific to mean-field gamesand that combine both equations. These bounds together with the estimates for theHamilton–Jacobi equation or the Fokker–Planck equation improve earlier results.

Chapter 7 is devoted to stationary models. There, we develop a priori estimatesfor three different problems. First, we consider MFGs with polynomial dependenceon m. To get Sobolev regularity, we combine the integral Bernstein estimate inChap. 3 with the first-order estimates in Chap. 6. Next, we investigate two MFGswith singularities: the congestion problem and the logarithmic nonlinearity.

In Chaps. 8 and 9, we explore time-dependent MFGs. In the first of these twochapters, we consider models without singularities and illustrate two regularityregimes. The first regime corresponds to subquadratic Hamiltonians. In this case,the main tool is the Gagliardo–Nirenberg estimate discussed in Chap. 3. The secondregime corresponds to quadratic and superquadratic Hamiltonians. For these, weget the regularity using the nonlinear adjoint method from Chap. 5. Time-dependentMFGs with singularities present substantial challenges and are examined in Chap. 9.There, we investigate logarithmic nonlinearities in the subquadratic setting and theshort-time congestion problem.

Chapters 10 and 11 examine MFGs in the nonlocal and local cases, respectively.We use fixed-point methods to get the existence of solutions for nonlocal problemsin both first-order and second-order cases. Besides their independent interest,nonlocal MFGs are used later to study local problems through a regularizationprocedure. Next, in Chap. 11, we present two techniques to prove the existence ofsolutions to MFGs. First, we discuss the regularization method. Then, we examinecontinuation arguments for both stationary and time-dependent problems.

Preface ix

Thanks

This book benefited immensely from the input from various colleagues and students.In particular, the authors are especially grateful to David Evangelista, GustavoMadeira, José Ruidival dos Santos, Levon Nurbekyan, Lucas Fabiano Lima, MarcSedjro, Mariana Prazeres, Renato Moura, Roberto Velho, and Teruo Tada, who readparts of the original manuscript and gave invaluable feedback. D. Gomes and V.Voskanyan were supported by KAUST baseline and start-up funds and KAUST SRI,Center for Uncertainty Quantification in Computational Science and Engineering.E. Pimentel was supported by FAPESP (Grant 2015/13011-6) and baseline funds ofUFSCar Graduate Program in Mathematics (PPGM-DM-UFSCar).

Bibliographical Notes

Mean-field games were introduced independently and around the same time in theengineering community in [142, 143] and in the mathematics community in [164–167]. Many mathematical aspects of the theory were developed in [174], a coursetaught by Lions, and several of the techniques in this book can be traced to ideasoutlined there.

Before the introduction of MFGs, systems combining a Hamilton–Jacobi equa-tion with a Fokker–Planck or transport equation that resemble MFGs were con-sidered in various settings. For example, the PDE approach to the Aubry–Mathertheory [93–95], the problems in [89, 90], and the Benamou–Brenier formulation ofthe optimal transport problem [29] are forerunners of MFGs. The entropy-penalizedscheme in [122] can be reinterpreted as a discrete-time mean-field game.

The goal of this book is to develop the regularity theory for MFGs. Theseproblems have been investigated intensively in the last few years, and we givedetailed references at the end of each chapter on the different models and problems.Due to space and time constraints, we cannot discuss the numerous applications ofMFGs in engineering and in economics and the many recent results on stochasticmethods, numerical analysis, and other MFG models. To make up for theseomissions, next, we give a brief bibliography and refer the reader to the booksand surveys [30, 61, 121, 133] for more material and references. Also, here, wedo not develop the theory of weak solutions to MFGs and instead refer the readerto the following papers [62–64, 68, 98, 195, 196]. Furthermore, we do not discussnumerical methods for MFGs here; for that, see, for example, [1–4, 54, 70, 138–140, 160].

In the engineering community, emerging research includes power grids andenergy management [14, 14, 148–150, 179], adaptive control [147, 184] and risk-sensitive or robust control [85, 86, 208, 210], robust MFGs [26, 209], learning [214],and networks [141], among several others [144, 206, 207]. Traffic and crowd models

x Preface

are an important and natural area of application of MFGs [42, 44, 45, 76, 83], as wellas related problems on networks and graphs [27, 56, 57].

Some of the first MFG models were motivated by economic growth [167–169].Subsequently, various problems in economics and finance have been consideredin the literature, including socioeconomic models [37, 145, 185], inspection andcorruption [153, 156] systemic risk [103], price formation [41, 46, 48, 49, 178],social dynamics [24], consensus [39, 185, 186, 198], and opinion dynamics [23, 36,201, 202]. In the context of heterogeneous agent models (see [159]) with rationalexpectations (see [175]), MFGs became a popular modeling tool [5, 6, 176, 187].An earlier model that predates the emergence of MFGs is the Aiyagari–Bewley–Huggett model [7, 38, 146]. A recent book [133] describes several MFG models inmathematical economics.

MFGs where the agents are subjected to correlated random forces were studiedby stochastic methods in [71, 72, 74, 75, 154, 161–163]. The master equation wasused to study problems with correlations in [33, 73] and deterministic problems in[105, 124]. An important tool in the study of the N player limit with or withoutcorrelations is the theory of nonlinear Markov processes [152]. Some applicationsof these methods were developed in [155, 157, 158]. Finally, minimax methods wereconsidered in [11–13].

Several authors considered extensions of the original MFG framework. Theseinclude finite state mean-field games [21, 99, 113, 126, 128, 131, 132], problemswith major and minor agents [183], multi-population models [77, 79, 97], extendedMFGs [124, 129, 213], logistic population effects [120], problems with densityconstraints [180, 199], and obstacle-type problems [116].

Thuwal, Saudi Arabia Diogo A. GomesSão Carlos, Brazil Edgard A. PimentelThuwal, Saudi Arabia Vardan Voskanyan

Acknowledgments

D.A. Gomes was partially supported by KAUST baseline and start-up fundsand KAUST SRI, Uncertainty Quantification Center in Computational Scienceand Engineering. E.A. Pimentel was partially supported by FAPESP (Grant #2015/13011-6) and by baseline funds from UFSCar Graduate Program in Mathe-matics.

xi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Derivation of MFG Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Explicit Solutions, Special Transformations, and Further Examples . 92.1 Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 The Hopf–Cole Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Gaussian-Quadratic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Interface Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Estimates for the Hamilton–Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Control Theory Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Integral Bernstein Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Integral Estimates for HJ Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5 Gagliardo–Nirenberg Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Estimates for the Transport and Fokker–Planck Equations . . . . . . . . . . . 394.1 Mass Conservation and Positivity of Solutions . . . . . . . . . . . . . . . . . . . . . 394.2 Regularizing Effects of the Fokker–Planck Equation. . . . . . . . . . . . . . . 414.3 Fokker–Planck Equation with Singular Initial Conditions . . . . . . . . . 434.4 Iterative Estimates for the Fokker–Planck Equation . . . . . . . . . . . . . . . . 444.5 Relative Entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.7 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 The Nonlinear Adjoint Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1 Representation of Solutions and Lipschitz Bounds . . . . . . . . . . . . . . . . . 645.2 Conserved Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3 The Vanishing Viscosity Convergence Rate . . . . . . . . . . . . . . . . . . . . . . . . 665.4 Semiconcavity Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.5 Lipschitz Regularity for the Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . 70

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5.6 Irregular Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.7 The Hopf–Cole Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.8 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6 Estimates for MFGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.1 Maximum Principle Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2 First-Order Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.3 Additional Estimates for Solutions of the Fokker–Plank Equation 816.4 Second-Order Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.5 Some Consequences of Second-Order Estimates . . . . . . . . . . . . . . . . . . . 846.6 The Evans Method for the Evans–Aronsson Problem . . . . . . . . . . . . . . 856.7 An Energy Conservation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.8 Porreta’s Cross Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.9 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7 A Priori Bounds for Stationary Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.1 The Bernstein Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.2 A MFG with Congestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987.3 Logarithmic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

8 A Priori Bounds for Time-Dependent Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.1 Subquadratic Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068.2 Quadratic Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.3 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

9 A Priori Bounds for Models with Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.1 Logarithmic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.2 Congestion Models: Local Existence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159.3 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

10 Non-local Mean-Field Games: Existence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12510.1 First-Order, Non-local Mean-Field Games. . . . . . . . . . . . . . . . . . . . . . . . . . 12510.2 Second-Order, Non-local Mean-Field Games . . . . . . . . . . . . . . . . . . . . . . 12810.3 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

11 Local Mean-Field Games: Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13111.1 Bootstrapping Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13111.2 Regularization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13311.3 Continuation Method: Stationary Problems. . . . . . . . . . . . . . . . . . . . . . . . . 13411.4 Continuation Method: Time-Dependent Problems . . . . . . . . . . . . . . . . . 13811.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Chapter 1Introduction

Lasry and Lions and, more or less simultaneously, Caines, Huang and Malhaméintroduced a class of models called mean-field games (MFGs) to study systemswith large numbers of identical agents in competition. In these games, the agents arerational and seek to optimize a value function by selecting appropriate controls. Theinteractions between them are determined by a mean-field coupling that aggregatestheir individual contributions. Many of these models are formed by a Hamilton–Jacobi equation coupled with a Fokker–Planck equation.

Hamilton–Jacobi and Fokker–Planck equations have been the subject of exten-sive research. Yet, in MFGs, the coupling between these two equations leads tonon-trivial existence, regularity, and uniqueness questions. Here, we focus on theregularity theory for MFGs. For pedagogical reasons, we illustrate our methods withelementary examples. These include the two systems of partial differential equations(PDEs) described next and closely related examples.

We consider a large population of agents. The state of each of them is given bya point x 2 R

d or, in the periodic setting, by a point x 2 Td, where T

d D Rd=Zd is

the standard d-dimensional torus. We denote by P.Rd/ (or P.Td/) the set of Borelprobability measures on R

d (resp. Td). The statistical distribution of the agents isdescribed by a probability measure, m 2 P.Rd/ (or P.Td/). Each agent has spatialpreferences that are determined by a C1 function, V W Rd ! R (or V W Td ! R).Next, we fix a real-valued function, F; that encodes the interactions between eachagent and the mean field. The domain of F is either Rd � R

C (resp. Td � RC), the

local case, or Rd � P.Rd/ (resp. Td � P.Td/), the non-local case. In the presentdiscussion, we consider the non-local case. We assume that F is continuous (withrespect to the weak convergence in P.Rd/ or P.Td/).

Let � � 0. The workhorse of MFG theory is the following system:(�ut C jDuj2

2C V.x/ D � �u C F.x;m/;

mt � div.mDu/ D � �m;(1.1)

© Springer International Publishing Switzerland 2016D.A. Gomes et al., Regularity Theory for Mean-Field Game Systems,SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-38934-9_1

1

2 1 Introduction

with initial and terminal conditions(u.x;T/ D uT.x/

m.x; 0/ D m0.x/:(1.2)

Here, m0 and uT are given functions, m0 � 0 withRRd m0dx D 1. Our main goal is

to show the existence of solutions, u;m W Rd � Œ0;T� ! R; with m � 0. Becausem is a probability density at t D 0, it remains a probability for all positive times(see Sect. 4.1). To avoid technical difficulties, it is common to work with periodicboundary conditions. In this case, the domain of u and m is T

d � Œ0;T�. In moregeneral problems, the terminal condition, uT ; may depend upon m.�;T/, i.e., it hasthe form uT.x;m.�;T//.

The corresponding stationary MFG is given by( jDuj22

C V.x/ D � �u C F.x;m/C H;

� div.mDu/ D ��m;(1.3)

and the solution is a triplet .u;m;H/. In the periodic case, u;m W Td ! R and

H 2 R. We require m � 0 andRTd mdx D 1.

In (1.1) and (1.3), the equations are coupled through the vector field, Du, inthe Fokker–Planck equation, and the term F in the Hamilton–Jacobi equation. For� D 0, (1.1) and (1.3) are called first-order or deterministic MFGs. Otherwise, if� > 0, (1.1) and (1.3) determine, respectively, second-order parabolic and ellipticMFGs.

In the rest of this chapter, we present a brief derivation of deterministic MFGsand examine uniqueness.

1.1 Derivation of MFG Models

Here, we present a heuristic derivation of a time-dependent deterministic mean-fieldgame that corresponds to � D 0 in (1.1). The case � > 0 is handled in a similar wayusing stochastic control methods.

1.1.1 Optimal Control and Hamilton–Jacobi Equations

We begin by examining the terminal value deterministic optimal control problem.We fix T > 0 and consider an agent whose state is x.t/ 2 R

d for 0 � t � T .Let W D L1.Œt;T�;Rd/. Agents can change their state by choosing a control in W .For each control v 2 W , the state evolves according to

Px.t/ D v.t/: (1.4)

1.1 Derivation of MFG Models 3

We fix a Lagrangian QL W Rd � Rd � Œ0;T� ! R, with v 7! L.x; v; t/ uniformly

convex. For example,

QL.x; v; t/ D jvj22

� V.x/C QF.x; t/; (1.5)

with QF W Rd � Œ0;T� ! R a continuous function bounded from below. Next, we

choose a bounded continuous function, uT W Rd ! R; called the terminal cost.Agents have preferences that are encoded in the action functional,

J.vI x; t/ DZ T

t

QL.x.s/; v.s/; s/ds C uT.x.T//;

where x solves (1.4) with the initial condition x.t/ D x. Each agent seeks tominimize J among all possible controls in W . The infimum over all controls,

u.x; t/ D infv2W J.vI x; t/; (1.6)

is called the value function.Recall that the Legendre transform, QH; of QL is given by

QH.x; p; t/ D supv2Rd

��p � v � QL.x; v; t/� : (1.7)

The function QH is called the Hamiltonian. By the uniform convexity of QL in thesecond coordinate, the maximum in the previous inequality is achieved at a uniquepoint, v�. For each .x; t/, v� is determined by

v� D �Dp QH.x; p; t/: (1.8)

If QL is given by (1.5), then

QH.x; p; t/ D jpj22

C V.x/ � QF.x; t/:

A classical result in control theory states that if u 2 C1.Rd �Œt0;T�/, then u solvesthe Hamilton–Jacobi equation,

� ut.x; t/C QH.x;Dxu.x; t/; t/ D 0: (1.9)

Further, as we prove next, the optimal control, v�.t/; is determined in feedbackform by

v�.t/ D �Dp QH.x�.t/;Dxu.x�.t/; t/; t/: (1.10)

4 1 Introduction

In general, the value function is not differentiable. However, it solves (1.9) in aweaker sense—as a viscosity solution. Here, we do not develop this theory. Instead,we show the converse of that statement, namely: if Qu solves (1.9) and satisfies theterminal condition

Qu.x;T/ D uT.x/; (1.11)

then Qu is the value function in (1.6).

Theorem 1.1 (Verification Theorem). Let Qu 2 C1.Rd � Œt0;T�/ solve (1.9) withthe terminal condition (1.11). Let

v�.t/ D �Dp QH.x�.t/;Dx Qu.x�.t/; t/; t/ (1.12)

and x�.t/ solve (1.4). Then, v�.t/ is an optimal control for (1.6) and Qu.x; t/ D u.x; t/,where u is the value function in (1.6).

Proof. First, we observe that for any v.s/ and any trajectory x solving (1.4), we have

Qu.x.T/;T/ DZ T

t.Dx Qu.x.s/; s/ � v.s/C Qus.x.s/; s// ds C Qu.x.t/; t/: (1.13)

Because of (1.7), we have

.Dx Qu.x.s/; s/ � v.s/ � � QH.x;Dx Qu.x.s/; s/ � QL.x.s/; v.s//:Furthermore, the previous inequality is an identity for v D v� due to (1.12). Bycombining (1.13) with (1.7), we get

Qu.x; t/ �Z T

t

QL.x.s/; v.s/; s/ds C uT.x.T//: (1.14)

Finally, from (1.8), we conclude that the previous inequality is an identityif v D v�. ut

1.1.2 Transport Equation

Let b W Rd � Œ0;T� ! Rd be a Lipschitz vector field. Consider a population of agents

with dynamics given by (Px.t/ D b.x.t/; t/ t > 0;

x.0/ D x:(1.15)

The previous equation induces a flow, ˆt, in Rd that maps the initial condition,

x 2 Rd, at t D 0 to the solution of (1.15) at time t > 0.


Fix a probability measure, m0 2 P.Rd/. For 0 � t � T , let m.�; t/ be the push-forward by ˆt of m0, sometimes denoted by ˆt]m0; given by

ZRd�.x/m.x; t/dx D

ZRd��ˆt.x/

�m0dx: (1.16)

For 0 � t � T , m.�; t/ is a probability measure. Next, we derive a partial differentialequation for m.

Proposition 1.2. Let m be determined by (1.16) for some probability measurem0 2 P.Rd/. Assume that b.x; t/ is Lipschitz continuous in x. Let ˆt be the flowcorresponding to (1.15). Then, m 2 C.RC

0 ;P.Rd// and

(mt.x; t/ C div.b.x; t/m.x; t// D 0; .x; t/ 2 R

d � Œ0;T�;m.x; 0/ D m0.x/; x 2 R

d;(1.17)

in the distributional sense.

Proof. We recall that � solves (1.17) in the distributional sense if

�Z T

0

ZRd�.x; t/ .�t.x; t/C b.x; t/�x.x; t// dxdt D

ZRd�0.x/�.x; 0/dx;

for every � 2 C1c .R

d � Œ0;T//. Differentiating both sides of (1.16) with respect tot gives

ZRd�.x; t/mt.x; t/dx D

ZRd

�b.ˆt.x/; t/Dx�

�ˆt.x/; t

��m0.x/dx:

Therefore, ZRd�.x; t/mt.x; t/dx D

ZRd.b.x; t/Dx�.x; t//m.x; t/dx;

using the definition of ˆt. To conclude the proof, we integrate the previous identityby parts. ut

1.1.3 Mean-Field Models

The mean-field game framework was developed to study systems with an infinitenumber of rational agents in competition. In these systems, each agent seeks tooptimize an individual control problem that depends on statistical information aboutthe whole population. Further, the only information available to the agents is theprobability distribution of the agents’ states.

6 1 Introduction

Here, the interaction between the mean field and each agent is determined by therunning cost. First, we assume that for each time t, m.x; t/ is a probability density inR

d that gives the distribution of the agents in the different states. Next, we set

QL.x; v; t/ D L.x; v;m.�; t//:

The Lagrangian, L; is a real-valued map, L W Rd � R

d � RC ! R or

L W Rd � Rd � P.Rd/ ! R. The former case is called the local case and the latter

the non-local case. In the local case, we interpret L.x; v;m.�; t// as L.x; v;m.x; t//.We denote the Legendre transform of L by H. Finally, we suppose that each agentseeks to minimize the control problem (1.6). As a result, the value function, u; of arepresentative agent is determined by

�ut C H.x;Dxu;m/ D 0:

According to Theorem 1.1, if the previous equation has a solution, u, the vectorfield, b D �DpH.x;Dxu.x; t/;m/; gives an optimal strategy. Because all agents arerational, they use this strategy. u and m are thus determined by

(�ut C H.x;Dxu;m/ D 0

mt � div.DpHm/ D 0:(1.18)

In addition, if uT W Rd ! R is the terminal value function for the agents and theirinitial distribution is m0 W Rd ! R

C0 with

RRd m0 D 1, we supplement (1.18) with

the initial-terminal conditions (1.2).

1.1.4 Extensions and Additional Problems

In some applications, deterministic models are unsuitable due to random perturba-tions. In these cases, stochastic optimal control replaces deterministic control. Forthe stochastic case, the MFG is given by a second-order, nonlinear parabolic system,

(�ut C H.x;Dxu;D2u;m/ D 0

mt � div.DpHm/ �Pij @ij.DMij Hm/ D 0;

(1.19)

coupled with the initial-terminal conditions (1.2). The system (1.1) is a particularinstance of (1.19). The general theory for systems like (1.19) is not yet completelyunderstood, especially if the dependence on second-order derivatives is nonlinear.However, many important cases can be studied rigorously as we will see here.


In some applications, the initial-terminal problem is replaced by the planningproblem: given two probability measures, m0 and mT ; we look for a pair .u;m/solving (1.19) under the boundary conditions

m.x; 0/ D m0; m.x;T/ D mT : (1.20)

In this book, we consider (1.19) in the whole space, Rd, or with periodic

boundary conditions. In this last case, we regard u and m as real-valued functionswith domain T

d � Œ0;T�. Often, our methods extend in a straightforward way toDirichlet and Neumann boundary conditions.

In the periodic stationary case, the equation H.x;Dxu;D2u;m/ D 0 may fail tohave solutions, or these may not be probability measures. Therefore, we introducea constant, H; that represents a long-time average running cost. This constant isknown as the effective Hamiltonian in the theory of homogenization. The stationaryversion of (1.19) becomes

(H.x;Dxu;D2u;m/ D H

� div.DpHm/ �Pij @ij.DMij Hm/ D 0;

(1.21)

where the solution is a triplet, .u;m;H/, with u;m W Td ! R, m � 0, and H 2 R.The constant H is chosen so that m is a probability measure.

1.1.5 Uniqueness

We continue our study of (1.1)–(1.2) by examining the uniqueness of solutions. Thefundamental element for the uniqueness is a monotonicity condition introduced byLasry and Lions. Here, we consider the non-local case, F W Rd � P.Rd/ ! R, asthe local case is analogous. We say that (1.1) satisfies the Lasry–Lions monotonicitycondition if Z

RdŒF.x;m1/ � F.x;m2/� .m1 � m2/dx > 0; (1.22)

for every m1; m2 2 P.Rd/ with m1 ¤ m2.The next theorem gives the uniqueness of classical solutions to (1.1)–(1.2).

Theorem 1.3 (Uniqueness of Classical Solutions). Assume that the mean-fieldcoupling, F; satisfies the Lasry–Lions monotonicity condition. Then, there existsat most one classical solution, .u;m/; for (1.1)–(1.2).

Proof. We argue by contradiction. Let .u1;m1/ and .u2;m2/ solve (1.1)–(1.2).Define

Qu WD u1 � u2

8 1 Introduction

and

Qm WD m1 � m2:

First, we subtract the equations for .u1;m1/ and .u2;m2/ to get

(�Qut C jDu1j2

2� jDu2j2

2D � �Qu C F.x;m1/ � F.x;m2/

Qmt � div.m1Du1/C div.m2Du2/ D � � Qm:

Next, we multiply the first equation by Qm and the second one by Qu. Thereafter, wesubtract them and integrate by parts. This leads to the identity

d

dt

ZTd

Qu Qm DZTd

Qm2

�jDu1j2 � jDu2j2

��ZTd

DQu .m1Du1 � m2Du2/ (1.23)

�ZTd.F.x;m1/ � F.x;m2// Qm:

Notice that

Qm2

�jDu1j2 � jDu2j2

�� DQu .m1Du1 � m2Du2/ D �m1 C m2

2jDu1 � Du2j2 :

(1.24)

Integrating (1.23) over Œ0;T�, using (1.2), (1.24) and the Lasry–Lions monotonicitycondition, we obtainZ

Td.F.x;m1/ � F.x;m2// .m1 � m2/C

ZRd

m1 C m2

2jDu1 � Du2j2 � 0:

Hence, m1 D m2. Finally, the uniqueness of solutions for (1.1) gives u1 D u2. ut

Chapter 2Explicit Solutions, Special Transformations,and Further Examples

Few mean-field games can be solved explicitly. However, examples for which closedsolutions are known illustrate essential features of the theory. Moreover, explicitsolutions to MFGs are a key ingredient in the continuation method discussed inChap. 11.

2.1 Explicit Solutions

We begin our study of explicit solutions by considering a first-order quadraticMFG with a logarithmic nonlinearity. While logarithmic nonlinearities pose severaltechnical challenges (see Chaps. 7 and 9), the model considered here can be solvedby elementary methods. This game is given by

( juxj22

C V.x/C b.x/ux D ln m C H;

�.m.Du C b.x///x D 0;(2.1)

with u;m W T ! R, m � 0, ZT

mdx D 1;

and, for definiteness, ZT

u dx D 0: (2.2)


9

10 2 Explicit Solutions, Special Transformations, and Further Examples

Moreover, we suppose that ZT

b.y/dy D 0:

If Du C b D 0, the second equation in (2.1) holds immediately. This suggests thatwe set

u.x/ D �Z x

0

b.y/ dy CZT

Z z

0

b.y/ dy dz:

Using the previous formula in the first equation, we get

m.x/ D eV.x/� b2.x/2R

TeV.y/� b2.y/

2 dy:

In particular, let W Td ! R be a periodic C1 function withRT dx D 0. Suppose

that b.x/ D x.x/. Then,

u.x/ D � .x/; m.x/ D eV.x/� 2x .x/2R

TeV.y/� 2x .y/

2 dy; and H D ln

�ZT

eV.y/� 2x .y/2 dy

solves (2.1).A related problem is the congestion model:(

u2x2m1=2

C V.x/ D ln m C H;

�.m1=2ux/x D 0:(2.3)

It is easy to see that u.x/ D 0, m.x/ D eV.x/RT

eV.y/ dyand H D ln

RT

eV.x/dx solve (2.3).

2.2 The Hopf–Cole Transform

The Hopf–Cole transform is a well-known technique to convert certain nonlinearequations into linear equations. Here, we illustrate an application to MFGs. ForP 2 R

d, consider the system(��u C 1

2jP C Duj2 C V.x/ D ln m

��m � div..P C Du/m/ D 0:(2.4)

Define m by the Hopf–Cole transform

m D ev�u2 ; (2.5)

2.3 Gaussian-Quadratic Solutions 11

where u and v solve (��u C 1

2jP C Duj2 C V.x/ D v�u

2

�v C 12jP C Dvj2 C V.x/ D v�u

2:

(2.6)

By a direct computation, the function, m; given by (2.5) solves

��m � div..P C Du/m/ D 0: (2.7)

To check this, it is enough to observe that

��m D m

�1

2�u � 1

2�v � jDu � Dvj2

4

D m�u � m

4

�jP C Duj2 � jP C Dvj2 C jDu � Dvj2�D .P C Du/ � Dm C m�u D div..P C Du/m/:

2.3 Gaussian-Quadratic Solutions

Gaussian-quadratic solutions to MFGs are relevant in several applications. Indimension d � 1, we consider the MFG in R

d given by

(��u C 1

2jDuj2 C ˇjxj2 D ln m C H

��m � div.mDu/ D 0:(2.8)

We set m D �e�u so that the second equation holds trivially. Next, we select

u D ˛jxj2:

Using the ansatz in the first equation of (2.8) gives that ˛ solves

2˛2 C ˛ C ˇ D 0:

If ˇ < 0, the preceding equation has a solution, ˛ > 0. Finally, we determine � bythe normalization condition,

RR

mdx D 1. To find H, we use the expressions for uand m in the first equation of (2.8).


2.4 Interface Formation

In this last example, we describe the formation of interfaces and the breakdown ofregularity. For � 2 R; we consider the MFG

( juxj22

C �V.x/ D m C H.�/;

�.mux/x D 0;(2.9)

with periodic conditions; that is, u;m W T ! R, m � 0, andRT

mdx D 1.First, we attempt to solve (2.9). The second equation in (2.9) implies that

mux D c, for some constant c. If c ¤ 0, then ux D cm . This is not possible becauseR

Tuxdx D 0 and m > 0. Therefore, mux D 0. Accordingly, u is constant in the set

m > 0. Hence, the second equation holds trivially. Moreover, we gather

m.x; �/ D �V.x/ � H.�/

on the set m > 0. In addition, on the set m D 0, the first equation gives

�V.x/ � H.�/ � 0:

Thus,

m.x; �/ D ��V.x/ � H.�/

�C:

The map

h 7!ZT

.�V.x/ � h/C dx

is monotone decreasing. Hence, there is a unique value, H.�/; for which

ZT

��V.x/ � H.�/

�Cdx D 1:

If � is small, the conditionRT

m D 1 gives

H.�/ D �

ZT

V � 1:

Thus,

m.x; �/ D 1C �

V.x/ �

ZT

V

�:

In contrast, for large j�j, the condition m > 0 fails.

2.5 Bibliographical Notes 13

Fig. 2.1 H.�/

-3 -2 -1 1 2 3λ

-1.0

-0.8

-0.6

-0.4

-0.2

H(λ)

Fig. 2.2 m.x; �/

Because m.x; �/C H.�/ � �V.x/ D .H.�/ � �V.x//C, we have

juxj22

D .H.�/ � �V.x//C:

The solution u to the preceding equation can fail to be a classical solution.Furthermore, as we show next, it may admit multiple solutions.

Figure 2.1 illustrates the behavior of H.�/ for V.x/ D sin.2x/, and Fig. 2.2depicts m.x; �/. In general, the solution, u; is not unique and may not be differen-tiable. In Figs. 2.3 and 2.4, we plot two two-periodic solutions .m; u/ for � D 2.

2.5 Bibliographical Notes

The explicit solution in Sect. 2.1 appeared in [8]. The Hopf–Cole transform wasintroduced in the context of MFGs in [174]. Similar ideas were used in [138] todevelop numerical methods and in [66] to show the existence of classical solutions.


Fig. 2.3 m.x; 2/

0.5 1.0 1.5 2.0x

0.5

1.0

1.5

2.0

2.5

m(x,2)

Fig. 2.4 u.x; 2/—twodistinct solutions

0.5 1.0 1.5 2.0x

0.1

0.2

0.3

0.4

0.5

u(x,2)

A remarkable extension of the Hopf–Cole transform was presented in [78]. TheHopf–Cole transform was used in [203] to convert an MFG into a system ofSchrödinger equations. Gaussian-quadratic solutions were discussed in [137] and,with more generality, in [15]. Moreover, they have applications in machine learning,in particular, in clustering and non-supervised learning [189, 190]. The N-playerlinear-quadratic counterpart was considered in [17, 18, 197]. Some applications oflinear-quadratic MFGs are presented in [22, 31, 32, 145, 184]. The discussion inSect. 2.3 is inspired by [130]. Explicit examples where MFG partial differentialequations are converted into ordinary differential equations were examined in[25, 182]. A further explicit example was studied in [204].

Chapter 3Estimates for the Hamilton–Jacobi Equation

In this chapter, we examine a priori estimates for solutions of Hamilton–Jacobiequations. We are interested in solutions of time-dependent problems, u W T

d �Œ0;T� ! R or u W Rd � Œ0;T� ! R of

� ut C H.x;Du/ D ��u: (3.1)

For stationary problems, we consider periodic solutions. In this case, a solution is apair, .u;H/ with H 2 R and u W Td ! R; satisfying

� ��u C H.x;Du/ D H in Td: (3.2)

The techniques we develop are critical to the study of MFGs. In those games,the Hamilton–Jacobi equation that determines the value function, u; depends onthe density of the agents, m. Often, we have estimates for m in low regularityspaces; Lebesgue or Sobolev spaces are examples. For this reason, a large part ofour discussion focuses on integral estimates of u.

3.1 Comparison Principle

A central tool in the theory of Hamilton–Jacobi equations is the comparisonprinciple stated in the next proposition. In the context of MFGs, the comparisonprinciple is frequently used to get lower bounds for solutions; see, for example,Sect. 6.1 in Chap. 6. Then, upper bounds can be proven by the methods discussed inSects. 3.2 and 3.4.


15

16 3 Estimates for the Hamilton–Jacobi Equation

Proposition 3.1 (Comparison Principle). Let u W Td � Œ0;T� ! R solve

� ut C H.x;Du/ � ��u � 0 in Td � Œ0;T/; (3.3)

and let v W Td � Œ0;T� ! R solve

� vt C H.x;Dv/ � ��v � 0 in Td � Œ0;T/: (3.4)

Suppose that u � v at t D T. Then, u � v in Td � Œ0;T/.

Proof. Let uı D u C ıt . We have

� uıt C H.x;Duı/ � ��uı > 0 in Td � Œ0;T/: (3.5)

Subtracting (3.4) from (3.5), we get

� .uı � v/t C H.x;Duı/ � H.x;Dv/ � ��.uı � v/ > 0 in Td � Œ0;T/: (3.6)

Consider the function uı � v and let .xı; tı/ be a point of minimum of uı � v onT

d � Œ0;T/. This minimum is achieved at some point, tı > 0. We claim that tı D T .If not, at .xı; tı/, we have

uıt � vt; Duı D Du; �uı � �v:

However, the earlier identities and inequality yield a contradiction in (3.6). Accord-ingly, the minimum of uı � v is attained at T . Hence, uı � v in T

d � Œ0;T�. Theconclusion follows by letting ı ! 0. ut

In the foregoing theorem, we assumed that u and v are, respectively, classicalsuper and subsolutions; that is, u and v are smooth enough, u satisfies (3.3) andv satisfies (3.4). However, by the theory of viscosity solutions, the comparisonprinciple holds with fewer regularity requirements. The interested reader can findadditional material on viscosity solutions in the references at the end of the chapter.

3.2 Control Theory Bounds

The Hamilton–Jacobi equations in MFGs are associated with control problems—deterministic control in the first-order case and stochastic control in the second-order case. Here, we consider C1 solutions, u W Rd � Œ0;T� ! R; of the Hamilton–Jacobi equation,

� ut C jDuj22

C V.x/ D 0; (3.7)

3.2 Control Theory Bounds 17

with the terminal condition,

u.x;T/ D uT.x/; (3.8)

and investigate the corresponding deterministic control problem.For convenience, we suppose that V is of class C2 and globally bounded. By

Theorem 1.1, a solution of (3.7) is the value function of the control problem

u.x; t/ D infx

Z T

t

jPx.s/j22

� V.x.s//ds C uT.x.T//; (3.9)

where the infimum is taken over all trajectories, x 2 W1;2.Œt;T�/; with x.t/ D x.In what follows, we prove the existence of optimal trajectories. Then, using thecontrol theory characterization, we obtain various bounds for u.

3.2.1 Optimal Trajectories

We begin our study of (3.9) by examining the existence of optimal or minimizingtrajectories. Because (3.9) has quadratic growth in Px, the natural space to look forminimizers is the Sobolev space, W1;2.Œt;T�/.

Proposition 3.2. Let V be a bounded continuous function. There exists a minimizer,x 2 W1;2.Œt;T�/; of (3.9).

Proof. Let xn be a minimizing sequence for (3.9); that is, a sequence such that

u.x; t/ D limn!1

Z T

t

jPxn.s/j22

� V.xn.s//ds C uT.xn.T//:

We have supn kPxnkL2.Œt;T�/ � C. By Poincaré’s inequality, we conclude that

supn

kxnkW1;2.Œt;T�/ < 1:

Next, by Morrey’s theorem, the sequence xn is equicontinuous and bounded (sincexn.t/ is fixed). Hence, by the Ascoli–Arzelà Theorem, there exists a uniformlyconvergent subsequence. We can extract a further subsequence that convergesweakly in W1;2 to a function, x. To prove that x is a minimum, it is enough to showweakly lower semicontinuity; that is,

lim infn!1

Z T

t

� jPxnj22

� V. xn/

ds C uT.xn.T// �

Z T

t

� jPxj22

� V.x/

ds C uT.x.T//

(3.10)


for any sequence xn * x in W1;2.Œt;T�/. By convexity,

Z T

t

� jPxnj22

� V.xn/

ds C uT.xn.T// (3.11)

�Z T

t

�V.x/ � V.xn/C

� jPxj22

� V.x/

C Px.Pxn � Px/

ds C uT.xn.T//:

Because Pxn * Px and Px 2 L2.Œt;T�/, we have

Z T

tPx.Pxn � Px/ ! 0:

From the uniform convergence of xn to x, we conclude that

Z T

tV.xn/ � V.x/ ! 0

and that

uT.xn.T// ! uT.x.T//:

Thus, by taking the lim inf in (3.11), we get (3.10). utThe minimizers of (3.9) are solutions to the Euler–Lagrange equation, which is

an ordinary differential equation that we derive next.

Proposition 3.3. Let V be a C1 function. Let x W Œt;T� ! Rd be a W1;2.Œt;T�/

minimizer for (3.9). Then, x 2 C2.Œt;T�/; and it satisfies the Euler–Lagrangeequation:

Rx C DxV.x/ D 0:

Moreover, set H.p; x/ D jpj22

C V.x/. Then, for p D �Px, we have that .x; p/ solvesthe Hamiltonian dynamics: (

Pp D DxH.p; x/

Px D �DpH.p; x/:(3.12)

Proof. Let x W Œt;T� ! Rd be a W1;2.Œt;T�/minimizer for (3.9). Fix ' W Œ0;T� ! R

d

of class C2 with compact support on .t;T/. Because x is a minimizer, the function

i.�/ DZ T

t

jPx C � P'j22

� V.x C �'/C uT.x.T//


has a minimum at � D 0. Because i is differentiable, we have i0.0/ D 0. Therefore,

Z T

tŒPx � P' � DxV.x/'� ds D 0: (3.13)

Next, set

p.t/ D p0 CZ T

t�DxV.x/ds;

with p0 2 Rn to be chosen later. For each ' 2 C2

c..t;T// taking values in Rd,

we have

Z T

t

d

dt.p � '/dt D p � ' ˇTt D 0:

Thus,

Z T

tDxV.x/' C p � P'dt D 0:

Using (3.13), we conclude that

Z T

t.p C Px/ � P'dt D 0:

Therefore, p C Px is constant. Thus, selecting p0 conveniently, we have

p D �Px:Since p is continuous, the above identity gives Px D �DpH.p; x/, and, for thatreason, Px is continuous. Moreover, we have

Pp D DxH.p; x/;

and thus, p is C1. Further, we have

Px D �DpH.p; x/:

Consequently, Px is C1. As a result, x is C2. ut

3.2.2 Dynamic Programming Principle

The dynamic programming principle that we prove next is a semigroup propertythat the value function satisfies.


Proposition 3.4. Let V be a bounded continuous function and u be given by (3.9).Then, for any t0 with t < t0 < T, we have

u.x; t/ D infx

Z t0

t

jPx.s/j22

� V.x.s//ds C u.x.t0/; t0/: (3.14)

Proof. Let

Qu.x; t/ D infx

Z t0

t

jPx.s/j22

� V.x.s//ds C u.x.t0/; t0/; (3.15)

and u be given by (3.9). Take an optimal trajectory, x1, for (3.15) and select anoptimal trajectory, x2, for u.x.t0/; t0/. Consider the concatenation of x1 with x2

given by

x3 D(

x1.s/ t � s � t0

x2.s/ t0 < s � T:

We have

u.x; t/ �Z T

t

jPx3.s/j22

� V.x3.s//ds C uT.x3.T// D Qu.x; t/:

Conversely, let x be an optimal trajectory in (3.9). Then,

u.x.t0/; t0/ �Z T

t0

jPx.s/j22

� V.x.s//ds C uT.x.T//:

Consequently,

Qu.x; t/ �Z t0

t

jPx.s/j22

� V.x.s//ds C u.x.t0/; t0/ � u.x; t/:

ut

3.2.3 Subdifferentials and Superdifferentials of the ValueFunction

Consider a continuous function, W Rd ! R. The superdifferential DCx .x/ of

at x is the set of vectors, p 2 Rd, such that


lim supjvj!0

.x C v/ � .x/ � p � vjvj � 0:

Consequently, p 2 DCx .x/ if and only if

.x C v/ � .x/C p � v C o.jvj/;

as jvj ! 0. Similarly, the subdifferential, D�x .x/, of at x is the set of vectors, p;

such that

lim infjvj!0

.x C v/ � .x/ � p � vjvj � 0:

Next, we show that if is differentiable, then

D�x .x/ D DC

x .x/ D fDx .x/g:

Therefore, we regard D˙ as one-sided derivatives.

Proposition 3.5. Let W Rd ! R be a continuous function and x 2 Rd. If both

D�x .x/ and DC

x .x/ are non-empty, then

D�x .x/ D DC

x .x/ D fpg:

In that case, is differentiable at x with Dx D p. Conversely, if is differentiableat x, we have

D�x .x/ D DC

x .x/ D fDx .x/g:

Proof. Suppose that D�x .x/ and DC

x .x/ are both non-empty. We claim that thesetwo sets agree and have a single point, p. To check this, take p� 2 D�

x .x/ andpC 2 DC

x .x/. Then,

lim infjvj!0

.x C v/ � .x/ � p� � vjvj � 0;

lim supjvj!0

.x C v/ � .x/ � pC � vjvj � 0:

Subtracting these two identities, we obtain

lim infjvj!0

.pC � p�/ � vjvj � 0:


In particular, by choosing v D �� pC�p�

jp��pCj , we get

�jp� � pCj � 0;

so, p� D pC � p. Consequently,

limjvj!0

.x C v/ � .x/ � p � vjvj D 0

and thus Dx D p.To prove the converse statement, it suffices to see that if is differentiable, then

.x C v/ D .x/C Dx .x/ � v C o.jvj/:ut

Proposition 3.6. Let

W Rd ! R

be a continuous function. Fix x0 2 Rd. If � W Rd ! R is a C1 function such that

.x/ � �.x/

has a local maximum (resp. minimum) at x0; then

Dx�.x0/ 2 DCx .x0/ .resp. D�

x .x0//:

Proof. Suppose that .x/ � �.x/ has a strict local maximum at 0. Without loss ofgenerality, we can assume that .0/��.0/ D 0 and �.0/ D 0. So, .x/��.x/ � 0

or, equivalently,

.x/ � p � x C .�.x/ � p � x/:

Thus, by setting p D Dx�.0/ and using

limjxj!0

�.x/ � p � x

jxj D 0;

we get Dx�.0/ 2 DCx .0/. The case of a minimum is similar. ut

Proposition 3.7. Let u be given by (3.9) and let x be a corresponding optimaltrajectory. Suppose that V is of class C2. Then, p D �Px satisfies

– p.t0/ 2 D�x u.x.t0/; t0/ for t < t0 � T;

– p.t0/ 2 DCx u.x.t0/; t0/ for t � t0 < T.


Proof. Let t < t0 � T . By the dynamic programming principle, we have

u.x; t/ DZ t0

t

jPxj22

� V.x/ds C u.x.t0/; t0/:

Furthermore,

u.x; t/ �Z t0

t

jPx C yt0�t j22

� V�

x C ys � t

t0 � t

�ds C u.x.t0/C y; t0/:

Let

ˆ.y/ D u.x; t/ �Z t0

t

jPx C yt0�t j22

� V�

x C ys � t

t0 � t

�ds:

Accordingly,

u.x.t0/C y; t0/ �ˆ.y/

has a minimum at y D 0. Thus, Dyˆ.0/ 2 D�x u.x.t0/; t0/. In addition,

Dyˆ.0/ D �Z t0

t

Pxt0 � t

� DxV.x/s � t

t0 � tD �Px.t0/ D p.t0/

after integrating by parts and using Proposition 3.3.To prove the second item in the theorem, we use the inequality

u.x C y; t/ �Z t0

t

jPx � yt0�t j22

� V

x C y

t0 � s

t0 � t

�ds C u.x.t0/; t0/:

Next, let

‰.y/ DZ t0

t

jPx � yt0�t j22

� V

x C y

t0 � s

t0 � t

�ds C u.x.t0/; t0/:

Then the function u.x C y; t/ � ‰.y/ has a maximum at y D 0. Thus, arguing asbefore, we get the second part of the theorem. ut

3.2.4 Regularity of the Value Function

A function, W Rd ! R; is semiconcave if there exists a constant, C; such that

� Cjxj2 is a concave function. Here, we prove that the value function of (3.9) isbounded, Lipschitz, and semiconcave.


Proposition 3.8. Let u.x; t/ be given by (3.9). Suppose that kVkC2.Rd/ � C. Then,there exist constants, C0, C1, and C2; depending only on uT and T � t, such that

– juj � C0 for all x 2 Rd; 0 � t � T:

– ju.x C y; t/ � u.x; t/j � C1jyj for all x; y 2 Rd; 0 � t � T:

– u.x C y; t/ C u.x � y; t/ � 2u.x; t/ � C2�1C 1

T�t

� jyj2 for all x; y 2 Rd; 0 �

t < T.

Proof. For the first claim, we take x.s/ D x for s 2 Œt;T�. Therefore,

u.x; t/ � �TZ

t

V.x/ds C uT.x.T// � .T � t/c1 C kuTk1:

Furthermore, for any trajectory, x, with x.t/ D x, we have

TZt

jPx.s/j22

� V.x.s//ds C uT. x.T// � �.T � t/kVk1 � kuTk1:

To prove that u is Lipschitz, take x; y 2 Rd. Proposition 3.2 gives the existence of

an optimal trajectory, x�, for any initial condition .x; t/. Therefore,

u.x; t/ DTZ

t

jPx�.s/j22

� V.x�.s//ds C uT.x�.T//

and, because x� C y is a sub-optimal trajectory, we have

u.x C y; t/ �TZ

t

jPx�.s/j22

� V.x�.s/C y/ds C uT.x�.T/C y/:

Then,

u.x C y; t/ � u.x; t/ � Œ.T � t/kVkC1 C kuTkC1 � jyj � .C.T � t/C C/jyj:

The previous estimate proves that u is uniformly Lipschitz in x.

3.3 Integral Bernstein Estimate 25

For the semiconcavity, we take x; y 2 Rd with jyj � 1, x� as above, y.s/ D y T�s

T�t ,

u.x ˙ y; t/ �TZ

t

jPx�.s/˙ Py.s/j22

� V.x�.s/˙ y.s//ds C uT.x�.T//:

Finally, we conclude that

u.x C y; t/C u.x � y; t/ � 2u.x; t/ � jyj2.T � t/

C kVkC2 jyj2 � C2

1C 1

T � t

�jyj2:

utRemark 3.9. If both V and uT have bounded C2 norms, the semiconcavity estimatesin the preceding proposition can be improved and do not depend on T � t.

3.3 Integral Bernstein Estimate

The integral Bernstein estimate is an important tool for the analysis of second-order stationary Hamilton–Jacobi equations with Lp data. Here, we examine theHamilton–Jacobi equation,

��u.x/C jDu.x/j22

C V.x/ D NH; (3.16)

with V 2 Lp. Our goal is to bound the norm of Du in Lq for some q > 1. Theseestimates are used in Chap. 7 to establish bounds for MFGs in Sobolev spaces. Then,by a bootstrapping argument, they give a priori smoothness for the solutions.

Before stating the main result, we prove two auxiliary estimates.

Lemma 3.10. Let uWTd ! R be a C3 function and let v D jDuj2. Suppose that Vis C1. Then, there exist positive constants, c and C, which do not depend on u or Vsuch that, for every p > 1;

�ZTdvp�vdx � 4pc

.p C 1/2

24Z

Tdv.pC1/d

d�2 dx

� d�2d

� C

ZTdvpC2dx

� pC1pC2

35

and

�2ZTd

DV � Du vpdx � 1

2

ZTd

ˇD2u

ˇ2vpdx C Cp

ZTd

jVj2 vpdx:


Proof. By integration by parts, we have the identity

�ZTdvp�vdx D

ZTd

pvp�1jDvj2dx D 4p

.p C 1/2

ZTd

jDv pC12 j2dx:

Next, we use Sobolev’s inequality to obtain

ZTd

jDv pC12 j2dx C

ZTdvpC1dx � c

��v pC12

��22�

D c

ZTdv.pC1/d

d�2 dx

� d�2d

;

where 2� D 2dd�2 is the Sobolev conjugated exponent of 2. Combining the above two

inequalities with

ZTdvpC1dx �

ZTdvpC2dx

� pC1pC2

;

we get the first estimate.For the second inequality, we integrate again by parts to get

�ZTd

DV � Du vpdx DZTd

V�u vpdx C pZTd

V vp�1Dv � Dudx: (3.17)

Next, we apply a weighted Cauchy inequality to each of the terms in the right-handside of the prior identity. First, for any > 0, we have

ZTd

V ��u vpdx �

ZTd

j�uj2 vpdx C C

ZTd

jVj2 vpdx

� 1

8

ZTd

ˇD2u

ˇ2vpdx C C

ZTd

jVj2 vpdx;

if we select a small enough . Next, because v D jDuj2 implies Dv D 2D2uDu, wehave

pZTd

V vp�1Dv � Dudx � 2pZTd

jVj vpjD2ujdx � 1

8

ZTd

ˇD2u

ˇ2vpdx C Cp

ZTd

jVj2 vpdx:

Using the two preceding bounds in (3.17), we get the second estimate. utTheorem 3.11. Let u be a C3 solution of (3.16). Suppose that V is C1. Then, forany p > 1; there exists a constant, Cp > 0; that depends only on j NHj, such that

kDukL2d.pC1/

d�2 .Td/� Cp

1C kVk

L2d.1Cp/

dC2p .Td/

�:

Note that �p D 2d.1Cp/dC2p ! d when p ! 1 and that �p is increasing when d > 2:

3.3 Integral Bernstein Estimate 27

Proof. We set v D jDuj2. Differentiating (3.16), we have

�uxi D 1

2vxi C Vxi : (3.18)

Thus,

��v D �2dX

i;jD1

�uxixj

�2 � 2

dXiD1

uxi�uxi D �2dX

i;jD1

�uxixj

�2 � 2

dXiD1

uxi

1

2vxi C Vxi

�:

(3.19)

By multiplying (3.19) by vp and integrating over Td, we have

�ZTdvp�vdx C 2

ZTd

ˇD2u

ˇ2vpdx (3.20)

D �ZTd

Du � Dv vpdx � 2

ZTd

DV � Du vpdx:

Lemma 3.10 provides bounds for the first term on the left-hand side and the lastterm on the right-hand side of (3.20).

Further, we observe that for all ı > 0; there exists a constant, Cı > 0; that doesnot depend on u; such that

�ZTd

Du � Dv vpdx � ı

ZTd

ˇD2u

ˇ2vpdx C Cı

p C 1

ZTdvpC2dx (3.21)

for every p > 1. To check (3.21), we integrate by parts and get

� 2ZTdvpDu � Dvdx D 2

p C 1

ZTdvpC1�udx

� C

p C 1

ZTdvpC1jD2ujdx � ı

ZTd

ˇD2u

ˇ2vpdx C Cı

p C 1

ZTdvpC2dx;

(3.22)

by Young’s inequality.Now, we claim that for any large enough p > 1, there exists Cp > 0 that does not

depend on u; such that

ZTdv

d.pC1/d�2 dx

� .d�2/d.pC1/

� Cp

ZTd

jVj2ˇp dx

� 1ˇp C Cp; (3.23)

where ˇp is the conjugate exponent of d.pC1/.d�2/p . Further, ˇp ! d

2when p ! 1:


To prove the previous claim, we combine (3.20) and the first estimate inLemma 3.10 to get

cp

ZTdv

d.pC1/d�2 dx

� d�2d

C 2

ZTd

ˇD2u

ˇ2vpdx � (3.24)

cp

ZTdvpC2dx

� pC1pC2

�ZTd

Du � Dv vpdx � 2

ZTd

DV � Du vpdx;

where cp WD 4p QC.pC1/2 , for some constant QC. From the second estimate in Lemma 3.10,

(3.21), and Young’s inequality, (zpC1pC2 � �z C Cp;�), we have

cp

ZTdv

d.pC1/d�2 dx

� d�2d

C�2 �

1

2C ı

�ZTd

ˇD2u

ˇ2vpdx � (3.25)

Cp

ZTd

jVj2 vpdx C

Cıp C 1

C ı

�ZTdvpC2dx C Cp;ı :

Next, using (3.16), we have

ZTd

ˇD2u

ˇ2vpdx � 1

d

ZTd

j�uj2 vpdx D 1

d

ZTd

ˇv2

C V � Hˇ2vpdx (3.26)

� 1

3d

ZTdv2vpdx � 1

d

ZTd

V2vpdx � 1

dCZvpdx

� cZTdvpC2dx � C

ZTd

V2vpdx � Cp;

where the second inequality follows from .a � b � c/2 � 13a2 � b2 � c2:

For a small ı and a large enough p, (3.25) and (3.26) yield

ZTdv

d.pC1/d�2 dx

� d�2d

� Cp

ZTd

jVj2 vpdx C Cp

� Cp

ZTdv

d.pC1/d�2 dx

� .d�2/pd.pC1/

ZTd

jVj2ˇp dx

� 1ˇp C Cp:

Hence,

ZTdv

d.pC1/d�2 dx

� .d�2/d.pC1/

� Cp

ZTd

jVj2ˇp dx

� 1ˇp C Cp:

This last estimate gives (3.23), and the theorem follows. ut

3.4 Integral Estimates for HJ Equations 29

3.4 Integral Estimates for HJ Equations

Here, we prove Lp estimates for positive subsolutions of Hamilton–Jacobi equations;that is, we consider functions, u W Td � Œ0;T� ! R; that satisfy

� ut C 1

�jDuj� C V.x/ � �u: (3.27)

The assumption that u is positive is not critical. Analogous results hold forsubsolutions that are bounded from below.

Usually, bounds from below result from the comparison principle in Sect. 3.1.In this section, our goal is to obtain bounds from above. Bounds for solutions ofHamilton–Jacobi equations are essential to get higher regularity. For example, inthe next section, we need L1 bounds to prove Sobolev regularity of solutions of(3.27).

Two distinct mechanisms give integrability for positive subsolutions of (3.27).The first corresponds to optimal control and is linked to the first-order term. Thesecond is given by stochastic effects and is associated with the Laplacian. In thesubquadratic case, � � 2, diffusion dominates; in the superquadratic case, � > 2,the optimal control is the primary source of regularity. Next, we isolate thesetwo effects and prove two estimates using elementary arguments. We begin byconsidering bounds that are derived through an optimal control technique.

Proposition 3.12. Suppose that V is continuous. Let p > dC1. Let u W Td�Œ0;T� !R, u of class C2, with u � 0, solve

� ut C 1

�jDuj� C V.x; t/ � 0: (3.28)

Suppose that u.x;T/ D uT.x/ is continuous. Then, there exists a constant, C > 0,depending only on kuTkL1.Td/, � , p, T, and d such that

kukL1.Td�Œ0;T�/ � C C CkVk�0p

dC�0p

Lp.Td�Œ0;T�/:

Proof. Let � 0 be determined by 1�

C 1� 0

D 1. Fix y 2 Rd and consider the trajectory

xy.s/ D x C y.s � t/

T � t: (3.29)

Recall that the Legendre transform of jpj��

is jvj�0

� 0

. Consequently, using a version ofthe representation formula (3.9) for (3.28), we have


u.x; t/ �Z T

t

24ˇ y

T�t

ˇ� 0

� 0 � V

x C y.s � t/

T � t; s

�35 ds C uT.x C y/:

Next, we average the prior upper bound with the Gaussian kernel

e� jyj

2

2�2

.2�2/d=2:

Because uT is bounded,RRd uT.x C y/ e

�

jyj

2

2�2

.2�2/d=2� C. We have

Z T

t

ZRd

ˇy

T�t

ˇ� 0

� 0e� jyj

2

2�2

.2�2/d=2dyds D C

��0

jT � tj� 0�1 ;

where C is a constant that does not depend on � .For p0 given by 1

p C 1p0

D 1, we have

��e� jyj

2

2�2

.2�2/d=2

��Lp0

.Rd/

D C

�d.p0�1/=p0

;

for some constant, C; independent of � . Then, by a change of variables,

�Z T

t

ZRd

V

x C y.s � t/

T � t; s

�e� jyj

2

2�2

.2�2/d=2dyds (3.30)

� C

�d.p0�1/=p0

Z T

tkV.�; s/kLp.Td/

ˇˇ .s � t/

T � t

ˇˇ�d=p

ds

� C

�d=pkV.�; s/kLp.Td�Œt;T�/

Z T

t

ˇˇ .s � t/

T � t

ˇˇ�dp0=p

ds

!1=p0

:

For p > d C 1, we have

Z T

t

ˇˇ .s � t/

T � t

ˇˇ�dp0=p

ds

!1=p0

D C.T � t/1=p0

:

Combining these identities gives

u.x; t/ � C��

0

jT � tj� 0�1 C CjT � tj1=p0

�d.p0�1/=p0

kVkLp.Td�Œt;T�/ C C:


By minimizing over � , we get

u.x; t/ � C C CkVk�0p

dC�0p

Lp.Td�Œ0;T�/:

We note that the constant, C; can be chosen uniformly for 0 � t � T . utRemark 3.13. The result in the previous proposition can be refined by replacing thesub-optimal trajectories in (3.29) by

xy.s/ D x C y.s � t/

.T � t/

and selecting > 0 conveniently. We leave this approach to the reader who canverify that, in the quadratic case, � D 2, it is enough to assume that V 2 Lp.Td �Œ0;T�/ with p > d=2 C 1. For a similar result, obtained by a different method, seeProposition 3.18 below.

Remark 3.14. If V 2 L1.Œ0;T�;Lp.Td// with p > d; we obtain a similar boundfrom (3.30). Using the method from the preceding remark, we see that V 2L1.Œ0;T�;Lp.Td// for p > d

2is enough to ensure L1 bounds in the quadratic case,

� D 2.

Proposition 3.15. Suppose that V is continuous. Let p > d2

C 1. Let u � 0, u ofclass C2, satisfy (3.27). Suppose that u.x;T/ D uT.x/ is continuous. Then, thereexists a constant, C > 0, that depends only on kuTkL1.Td/, � , p, and d, such that

kukL1.Td�Œ0;T�/ � C C CkVkLp.Td�Œ0;T�/:

Proof. Because u solves (3.27), we have

�ut C V.x; t/ � �u:

By the comparison principle,

u � w;

where w solves

�wt C V.x; t/ D �w;

with w.x;T/ D uT.x/. We write w D w0Cw1 where w0 solves the previous equationwith homogeneous terminal data and w1 solves the homogeneous backward heatequation,

�.w1/t D �w1;

with w1.x;T/ D uT . Clearly, w1 is bounded by kuTkL1.Td/.


To bound w0, we use the fundamental solution for the backwards heat equationwith homogeneous terminal data to get

w0.x; t/ DZ T

t

ZRd

V.x C y; s/e� jyj

2

4.T�s/

.4.T � s//d=2dyds:

We observe that for p0 given by 1p C 1

p0

D 1, we have

��Xk2Zd

e� jyCkj

2

4.T�s/

.4.T � s//d=2

��Lp0

.Td/

D C

jT � sjd.p0�1/=.2p0/:

By Hölder’s inequality,

jw0.x; t/j � CZ T

tkV.�; s/kLp.Td/

1

jT � sjd.p0�1/=.2p0/ds (3.31)

� CkVkLp.Td�Œt;T�/Z T

t

1

jT � sjd.p0�1/=2 ds

�1=p0

:

If d.p0 � 1/=2 < 1, the preceding integral converges and gives the desired bound.The prior condition is equivalent to p > d

2C 1. ut

Remark 3.16. The boundedness of u also holds if V 2 L1.Œ0;T�;Lp.Td// forp > d

2, as we can see from (3.31).

Next, we give an integral identity that is useful in the study of (3.27).

Lemma 3.17. Let � be an increasing C2 function. Let u � 0 solve (3.27). Then,

� d

dt

ZTd�.u/dx C

ZTd

��0.u/jDuj� C �00.u/jDuj2� dx �

ZTd

jV.x; t/j�0.u/dx:

Proof. The proof follows by multiplying (3.27) by �0.u/. utOur last result is an application of the foregoing lemma.

Proposition 3.18. Suppose that V is continuous. Fix p � minf d2; d�g. Let u be a C2

function, u � 0, solving (3.27). Suppose that u.x;T/ D uT.x/. Then, there exists aconstant, Cq > 0, depending only on kVkL1.Œ0;T�;Lp.Td//, kuTkL1.Td/, p, � , d, and qsuch that

kukL1.Œ0;T�;Lq.Td// � Cq

for any 1 � q < 1.


Proof. Let � > 0, to be selected later (see Eq. (3.33) for the case � � 2), and let

D e�u2 :

Using Lemma 3.17, we have

� d

dt

ZTd 2dx C c�

ZTd�1�� jD 2

� j�dx C 4

ZTd

jD j2dx � �

ZTd

jV.x; t/j 2dx;

where c� > 0 is a fixed constant. Integrating the previous identity in Œt;T�, we have

k .�; t/k2L2 C c�

Z T

t

ZTd�1�� jD 2

� j�dxds C 4

Z T

t

ZTd

jD j2dxds (3.32)

� �

Z T

t

ZTd

jV.x; t/j 2dxds C k .�;T/k2L2.Td/:

First, we consider the case � � 2. By Sobolev’s inequality,

k k2L2�

� CZTd 2 C jD j2dx;

where 2� D 2dd�2 . Therefore, by Hölder’s inequality,

ZTd

jV.x; t/j 2dx � CkVkL

d2 .Td/

ZTd 2 C jD j2dx

� CkVkL

d2 .Td/

k k2L2.Td/C CkVk

Ld2

ZTd

jD j2dx:

Applying the prior estimate on the right-hand side of (3.32), we get

k .�; t/k2L2.Td/C 4

Z T

t

ZTd

jD j2dxds

� �CkVkL1.Œ0;T�;L

d2 .Td//

.T � t/k k2L1.Œ0;T�;L2.Td//

CZ T

t

ZTd

jD j2dxds

�

C k .�;T/k2L2.Td/:

Let 0 � t� � T such that

k .�; t�/k2L2.Td/D k k2L1.Œ0;T�;L2.Td//

:


Next, we select a small enough � that may depend on V , such that

�C.1C .T � t//kVkL1.Œ0;T�;L

d2 .Td//

� 1

2: (3.33)

Then,

1

2k .�; t�/k2L2.Td/

C 2

Z T

t�

ZTd

jD j2dxds � k .�;T/k2L2.Td/:

Therefore, 2 L1.Œ0;T�;L2.Td/ and

Z T

t�

ZTd

jD j2dxds � C:

Next, we use 2 L1.Œ0;T�;L2.Td// to conclude that

sup0�t�T

Z T

t

ZTd

jD j2dxds � C:

The case � > 2 is similar because the Sobolev inequality gives

k 2� k�

L��.Td/� C

ZTd 2 C jD 2

� j�dx;

where �� D �dd�� . Next,

ZTd

jV.x; t/j 2dx � CkVkLd=� .Td/k 2kL��=� .Td/

� CkVkLd=� .Td/

ZTd 2 C jD 2

� j�dx:

Then, the proof proceeds as before. utRemark 3.19. The preceding result can be generalized for V 2 Lr.Œ0;T�;Ls.Td//.For that, we take �.z/ D zp

p , for p > 1. By Lemma 3.17, we have

� d

dt

ZTd

up

pdx C

ZTd

�up�1jDuj� C .p � 1/up�2jDuj2� dx D �

ZTd

Vup�1dx:

Then, through an iterative process, we can get bounds for u in various Lebesguespaces. Here, we do not pursue this direction and point to the references at the endof the chapter.

3.5 Gagliardo–Nirenberg Estimates 35

3.5 Gagliardo–Nirenberg Estimates

In the analysis of Hamilton–Jacobi equations in MFGs, the following line ofreasoning is frequently used: first, the comparison principle gives lower bounds forthe solutions; second, the results in the previous sections provide upper bounds;finally, the methods outlined next give the regularity of solutions. These methodsare based on the Gagliardo–Nirenberg inequality. We consider

� ut C 1

�jDuj� C V.x/ D �u; (3.34)

with 1 < � < 2. In this range of parameters, the nonlinearity jDuj� can beregarded as a perturbation of the heat equation. Then, the Gagliardo–Nirenberginequality together with the earlier bounds in L1 and the regularity of the heatequation give estimates for the solutions in Sobolev spaces.

Proposition 3.20. Let u 2 W2;p.Td/. Then, for 1 < r < 1, there exists a constant,C > 0; such that

kDukL�r.Œ0;T�;L�p.Td// � C��D2u

�� 12Lr.Œ0;T�;Lp.Td//

kuk 12

L1.Td�Œ0;T�/ :

Proof. The Gagliardo–Nirenberg inequality gives

kDukL2r.Œ0;T�;L2p.Td// � C��D2u

�� 12Lr.Œ0;T�;Lp.Td//

kuk 12

L1.Td�Œ0;T�/ :

Because 1 < � < 2, we have

kDukL�r.Œ0;T�;L�p.Td// � C kDukL2r.Œ0;T�;L2p.Td// :

Therefore, by combining these two inequalities, we conclude the proof. utNext, we recall a standard result for the heat equation.

Lemma 3.21. Let u be a solution of (3.34) with 1 < � < 2. Then, for 1 < r,p < 1, there exists a constant C > 0 such that

kutkLr.Œ0;T�;Lp.Td// ;��D2u

��Lr.Œ0;T�;Lp.Td// � C kVkLr.Œ0;T�;Lp.Td//C C kDuk�

L�r.Œ0;T�;L�p.Td/ //:

By combining Proposition 3.20 with Lemma 3.21, we get the following Hessianintegrability estimate:

Proposition 3.22. Let u be a solution of (3.34) with 1 < � < 2. Fix 1 < r,p < 1. Then, there exists a constant, C > 0; such that

��D2u��

Lr.Œ0;T�;Lp.Td// � C kVkLr.Œ0;T�;Lp.Td// C C kuk�

2��

L1.Td�Œ0;T�// :


Proof. We start by combining Lemma 3.21 with Proposition 3.20 to obtain

��D2u��

Lr.Œ0;T�;Lp.Td// � C��D2u

�� 2Lr.Œ0;T�;Lp.Td//

kuk�2

L1.Td�Œ0;T�//

C C kVkLr.Œ0;T�;Lp.Td/ / :

Because 1 < � < 2, a straightforward application of a weighted Young’s inequalityyields the result. utTheorem 3.23. Let u be a solution of (3.34) with 1 < � < 2 and assume thatV 2 Lr.0;TI Lp.Td// for 1 < r; p < 1. Then, there exists a constant, C > 0;

such that

kDukL�r.Œ0;T�;L�p.Td/ / � C C C kVkLr.Œ0;T�;Lp.Td// C C kuk2

2��

L1.Td�Œ0;T�/ :

Proof. By combining Propositions 3.20 and 3.22, we have

kDukL�r.Œ0;T�;L�p.Td/ / � C kVk 12

Lr.Œ0;T�;Lp.Td/ /kuk 1

2

L1.Td�Œ0;T�/ C C kuk2

2��

L1.Td�Œ0;T�/ :

Notice that kVkLr.Œ0;T�;Lp.Td/ / < C for some C > 0. In addition,

1

2<

2

2 � � ;

for 1 < � < 2. The result follows from a weighted Young’s inequality. utBy the estimates in Sect. 3.4, positive solutions of (3.34) are bounded in L1.Td �

Œ0;T�/, provided that V 2 Lp.Td � Œ0;T�/ for p satisfying

p >d

2C 1: (3.35)

As a result, the foregoing Theorem gives uniform estimates for positive solutions of(3.27) that we state next.

Corollary 3.24. Suppose that uT � 0, V � 0, and V 2 Lp.Td � Œ0;T�/ for some psatisfying (3.35). Let u solve (3.34). Then, there exists a constant, C, such that

kDukL�p.Td�Œ0;T�/ � C:



Hamilton–Jacobi equations arise in various contexts, including classical mechanics[10, 108], control theory [28, 87, 101, 177], and front propagation [43, 106, 188,200]. Maximum and comparison principles are essential to the study of ellipticand parabolic equations, see [47, 107]. In Hamilton–Jacobi equations, comparisonproperties are at the heart of the theory of viscosity solutions [16, 19, 59, 80, 102,112, 151, 172]. A classical introduction to calculus of variations is [40]. A morecontemporary approach is described in [81] and [82]. Minimizers of calculus ofvariation problems with Sobolev potentials are examined in [100]. Further boundson Hamilton–Jacobi equations using control theory methods are discussed in [16]and [59] for first-order equations, and in [102] for second-order equations. S.Bernstein introduced certain a priori estimates for the Dirichlet problems in [34, 35].Some of these estimates were generalized for other problems. In particular, theintegral Bernstein estimates considered here were developed in [173]. Differentauthors considered integral estimates for Hamilton–Jacobi equations and second-order elliptic problems. The exponential transform in Sect. 3.4 appeared in [151].The estimate mentioned in Remark 3.13 was developed in [193]. The result inRemark 3.19 was discussed in [68]. Here, we did not study the Hölder estimates forsuperquadratic problems and refer the reader to [20, 58, 60, 65, 69]. The statementand proof of the Gagliardo–Nirenberg inequality can be found in [104]. The methodsdescribed in the previous section can be improved to include the quadratic case [9].

Chapter 4Estimates for the Transport and Fokker–PlanckEquations

In this chapter, we turn our attention to the second equation in the MFG system, thetransport equation,

mt.x; t/C div.b.x; t/m.x; t// D 0 in Td � Œ0;T�; (4.1)

or the Fokker–Planck equation,

mt.x; t/C div.b.x; t/m.x; t// D �m.x; t/; in Td � Œ0;T�; (4.2)

where b W Td � Œ0;T� ! R

d is a smooth vector field. Both (4.1) and (4.2) areequipped with the initial condition

m.x; 0/ D m0.x/: (4.3)

We assume that m0 � 0 withR

m0 D 1; that is, m0 is a probability measure. Asbefore, we assume m0 to be of class C1 to simplify the presentation. Except for thediscussion of weak solutions, a solution to (4.1) or (4.2) is a positive C1 function,m. The choice of T

d as the spatial domain is of minor importance; many of ourresults extend to bounded domains with Dirichlet or Neumann boundary conditionsor to R

d if we assume enough decay of the solution.Our primary goal is to understand integrability and regularity properties of (4.1)

and (4.2) in terms of the vector field, b.

4.1 Mass Conservation and Positivity of Solutions

In this section, we examine two properties of solutions to (4.1) and (4.2), namelypositivity and mass conservation.


39

40 4 Estimates for the Transport and Fokker–Planck Equations

Proposition 4.1. Let m solve either (4.1) or (4.2) with the initial condition (4.3).Then, Z

Tdm.x; t/dx D 1

for all t � 0.

Proof. If m solves either (4.1) or (4.2), integration by parts yields

d

dt

ZTd

m.x; t/dx D �ZTd

div.bm/dx D 0:

For solutions of (4.2), we combine the former computation with

ZTd�m.x; t/dx D

ZTd

div.rm.x; t//dx D 0:

utRemark 4.2. Proposition 4.1 holds in bounded domains with homogeneous Neu-mann boundary conditions if b is orthogonal to the boundary. Those boundaryconditions encode a zero net flow through the boundary. For Dirichlet boundaryconditions, we have Z

Tdm.x; t/dx � 1:

Proposition 4.3. The transport equation (4.1) and the Fokker–Planck equation(4.2) preserve positivity: if m0 � 0 and m solves either (4.1) or (4.2), thenm.x; t/ � 0; 8.x; t/ 2 T

d � Œ0;T�:Proof. The proof is based on a duality argument. We take s 2 Œ0;T� and considerthe adjoint equation to (4.2):

(vt.x; t/C b � Dv.x; t/ D ��v.x; t/; 8.x; t/ 2 T

d � Œ0; s�v.x; s/ D �.x/;

(4.4)

where � 2 C1.Td/; �.x/ > 0; 8x 2 Td.

First, we claim that, by the comparison principle, v.x; t/ > 0; 8.x; t/ 2 Td �

Œ0; s�: Second, we multiply (4.2) by m and add (4.4) multiplied by v. Integrating theresulting expression in T

d, we get

d

dt

ZTd

m.x; t/v.x; t/dx D 0;

4.2 Regularizing Effects of the Fokker–Planck Equation 41

after integration by parts. Next, integrating in Œ0; s�, we have

ZTd

m.x; s/�.x/dx DZTdv.x; 0/m0.x/dx > 0:

Finally, since the previous identity holds for any positive �, we have m.x; s/ � 0: ut

4.2 Regularizing Effects of the Fokker–Planck Equation

To investigate the regularizing effects of the Fokker–Planck equation (4.2), werecord two useful identities.

Proposition 4.4. Let m be a smooth solution of (4.2) with m > 0 and assume that� 2 C2.R/. Then,

d

dt

ZTd�.m/dx C

ZTd

div .b/�m�0.m/ � �.m/� dx D �

ZTd�00.m/ jDmj2 dx;

(4.5)or, equivalently,

d

dt

ZTd�.m/dx �

ZTd

m�00.m/Dm � bdx D �ZTd�00.m/ jDmj2 dx: (4.6)

Proof. The two identities follow by multiplying (4.2) by �0.m/ and integrating byparts. ut

Next, we record some consequences of the preceding result.

Proposition 4.5. Let m be a smooth solution of (4.2) with m > 0. Then, there existC > 0 and c > 0, such that

d

dt

ZTd

1

m.x; t/dx � C

ZTd

jbj2m

dx � cZTd

jDmj2m3

dx; (4.7)

d

dt

ZTd

ln m.x; t/dx � �CZTd

jbj2dx C cZTd

jD ln mj2dx; (4.8)

and

d

dt

ZTd

m ln mdx �ZTd

jb.x; t/j jDmj dx �ZTd

jDmj2m

dx: (4.9)

Proof. For the first two assertions, we take, respectively, �.z/ � 1z2

and �.z/ � 1z

in Proposition (4.4) to get


d

dt

ZTd

1

m.x; t/dx D 2

ZTd

bm�2Dmdx � 2ZTd

m�3jDmj2dx

and

d

dt

ZTd

ln m.x; t/dx D �ZTd

bm�1Dmdx CZTd

m�2jDmj2dx:

Then, we use a weighted Cauchy inequality to get the results.For the last assertion, we use �.z/ D z ln z to get

d

dt

ZTd

m.x; t/ ln m.x; t/dx �ZTd

jb.x; t/j jDmj dx �ZTd

jDmj2m

dx:

utCorollary 4.6. Let m be a smooth solution of (4.2) with m > 0, m.x; 0/ D m0,RTd m0.x/dx D 1, and m0 > � > 0. Then, there exist constants, C, C� > 0; such

that Z T

0

ZTd

jD ln mj2dxdt � CZ T

0

ZTd

jbj2dxdt C C� : (4.10)

Proof. Because m0 is bounded from below by a positive quantity, the integralZTd

ln m.x; 0/dx

is bounded from below; moreover, Jensen’s inequality bounds it from above. Hence,ˇˇZTd

ln m.x; 0/dx

ˇˇ � C:

Next, an additional application of Jensen’s inequality gives

ZTd

ln m.x;T/dx � 0:

Finally, we integrate (4.8) in Œ0;T� and use the earlier bounds to get (4.10). utCorollary 4.7. Let m be a smooth solution of (4.2) with m > 0, m.x; 0/ D m0,RTd m0.x/dx D 1, m0 > 0. Then,

ZTd

m.x;T/ ln m.x;T/dx CZ T

0

ZTd

jDmj22m

dxdt

� 1

2

Z T

0

ZTd

jbj2mdxdt CZTd

m.x; 0/ ln m.x; 0/dx:

4.3 Fokker–Planck Equation with Singular Initial Conditions 43

Proof. First, we integrate (4.9) in Œ0;T�. The result follows from the estimate

Z T

0

ZTd

jbjjDmj �Z T

0

ZTd

jbj2m2

C jDmj22m

:

ut

4.3 Fokker–Planck Equation with Singular Initial Conditions

Let � solve (4.2) with a Dirac delta as the initial condition; that is, � solves

(�t.x; t/C div .b.x; t/�.x; t// D ��.x; t/; 8.x; t/ 2 T

d � Œ0;T��.x; 0/ D ıx0 ;

(4.11)

in the sense of distributions. If b is regular, then � is a function for t > 0 (seeRemark 4.9). In the next proposition, we give integral estimates on the derivativesof �.

Proposition 4.8. Let � solve (4.11). Then, for any 0 < ˛ < 1, there exists aconstant C > 0 that does not depend on the solution, such that

Z T

0

ZTd

jD.� ˛2 /j2dxdt � C C C

Z T

0

ZTd

jbj2�˛.x; t/dxdt:

Proof. Using Proposition (4.4) for m D � and �.z/ � z˛ and integrating on Œ0;T�,we obtain

c˛

Z T

0

ZTd

jD.� ˛2 /j2dxdt D 1

˛

ZTd.�˛.x;T/ � �˛.x; 0//dx

C .1 � ˛/Z T

0

ZTd�˛�1b � D�dxdt

�C C "

Z T

0

ZTd

jD.� ˛2 /j2dxdt C C"

Z T

0

ZTd

jbj2�˛dxdt

for any " > 0; where c˛ D 4.1�˛/˛2

: Here, we useZTd�˛.x; 0/dx � 1;

ZTd�˛.x;T/dx � 1;

and Jensen’s inequality. Taking a small enough " yields the result. utRemark 4.9. To justify rigorously the above computations, it suffices to argue byapproximation. For that, we consider a family of solutions, �", of (5.2) with the


initial value ", where "WTd ! R is a family of smooth, compactly supportedfunctions with

RTd ".x/dx D 1 and " * ıx0 , as " ! 0. Then, we carry out the

preceding computations with �" and, finally, we let " ! 0.

4.4 Iterative Estimates for the Fokker–Planck Equation

Because (4.2) preserves mass and positivity of the initial conditions, solutions withinitial data in L1.Td/ satisfy m.�; t/ 2 L1.Td/ for every t 2 Œ0;T�. However, in manyapplications, we need a higher regularity for m, and it is critical to ensure that

m.�; t/ 2 Lp.Td/;

for some p > 1. Here, we study the integrability of solutions of the Fokker–Planckequation using an iterative argument.

First, we consider solutions to (4.2) that satisfy the estimateZ T

0

ZTd

jdiv b.x; t/j2 mdxdt � C:

Often, the previous estimate holds in MFGs (see Sects. 6.4 and 6.5 in Chap. 6). Next,we investigate estimates for m that depend polynomially on Lp norms of the driftterm. b. Because the cases p < 1 and p D 1 require distinct treatment, these arepresented separately.

4.4.1 Regularity by Estimates on the Divergence of the Drift

In this section, we consider how to use estimates on div b to get a higher integrabilityfor the solutions of (4.2)–(4.3). Before we proceed, we recall the following versionof the Poincaré inequality:

Proposition 4.10. For any 0 < r < 2; there exists a constant, Cr > 0; such that,for any f 2 W1;2.Td/; we have

ZTd

jf j2�

dx

�1=2�

� Cr

"ZTd

jDf j2dx

�1=2CZ

Tdjf jrdx

�1=r#:

Proof. The proof uses the compactness argument of the Poincaré inequality. By theSobolev theorem, it suffices to show that

ZTd

jf j2dx

�1=2� Cr

"ZTd

jDf j2dx

�1=2CZ

Tdjf jrdx

�1=r#:

4.4 Iterative Estimates for the Fokker–Planck Equation 45

If the inequality in the statement is not true, then we can find a sequence, fn, suchthat

ZTd

jfnj2dx

�1=2D 1

and "ZTd

jDfnj2dx

�1=2CZ

Tdjfnjrdx

�1=r#<1

n:

By the Rellich–Kondrachov theorem, we can extract a subsequence such thatfn ! Qf , strongly in L2, and therefore

ZTd

jQf j2dx

�1=2D 1:

However, DQf D 0 and, by Fatou’s Lemma,

ZTd

jQf jrdx � lim infn!1

ZTd

jfnjrdx D 0:

Hence, Qf D 0, which is a contradiction. utTheorem 4.11. Let m be a solution of (4.2)–(4.3) and assume that

Z T

0

ZTd

jdiv b.x; t/j2 mdxdt < QC (4.12)

for some QC > 0. Then, there exists a constant, Cr > 0; that depends only on QC, rand the dimension d such that

1. if d > 2,

km�kL1.Œ0;T�;Lr.Td// � Cr

for 1 � r < 2�

2I

2. if d � 2,

km�kL1.Œ0;T�;Lr.Td// � Cr

for any 1 � r < 1.

Proof. To prove the Theorem, we argue by induction. For that, we define anincreasing sequence, ˇn; for which we prove

km.�; t/kL1Cˇn .Td/ � C


for some C D C.n/ > 0. Set ˇ0 D 0; thus, we have

km.�; t/kL1Cˇ0 .Td/ D 1 � C

for every t 2 Œ0;T�.We begin with the case d > 2 and define ˇnC1 by

ˇnC1 D 2

d.ˇn C 1/:

Because ˇn is the nth partial sum of the geometric series with term2n

dnand ˇ1 D 2

d ,we get

limn!1ˇn D 2

d � 2 D 2�

2� 1:

Next, we set

qn D 2�

2.ˇnC1 C 1/ D d

d � 2.ˇnC1 C 1/:

Because ˇn <2

d�2 , we have

qn >d

d � 2ˇnC1 C ˇnC1 C 1 > 2ˇnC1 C 1:

Hence, Hölder’s inequality implies that

km.�; t/kL2ˇnC1C1.Td/

� km.�; t/k1��n

L1Cˇn .Td/km.�; t/k�n

Lqn .Td/;

where 0 < �n < 1 is defined via

�n

qnC 1 � �n

1C ˇnD 1

2ˇnC1 C 1: (4.13)

The previous equation gives

�n D qn

qn � ˇn � 12ˇnC1 � ˇn

1C 2ˇnC1D ˇnC1 C 1

1C 2ˇnC1:

Because km.�; t/kL1Cˇn .Td/ � C; it follows that

kmk2ˇnC1C1L2ˇnC1C1

.Td/� Ckmk�n.2ˇnC1C1/

Lqn .Td/D CkmkˇnC1C1

Lqn .Td/: (4.14)


Next, using �.z/ D zˇC1, with ˇ > 0, Proposition 4.4 gives

d

dt

ZTd

mˇC1dx D �ˇZTd

div.b.x; t//mˇC1dx � ˇ.ˇ C 1/

ZTd

mˇ�1jDxmj2dx:

Integrating the previous equation on Œ0; ��, we get

ZTd

mˇC1.x; �/dx DZTd

mˇC1.x; 0/dx � ˇZ �

0

ZTd

div.b.x; t//mˇC1dxdt

� 4ˇ

ˇ C 1

Z �

0

ZTd

jDxmˇC12 j2dxdt:

By rearranging the terms, we have

ZTd

mˇC1.x; �/dx C 4ˇ

ˇ C 1

Z �

0

ZTd

jDxmˇC12 j2dxdt

DZTd

mˇC1.x; 0/dx � ˇZ �

0

ZTd

div.b.x; t//mˇC1dxdt:

(4.15)

In addition,

ZTd

j div.b.x; t//mˇC1jdx �Z

Tdj div.b.x; t//j2mdx

�1=2 ZTd

m2ˇC1dx

�1=2

� Cı

ZTd

j div.b.x; t//j2mdx

�C ı

ZTd

m2ˇC1dx

�;

(4.16)

where all integrals are evaluated at a fixed time, t.In (4.15), we set ˇ D ˇnC1. Accordingly, (4.14) and (4.16) imply that

ZTd

mˇnC1C1.x; �/dx C 4ˇnC1ˇnC1 C 1

Z �

0

ZTd

jDxmˇnC1C1

2 .x; t/j2dxdt

�ZTd

mˇnC1C1.x; 0/dx C Cı

Z �

0

ZTd

j div.b.x; t//j2mdxdt

C ı

Z �

0

kmkˇnC1C1Lqn .Td/

dt (4.17)

for any � 2 Œ0;T�.Because qn D 2�

2.ˇnC1 C 1/ and

RTd m.x; t/dx D 1, for all 0 � t � T ,

Proposition 4.10 gives


kmkˇnC1C1Lqn .Td/

D kmˇnC1C1

2 k2L2� .Td/

� C C CZTd

jDmˇnC1C1

2 .x; t/j2dx:

Combining the previous bound with (4.17) and choosing a small enough ı, weconclude that there exists ı1 > 0 such thatZ

TdmˇnC1C1.x; �/dx C ı1

Z �

0

kmkˇnC1C1qn dt

�C C CZTd

mˇnC1C1.x; 0/dx C CZ �

0

ZTd

j div.b.x; t//j2mdxdt:

Because of (4.12), the last term on the right-hand side is bounded; this concludesthe proof of the first assertion of the Theorem.

Now, we consider the case when d D 2. As before, we define ˇn inductively.We start with ˇ0 D 0. Next, we fix p > 1 and set

ˇnC1 WD p � 1p

.ˇn C 1/:

Then, ˇn is the nth partial sum of the geometric series with term

.p � 1/npn

:

Therefore,

limn!1ˇn D p � 1:

Let

qn D p.ˇnC1 C 1/:

For �n as in (4.13), we obtain

kmkL2ˇnC1C1.Td/

� kmk1��n

L1Cˇn .Td/kmk�n

Lqn .Td/:

Hence, (4.13) leads to

�n.2ˇnC1 C 1/ D 1C ˇnC1:

Because kmkL1Cˇn .Td/ � C; it follows that

ZTd

m2ˇnC1C1dx D kmk2ˇnC1C1L2ˇnC1C1

.Td/� Ckmk�n.2ˇnC1C1/

Lqn .Td/D Ckmk1CˇnC1

Lqn .Td/: (4.18)


By gathering (4.15), (4.16), and (4.18), we obtain

ZTd

mˇnC1C1.x; �/dx C 4ˇnC1ˇnC1 C 1

Z �

0

ZTd

jDxmˇnC1C1

2 .x; t/j2dxdt

�ZTd

mˇnC1C1.x; 0/dx C Cı

Z �

0

ZTd

j div.b.x; t//j2dxdt C ı

Z �

0

kmk1CˇnC1

Lqn .Td/dt

(4.19)

for any � 2 Œ0;T�. As before, Proposition 4.10 gives

kmk1CˇnC1

Lqn .Td/D km

ˇnC1C1

2 k2L

2qnˇnC1C1

.Td/

� CZTd

jDxmˇnC1C1

2 j2dx C C: (4.20)

In light of (4.19) and (4.20), we choose a small enough ı > 0. Then, for someı1 > 0, we haveZ

TdmˇnC1C1.x; �/dx C ı1

Z �

0

kmkˇnC1C1Lqn .Td/

dt

� C C CZTd

mˇnC1C1.x; 0/dx C CZ �

0

ZTd

j div.b.x; t//j2dxdt:

Finally, as before, (4.12) provides an upper bound for the last term on the right-handside. This reasoning concludes the proof of the Theorem. ut

4.4.2 Polynomial Estimates for the Fokker–PlanckEquation, p < 1

Next, we investigate a priori bounds for Lp norms of the solutions of (4.2) thatare polynomial in norms of the drift, b. These bounds are essential in the study ofregularity of MFGs. Here, we consider the case when p < 1. This case is used forthe study of subquadratic MFGs in Chap. 8. The case p D 1 is examined in thenext section and used in the study of superquadratic MFGs.

Proposition 4.12. Let m be a smooth solution of (4.2) and assume that ˇ > 1.Then, there exist non-negative constants, C and c, such that

d

dt

ZTd

mˇ.x; t/dx � C��b2

��Lp.Td/

��mˇ��

Lq.Td/� c

ZTd

ˇD�

mˇ2

�ˇ2dx;

where

1 � p; q � 1 and1

pC 1

qD 1:


Proof. Let �.z/ � zˇ . According to Proposition (4.4), we have

d

dt

ZTd

mˇ dx D ˇ.ˇ � 1/ZTd

mˇ�1b � Dmdx � ˇ.ˇ � 1/ZTd

mˇ�2jDmj2dx

� CZTd

jbj2mˇdx � cZTd

mˇ�2jDmj2dx:

The result follows from Hölder’s inequality. utNext, to proceed with our study, we define the sequence .ˇn/n2N as

ˇnC1 WD �ˇn (4.21)

for some � > 1 to be fixed later. We choose ˇ0 > 1 to be any number such that

ZTd

mˇ0.x; t/dx < 1

for every t 2 Œ0;T�.Remark 4.13. In some applications in MFGs, we have a priori bounds onR T0

RTd j div bj2m. Hence, by the results in the preceding section, ˇ0 can be chosen

as close to dd�2 as desired.

For convenience, we take

1 < q <d

d � 2 : (4.22)

Next, we choose 0 � � 1 such that

1

qˇnC1D

ˇnC 2.1 � /

2�ˇnC1:

From the previous identity, we have

D d C 2q � dq

q Œ.� � 1/d C 2�: (4.23)

For later reference, we record in the next Lemma an elementary inequality.

Lemma 4.14. Let m W Td ! R be a smooth, non-negative function withR

Td mdx D 1. Then, there exists a constant, C > 0, such that

��mˇnC1��

Lq.Td/� C

ZTd

mˇn dx

�� 241C Z

Td

ˇˇD

mˇnC12

�ˇˇ2

dx

!1�35 :


Proof. First, we notice that Hölder’s inequality gives

ZTd

mqˇnC1dx

� 1qˇnC1 �

ZTd

mˇn

� ˇnZ

Tdm

2�ˇnC12

� 2.1�/

2�ˇnC1

:

Therefore,

��mˇnC1��

Lq.Td/�Z

Tdmˇn dx

�� ZTd

m2�ˇnC1

2 dx

� 2.1�/

2�

: (4.24)

BecauseRTd mdx D 1, Proposition 4.10 gives

��mˇnC12

��2.1�/

L2� .Td/

� C

ZTd

ˇˇD

mˇnC12

�ˇˇ2

dx

!1�C C: (4.25)

The Lemma follows by combining (4.24) with (4.25). utProposition 4.15. Let m be a solution of (4.2). Let and q be as in (4.23) and(4.22). Define p; r > 1 by

1

pC 1

qD 1 and r D 1: (4.26)

Then, there exists a constant, C > 0; such that

d

dt

ZTd

mˇnC1dx � C

"1C

��jbj2��r

Lp.Td/

ZTd

mˇn

��#:

Remark 4.16. Elementary computations show that for p > d2

and r > 2p2p�d , there

exists �; q > 1 such that (4.22), (4.23), and (4.26) hold simultaneously.

Proof. By Proposition 4.12, we have

d

dt

ZTd

mˇnC1dx ��jbj2

��Lp.Td/

��mˇnC1��

Lq.Td/� c

ZTd

ˇˇD

mˇnC12

�ˇˇ2

:

Because of Lemma 4.14, the above inequality becomes

d

dt

ZTd

mˇnC1 � C��jbj2

��Lp.Td/

ZTd

mˇn

�� 241C Z

Td

ˇˇD

mˇnC12

�ˇˇ2!1�35

� cZTd

ˇˇD

mˇnC12

�ˇˇ2

:


Let r0 be given by

1

rC 1

r0 D 1:

Taking into account that r D 1

, we have

r0.1 � / D 1:

Hence, we get

d

dt

ZTd


��Lp.Td/

ZTd

mˇn

�� ZTd

ˇˇD

mˇnC12

�ˇˇ2!1�

C C��jbj2

��Lp.Td/

ZTd

mˇn

�� c

ZTd

ˇˇD

mˇnC12

�ˇˇ2

� C��jbj2

��Lp.Td/

ZTd

mˇn

��C C

��jbj2��r

Lp.Td/

ZTd

mˇn

��

C ı

ZTd

ˇˇD

mˇnC12

�ˇˇ2

� cZTd

ˇˇD

mˇnC12

�ˇˇ2

;

where the last inequality follows from Young’s inequality weighted by ı. Bychoosing a small enough ı and using that < 1, the result follows. utProposition 4.17. Let m be a solution of (4.2) and let r and p be as in Remark 4.16.Then, there exist constants, C > 0 and � > 1; such thatZ

Tdmˇn � C C C

��jbj2��rn

Lr.Œ0;T�;Lp.Td//;

where ˇn is given by (4.21) and

rn WD r�n � 1� � 1 :

Proof. We argue by induction in n. For n D 1, Proposition 4.15 yields

d

dt

ZTd

mˇ1.x; t/dx � C

"1C

��jbj2��r

Lp.Td/

ZTd

mˇ0.x; t/dx

��#:

Because ZTd

mˇ0.x; t/dx � C;


we haveZTd

mˇ1.x; �/dx � C

1C

Z �

0

��jbj2��r

Lp.Td/dt

�� C

1C

��jbj2��r

Lr.0;TILp.Td//

�

for � 2 .0;T�. The result holds for n D 1.For � 2 .0;T�, the induction hypothesis implies that

Z �

0

d

dt

ZTd

mˇnC1dxdt � C C CZ �

0

��jbj2��r

Lp.Td/

1C

��jbj2��rn

Lp.Td/

��dt:

Therefore,ZTd

mˇnC1 .x; �/dx � CZ �

0

��jbj2��r

Lp.Td/

��jbj2��rn�

Lr.0;TILp.Td//dt C C

Z �

0

��jbj2��r

Lp.Td/dt C C

� C

1C

��jbj2��rCrn�

Lr.0;TILp.Td//

�:

ut

4.4.3 Polynomial Estimates for the Fokker–PlanckEquation, p D 1

We end this section by obtaining estimates for m in L1.Œ0;T�;Lp.Td// that arepolynomial in the L1-norm of the vector field, b.

As previously, fix ˇ0 > 1 for which

ZTd

mˇ0.x; t/dx < 1:

Let .ˇn/n2N be as in (4.21). We consider here only the case when d > 2 as the cased D 2 is similar.

Lemma 4.18. Let m W Td ! R, d > 2, be a smooth, non-negative function. Then,

ZTd

mˇnC1 .�; x/dx �Z

Tdmˇn .x; �/ dx

�� ZTd

m2�ˇnC1

2 .x; �/ dx

� 2.1�/

2�

;

where is given by

D 2

d .� � 1/C 2: (4.27)


Proof. Hölder’s inequality yields

ZTd

mˇnC1

� 1ˇnC1 �

ZTd

mˇn

� ˇnZ

Tdm

2�

2 ˇnC1

� .1�/

2�2 ˇnC1

provided that

1

�ˇnD

ˇnC 2 .1 � /

2��ˇn: (4.28)

The statement follows by rearranging the exponents. Solving (4.28) for gives(4.27) with 0 � � 1. utProposition 4.19. Let m W Td ! R, d > 2, be a smooth non-negative function withRTd mdx D 1. Then, there exists a constant, C > 0; such that

ZTd

mˇnC1dx ��Z

Tdmˇn dx

�C

241C

ZTd

ˇˇDx

m

ˇnC12

�ˇˇ2

dx

!.1�/35for as in (4.27).

Proof. The result follows by combining Proposition 4.10 with Lemma 4.18. utNext, we produce an upper bound for

d

dtkmkˇnC1

LˇnC1 .Td/:

Proposition 4.20. Let m be a solution of (4.2) with d > 2. Let be given by (4.27)and r D 1

. Then,

d

dt

ZTd

mˇnC1dx � C C C��jbj2

��r

L1.Td/

ZTd

mˇn dx

��: (4.29)

Proof. By Proposition 4.4, we have

d

dt

ZTd

mˇnC1 .x; t/dx � C��jbj2

��L1.Td/

ZTd

mˇnC1 .x; t/dx � cZTd

ˇˇDx

m

ˇnC12

�ˇˇ2

dx:


Using Proposition 4.19 in the previous inequality gives

d

dt

ZTd


��L1.Td/

ZTd

mˇn

�� 24C

ZTd

ˇˇDx

mˇnC12

�ˇˇ2 dx

!.1�/C C

35

� cZTd

ˇˇDx

mˇnC12

�ˇˇ2

� C��jbj2

��L1.Td/

ZTd

mˇn

�� ZTd

ˇˇDx

mˇnC12

�ˇˇ2!.1�/

C C��jbj2

��L1.Td/

ZTd

mˇn

�� c

ZTd

ˇˇDx

mˇnC12

�ˇˇ2

� C��jbj2

��L1.Td/

ZTd

mˇn

��C C

��jbj2��r

1

ZTd

mˇn

��

� C C C��jbj2

��r

L1.Td/

ZTd

mˇn

��;

where we used Young’s inequality weighted by " for the conjugate exponents,r and r0; given by

r0 D 1

1 � and r D 1

:

utCorollary 4.21. Let m be a solution of (4.2)–(4.3), d > 2 and assume that

m 2 L1 �Œ0;T�;Lˇ0

�T

d��

for some ˇ0 � 1. Consider the sequence .ˇn/n2N given by (4.21) for � > 1, and letr be as in Proposition 4.20. Then,

ZTd

mˇnC1 .�; x/dx � C C C��jbj2

��rnC1

L1.Td�Œ0;T�/ ;

where .rn/n2N is given by

rn D r

�n � 1� � 1

�:

Proof. We use an induction argument. Integrating (4.29) on .0; �/ and rearrangingthe exponents, we getZ

TdmˇnC1 .�; x/ dx � C

��jbj2��r

L1.Td�Œ0;T�/

Z �

0

ZTd

mˇn dx

��dt C C: (4.30)


First, we check the statement for n D 0. In this case, we haveZTd

mˇ0 .�; x/ dx � C:

Consider the induction hypothesis:

ZTd

mˇn dx � C C C��jbj2

��rn

L1.Td�Œ0;T�/ :

Hence,ZTd

mˇnC1 .�; x/ dx � C��jbj2

��r

L1.Td�Œ0;T�/ C C��jbj2

��rCrn�

L1.Td�Œ0;T�/ C C

� C��jbj2

��rCrn�

L1.Td�Œ0;T�/ C C;

where we have used a weighted Young’s inequality. Therefore,

rnC1 D r C rn�:

Finally, we have

r C r�n � 1� � 1 � D r� � r C r�nC1 � r�

� � 1 D r�nC1 � 1� � 1 D rnC1;

which completes the proof. ut

4.5 Relative Entropy

Let b W Td � Œ0;T� ! Rd be a smooth vector field. Let m solve

mt C div.bm/ D �m: (4.31)

Suppose that we have bounds on Lp norms of m. Often, this is the case in MFGs(see, for example, the first-order and second-order estimates in Chap. 6). Here, weinvestigate some consequences of this integrability to other solutions to the Fokker–Planck equation.

Let � solve

�t C div.b�/ D ��; (4.32)

4.5 Relative Entropy 57

and suppose that

� D �m; (4.33)

for some function � W Td � Œ0;T� ! R.The relative entropy between � and m is the integral

ZTd� ln �mdx:

More generally, for a convex function �; the �-entropy is

ZTd�.�/mdx:

Next, we derive a PDE for � and �.�/. Then, we examine the integrability of �with respect to the measure, m; and get further integrability for �.

Lemma 4.22. Let m be a solution to (4.31) with m > 0. Let .�; �/ solve(4.32)–(4.33). Then,

�t C b � D� � 2Dm

mD� D ��: (4.34)

Furthermore, for any convex function, �.z/, we have

.�.�//t C

b � 2Dm

m

�D .�.�// D �.�.�// � �00.�/ jD� j2 : (4.35)

In particular, for �.z/ D zp and p > 1; we have

.�p/t C

b � 2Dm

m

�D .�p/ D �.�p/ � p.p � 1/�p�2 jD� j2 : (4.36)

Proof. From (4.33), we have

�t D �tm C �mt;

div .b � �/ D div .b � m/ � C bmD�;

and

�� D ��m C�m� C 2D�Dm:

From these three identities, we readily obtain (4.34). To establish (4.35), we multiply(4.34) by �0.�/. The identity (4.36) corresponds to �.�/ D �p. ut


Lemma 4.23. Let m solve (4.31) and .�; �/ solve (4.32)–(4.33). Then,

d

dt

ZTd�pmdx D �Cp

ZTd

ˇD�

p2

ˇ2mdx; (4.37)

for some constant, Cp > 0.

Proof. Multiplying (4.36) by m, (4.31) by �p and adding these expressions, we have

d

dt.�pm/ D

�b C 2Dm

m

�D .�p/m � div .b � m/ �p

C�.�p/m C �p�m � p.p � 1/�p�2 jD� j2 m

D � div.b � �pm/C�.�pm/ � Cp

ˇD�

p2

ˇ2m:

By integrating over Td, we obtain

d

dt

ZTd�pmdx D

ZTd�.�pm/ dx �

ZTd

div .b � .�pm// dx

� Cp

ZTd

ˇD�

p2

ˇ2mdx D �Cp

ZTd

ˇD�

p2

ˇ2mdx:

utLemma 4.24. Let m solve (4.31), with m > 0, and let � solve (4.32). Assume that(4.33) holds. Then,Z

Td�qdx D

ZTd�qmqdx � C

ZTd�pmdx C C

ZTd

mrdx;

where

q D pr

r C p � 1 > 1 (4.38)

and

r > 1: (4.39)

Proof. Let 0 < a < 1. Then,

�qmq D �qmaCq�a:

Next, we notice that

�qmq � �pm C mr; (4.40)

4.6 Weak Solutions 59

where

p D q

a(4.41)

and

r D q � a

1 � a: (4.42)

According to (4.42), we have that r > 1. By combining (4.41) and (4.42), we get(4.38). Integrating (4.40) over Td yields the result. utCorollary 4.25. Let m solve (4.31), with m > 0, and let � solve (4.32). Supposethat p, q, and r satisfy (4.41)–(4.42). Assume that

ZTd�p.x; 0/m.x; 0/dx � C

and m 2 L1.Œ0;T�;Lr.Td//. Then,

ZTd�qdx � C;

on 0 � t � T.

4.6 Weak Solutions

If the vector field, b; has low regularity, the Fokker–Planck equation may not haveC2 solutions. To study (4.2), we thus need to consider weak solutions. We say thatm W Td � Œ0;T� ! R

C0 , m 2 L1.Td � Œ0;T�/, is a weak solution of (4.2) if mjbj2 2 L1

and

Z T

0

ZTd

m.��t C bD� ��/dxdt DZTd

m0.x/�.x; 0/dx: (4.43)

For (4.43) to be well defined, it is enough that mjbj 2 L1. However, the additionalintegrability requirement mjbj2 2 L1 makes it possible to obtain further properties.Here, we investigate the uniqueness of solutions.

To establish the uniqueness of weak solutions, we introduce the approximate dualproblem. We fix a vector field, Qb W Td � Œ0;T� ! R

d; and consider the PDE

� vt C QbDv D �v C ; (4.44)


together with the terminal condition v.x;T/ D 0. If Qb is C1, the previous equationadmits a solution. By the maximum principle, if is bounded, so is v.

Proposition 4.26. Let v solve (4.44) and let m be a weak solution of (4.2). Supposethat 2 L1. Then,

Z T

0

ZTd

mjDvj2dxdt � Ck k211C

Z T

0

ZTd

mjb � Qbj2dxdt

�:

Proof. Let v solve (4.44). Because m is a weak solution of (4.2), we have

Z T

0

ZTd

m.�.v2/t C bD.v2/ ��.v2//dxdt DZTd

m0.x/.v.x; 0//2dx:

Thus,

1

2

ZTd

m0.x/.v.x; 0//2dx C

Z T

0

ZTd

�mv � vm.b � Qb/ Dv C mjDvj2dxdt D 0:

Because m0 � 0 and kvk1 � Ck k1, the result follows from a weighted Cauchyinequality. utProposition 4.27. Let b 2 L2.Td � Œ0;T�/. Then, there exists at most one weaksolution of the Fokker–Planck equation (4.2) with the initial condition (4.3).

Proof. Let m1 and m2 be two weak solutions to (4.2) with the initial condition (4.3).Consider a sequence, b�; of smooth vector fields converging to b in L2 with respectto the measure 1C m1 C m2; that is,

Z T

0

ZTd

jb� � bj2.1C m1 C m2/dxdt ! 0:

Denote by m either m1 or m2. Because m is a weak solution of (4.2) and v� solves(4.44), we get

Z T

0

ZTd

m C m.b � b�/Dv� D

ZTd

m0.x/v�.x; 0/dx:

By the estimates in the preceding proposition, we have

Z T

0

ZTd

m.b � b�/Dv� ! 0:


The above discussion thus gives

Z T

0

ZTd.m1 � m2/ D 0

for any . Accordingly, m1 D m2. ut


The singular initial condition for the Fokker–Planck equation was used in [130, 135]to establish L1 bounds for Hamilton–Jacobi equations. The iterative estimates inSect. 4.4 follow [134] and [135]. The books [191] and [192] discuss applicationsof the transport equation and the Fokker–Planck equation in mathematical biology.The book [192] includes a discussion of relative entropy and some applications. Ourdiscussion of weak solutions is inspired by [195, 196].

Chapter 5The Nonlinear Adjoint Method

The nonlinear adjoint method was introduced by L.C. Evans as a tool to studyHamilton–Jacobi equations. This method draws on earlier research on Aubry–Mather and weak KAM theories and has many applications including the vanishingviscosity limit, the infinity Laplacian, non-convex Hamilton–Jacobi equations,and MFGs. We have already encountered a duality argument in the proof ofProposition 4.3. Here, we consider the Hamilton–Jacobi equation,

� ut C jDuj22

C V.x; t/ D �u; (5.1)

develop the nonlinear adjoint method, and derive several estimates for the solu-tion, u. These estimates highlight multiple regularity mechanisms and generalizesome of the earlier results.

We assume that V 2 C1.Td � Œ0;T�/: For x0 2 Td, we introduce the adjoint

variable, �; as the solution of

(�t � div.Du.x/�/ D ��;

�.x; 0/ D ıx0 :(5.2)

The above equation is equivalent to (4.11) for b D �Du. Hence, from the results inthe previous chapter, we have � � 0 and

RTd �.x; t/dx D 1. Our first goal is to derive

a representation formula for solutions of (5.1) in terms of �. This is the central ideain the nonlinear adjoint method.


63

64 5 The Nonlinear Adjoint Method

5.1 Representation of Solutions and Lipschitz Bounds

An essential tool for the study of first-order PDEs is the method of characteristics.This method does not extend in a straightforward way to second-order equations.A possible extension uses backward–forward stochastic differential equations. Ourapproach uses the adjoint equation (5.2) to represent solutions of (5.1) by integralswith respect to �. The results and methods here complement the comparisonprinciple in Sect. 3.1 and the Lipschitz and semiconcavity bounds in Sect. 3.2.

Proposition 5.1. Let u and � solve (5.1) and (5.2), respectively. Then, forany T > 0;

u.x0; 0/ DZ T

0

ZTd

� jDuj22

� V.x; t/

�.x; t/dxdt C

ZTd

uT.x/�.x;T/dx:

Proof. We multiply (5.1) by � and integrate by parts using (5.2). utNext, we consider a distribution, Q�; solving

(Q�t � div.b.x; t/ Q�/ D � Q�;Q�.x; 0/ D ıx0 :

(5.3)

In the following proposition, we give an upper bound for u.

Proposition 5.2. Let u and Q� solve (5.1) and (5.3), respectively. Then,

u.x0; 0/ �Z T

0

ZTd

� jbj22

� V.x; t/

Q�.x; t/dxdt C

ZTd

uT.x/ Q�.x;T/dx: (5.4)

Proof. We multiply (5.1) by Q� and integrate by parts using (5.3) to get

u.x0; 0/ DZ T

0

ZTd

�b � Du � jDuj2

2� V.x; t/

Q�.x; t/dxdt C

ZTd

uT.x/ Q�.x;T/dx:

The result follows from Cauchy’s inequality. utThe bound in the preceding proposition results from a stochastic control interpreta-tion of (5.1). By Proposition 5.1, the vector field b D Du gives an equality in (5.4),whereas any other vector field gives an inequality. We thus interpret b as the optimaldrift; the reader can compare this result with Theorem 1.1.

Remark 5.3. If V is bounded, Proposition 5.1 gives

u.x; t/ � �kVkL1.T � t/ � kuTkL1 :

5.2 Conserved Quantities 65

At the same time, Proposition 5.2 for b D 0 gives

u.x; t/ � kVkL1.T � t/C kuTkL1 :

These two bounds can also be proved by the Comparison Principle, Proposition 3.1.

Next, we give a representation formula for the derivatives of u:

Proposition 5.4. Let u and � solve (5.1) and (5.2), respectively. Then,

Dxi u.x0; 0/ D �Z T

0

ZTd

Dxi V.x; t/�.x; t/dxdt CZTd

Dxi uT.x/�.x;T/dx:

Proof. The proof follows by differentiating (5.1) and integrating with respectto �. ut

5.2 Conserved Quantities

Conserved quantities are essential features in Hamiltonian mechanics. Here, weshow that some of these quantities have a counterpart in the adjoint method.Moreover, in MFGs, such quantities play an essential role in the analysis of thelong-time behavior. This matter is examined in Sect. 6.7.

First, we consider a C1 Hamiltonian, H W Rd � R

d ! R. The Hamiltoniandynamics associated with H is given by the following ordinary differential equation:

(Px D �DpH.x;p/

Pp D DxH.x;p/:(5.5)

We say that a C1 function, F W Rd � Rd ! R, is a conserved quantity by (5.5) if

d

dtF.x;p/ D 0:

For example, is the Hamiltonian, H, is the total energy of the system and is aconserved quantity because

d

dtH.x;p/ D DxH.x;p/Px C DpH.x;p/ Pp D 0

using (5.5).


Next, we define the Poisson bracket between two C1 functions, F;G W Rd �Rd !

R, as

fF;Gg DdX

iD1

@F

@xi

@G

@pi� @F

@pi

@G

@xi:

A straightforward computation shows that F is conserved by (5.5) if and only iffF;Hg D 0. Accordingly, H is conserved because fH;Hg D 0.

Proposition 5.5. Let u 2 C2.Td � Œ0;T�/ solve

�ut C H.x;Dxu/ D 0;

and suppose that � solves the adjoint equation

�t � div.DpH.x;Du/�/ D 0:

Then,

d

dt

ZRd

H.x;Du/ �dx D 0:

Furthermore, if F W Rd � Rd ! R satisfies fH;Fg D 0, then

d

dt

ZRd

F.x;Du/ �dx D 0:

Proof. We have

d

dt

ZRd

F � DZRd

F�t C DpF.x;Du/Dut�dx

DZRd

F div.DpH�/C DpF.x;Du/D.H.x;Du//�dx

DZRd

fH;Fg�dx

after integrating by parts. ut

5.3 The Vanishing Viscosity Convergence Rate

We consider the following equation on Td � Œ0;T� W

(�u�t C jDu� j2

2C V.x/ D ��u�;

u�.x;T/ D u�T.x/;(5.6)

5.3 The Vanishing Viscosity Convergence Rate 67

where we assume that V and uT are C1 functions. The limit � ! 0 in thepreceding equation is called the vanishing viscosity limit, and its study was oneof the first applications of the nonlinear adjoint method. Our aim is to investigatethe convergence rate of the solution, u� , as � ! 0.

We assume that u� is differentiable in � and let w� D @@�

u� . Then, differentiating(5.6) in �, we get the following equation for w�:

(�w�t C Du�Dw� D ��w� C�u�;

w�.x;T/ D 0:(5.7)

Next, we use the adjoint method to prove that w� is O.��1=2/. Once we establish thisbound, we get

ju�1 � u�2 j � C

1p�1

C 1p�2

�j�1 � �2j;

which gives the convergence rate as � ! 0.Let �� be the adjoint variable defined as before by

(��t � div.Du�.x/��/ D ��;

��.x; s/ D ıx0 :(5.8)

By the discussion in Sect. 4.1 and in Sect. 5.1 (see Remark 5.3 and Proposi-tion 5.4), we have

ju�j; jDu�j � C; �� 0;

ZTd��.x; t/dx D 1: (5.9)

We begin with an auxiliary result.

Lemma 5.6. Let u� be a classical solution of (5.6) and �� solve the adjointequation (5.8). Then, there exists a constant, C > 0, such that

�

Z T

s

ZTd

jD2u�j2��dxdt � C:

Proof. Let v� D jDu� j22

. Differentiating the first equation in (5.6) and multiplying itby Du� , we get

�v�t C Du� � Dv� C DVDu� D ��v� � �jD2u�j2:


Next, we multiply the previous identity by �� , integrate by parts in x and t, and use(5.8). Accordingly, we get

�

Z T

s

ZTd

jD2u� j2��dxdt D �Z T

s

ZTd

DVDu��dx CZTd��.x;T/v�.x;T/dx � v�.x; 0/:

Consequently, by (5.9),

�

Z T

s

ZTd

jD2u�j2��dxdt � C:

utRemark 5.7. The proof of the previous lemma does not depend on the convexity of

the Hamiltonian, H.x; p/ D jpj22; in (5.6). Similar bounds therefore hold for non-

convex Hamiltonians. In the convex case, including (5.6), we can prove a strongerestimate. This estimate is discussed in Theorem 5.9, and the next theorem can beimproved accordingly.

Theorem 5.8. Let w� be as before. Then, there exists a constant, C > 0, such that

supŒ0;T��Td

jw�j � C

T

�

� 12

:

Consequently, u� converges as � ! 0 and

supŒ0;T��Td

ju� � uj � C .T�/12 ;

where u WD lim�!0 u�:

Proof. Multiply (5.7) by �� and integrate by parts to get

w�.x; s/ DZ T

s

ZTd�u��dxdt

�Z T

s

ZTd

j�u�j2��dxdt

� 12Z T

s

ZTd��dxdt

� 12

� C

T

�

� 12

;

where we used Hölder’s inequality and Lemma 5.6. Furthermore,

ju� � uj �Z �

0

jw�jd� � C .T�/12 :

ut

5.4 Semiconcavity Estimates 69

5.4 Semiconcavity Estimates

Consider the Hamilton–Jacobi equation:

(�ut C jDuj2

2C V.x/ D ��u;

u.x;T/ D uT.x/:(5.10)

In general, it is not possible to get bounds for the second derivatives of u that areuniform in �. However, in this instance, solutions are semiconcave; that is, second-derivatives satisfy unilateral bounds. Our estimates extend the ones in Sect. 3.2.4.

Theorem 5.9. Assume that V and uT are C2. Let .u;m/ solve (5.10). Then, thereexists a constant, C > 0; such that

D2u.x; t/ � CI; .x; t/ 2 Œ0;T� � Td:

Without the assumption that uT is C2, we have the bound

D2u.x; t/ � C

T � t C 1

T � t

�I; .x; t/ 2 Œ0;T/ � T

d:

Remark 5.10. The constants in the previous theorem do not depend on �.

Proof. Because the viscosity coefficient, �; plays no role in the proof, we set � D 1.Take any vector, � 2 R

d with j�j D 1. It is enough to prove that �TD2u� D u�� Cfor all .x; t/ 2 Œ0;T�� T

d: Let w D u�� . Differentiating (5.10) twice in the directionof � gives

�wt C DuDw C jDu� j2 C V�� D �w:

Integrating the previous identity with respect to the adjoint variable, �, we obtain

w.x0; s/CZ T

s

ZTd

jDu� j2�dxdt D �Z T

s

ZTd

V��dxdt CZTd.uT/��.x/�.x;T/dx:

BecauseR T

s

RTd jDu� j2�dxdt � 0, we get w.x0; s/ � C:

For the second bound, let Qw D �T�tT�s

�2w. Then,

� Qwt � 2 T � t

.T � s/2w C DuD Qw C

T � t

T � s

�2jDu� j2 C

T � t

T � s

�2V�� D � Qw:

Hence, taking into account that jwj D j�Du� j � jDu� j, we have


w.x0; s/CZ T

s

ZTd

T � t

T � s

�2jwj2�dxdt

�Z T

s

ZTd

jV�� j�dxdt CZ T

s

ZTd2

T � t

.T � s/2jwj�dxdt:

Thus,

w.x0; s/CZ T

s

ZTd

T � t

T � s

�2jwj2�dxdt

� C.T � s/CZ T

s

ZTd

T � t

T � s

�2jwj2�dxdt C

Z T

s

ZTd

1

.T � s/2�dxdt:

Consequently,

w.x0; s/ � C

T � s C 1

T � s

�:

ut

5.5 Lipschitz Regularity for the Heat Equation

Next, we give Lipschitz estimates for solutions of the heat equation. In MFGs, theseestimates are used to prove the regularity of solutions of

�ut C jDuj��

D �u C F.m/

once we know that jDuj� and F.m/ have enough integrability.

Theorem 5.11. Let u be a solution of

(ut.x; t/C�u.x; t/ D f in T

d � Œ0;T�;u.x;T/ D uT.x/ in T

d;(5.11)

with uT 2 W1;1.Td/ and f 2 La.Td � Œ0;T�/ with a > d C 2. Then,

kDukL1.Td�Œ0;T�/ � Ckf kLa.Td�Œ0;T�/:

Proof. Consider the adjoint equation to (5.11); that is,

�t.x; t/ ��.x; t/ D 0 (5.12)

5.5 Lipschitz Regularity for the Heat Equation 71

equipped with initial condition

�.�; �/ D ıx0 ; (5.13)

for arbitrary � 2 Œ0;T/ and x0 2 Td. Select � with

0 < � < 1: (5.14)

Proposition 4.8, with b D 0 yields

Z �

0

ZTd

jD��=2j2dx dt � �

4.1 � �/ < C; (5.15)

where the last inequality follows from the fact that 0 < � < 1.Next, we fix a unit vector, � 2 R

d. Accordingly, we have

u� .x0; �/ DZTd.uT/��.x;T/dx C

Z �

0

ZTd

f��.x; t/dxdt: (5.16)

Clearly,

ˇˇZTd.uT/�.x/�.x;T/

ˇˇ dx � kuTkW1;1.Td/:

Hence, it remains to bound Z �

0

ZTd

f��.x; t/dxdt:

For 0 < � < 1, we haveˇˇZ �

0

ZTd

f��

ˇˇ �

Z �

0

ZTd

jf j�1� �2 j� �

2�1D�j

�kf kLa.Td�Œ0;��/k�1� �2 kLb.Td�Œ0;��/kD�

�2 kL2.Td�Œ0;��/;

with

1

aC 1

bD 1

2; a; b � 1: (5.17)

In light of (5.15), it suffices to estimate

k�1� �2 kLb.Td�Œ0;��/:


For that, assume that there exists such that

1 � C 2

2��D

�; 0 � � 1: (5.18)

Set

b D �

.1 � �2/: (5.19)

By Hölder’s inequality, we have

� ZTd�b.1� �

2 /� 1

b.1� �2 / D

� ZTd�b.1� �

2 /� � �

� ZTd��1�� Z

Td�2��2

� 22��:

Sobolev’s inequality yields

ZTd�2��2

� 22� � C C C

ZTd

jD� �2 j2:

Hence, ZTd�b.1� �

2 / � C C CZTd

jD� �2 j2:

The above inequality implies that

Z �

0

ZTd�b.1� �

2 / � C C CZ �

0

ZTd

jD� �2 j2 � C:

The previous inequality concludes the proof once we check the existence of �, ,and b such that (5.14), (5.17), (5.18), and (5.19) hold. Elementary computationsshow the existence of those numbers for a > d C 2. ut

5.6 Irregular Potentials

Here, we prove Lipschitz estimates for the solutions of (5.1), assuming only suitableintegrability on V . As usual, we work with classical (smooth) solutions and regularpotentials, V . However, the bounds that we obtain depend only on the integrabilityproperties of V .

Theorem 5.12. Let u solve (5.1). Fix r > d. Suppose that V � 0. For any Qı > 0,there exists a constant, C > 0; such that

Lip.u/ � C C C kVk2L1.Œ0;T�;Lr.Td// C Qı kukL1.Td�Œ0;T�/ :

5.6 Irregular Potentials 73

Proof. First, integrating (5.1), we get

Z T

0

ZTd

jDuj2dxdt � CkVkL1.Œ0;T�;Lr.Td// C CkukL1.Td�Œ0;T�/:

Next, the adjoint equation (5.2) gives the following representation formula for u:

u.x0; 0/ DZ T

0

ZTd

jDuj22

C V

��dxdt C

ZTd

u.x;T/�.x;T/dx: (5.20)

The positivity of V thus ensures that

Z T

0

ZTd

jDuj2�dxdt � CkukL1.Td�Œ0;T�/:

Let 0 < � < 1. Combining the preceding estimate with (5.2), we get

Z T

0

ZTd

jD� �2 j2dxdt � C

Z T

0

ZTd

jDuj2�C jDuj2 C Cdxdt

� C C CkVkL1.Œ0;T�;Lr.Td// C CkukL1.Td�Œ0;T�/:

Now, we set v WD Dxi u and observe that v solves

�vt C Du � Dv ��v D �Dxi.V/:

Integrating with respect to �, we get

v.x0; �/ D �Z T

�

ZTd

Dxi.V/�dxdt CZTdv.x;T/�.x;T/dx:

Therefore, we have

jv.x0; �/j � C CZ T

0

ˇˇZTd

Dxi.V/�dx

ˇˇ dt: (5.21)

Next, we estimate the last term on the right-hand side of (5.21). We have

ZTd

Dxi.V/�dx D �ZTd

VDxi.�/dx D 2

�

ZTd

V�1��=2Dxi.��=2/dx:

Thus,

Z T

0

ˇˇZTd

Dxi.V/�dx

ˇˇ dt � C

Z T

0

ZTd

V2�2��dxdt C CZ T

0

ZTd

jD.��=2/j2dxdt:


We bound the first term of the previous inequality as follows:

ZTd

V2�2��dx � kV2kL

r2 .Td/

k�2��kL

rr�2 .Td/

D kVk2Lr.Td/k�k2��

L.2��/r

r�2 .Td/

:

Because d < r, we can select � close to 1 such that

.2 � �/rr � 2 � 2��

2D �d

d � 2(in dimension d D 2, we replace 2� by a large enough p). Then, by Sobolev’sinequality,

k�kL2��2 .Td/

� CkD.��=2/k 2�

L2.Td/C C: (5.22)

Using Hölder’s inequality, we get

k�kL.2��/r

r�2 .Td/� k�k1��1

L1.Td/k�k�1

L2��2 .Td/

� CkD.��=2/k2�1�

L2.Td/C C; (5.23)

where �1 is defined by

r � 2.2 � �/r D 1 � �1

1C 2�1

2��:

As � ! 1; we have �1 ! dr . Moreover, for � > 1 � 1

d C 1r , we have

.2 � �/�1�

< 1:

Then, Young’s inequality yields

Z T

0

ˇˇZTd

Dxi .V/�dx

ˇˇ dt � C kVk2L1.Œ0;T�;Lr.Td//

C ı1

Z T

0

ZTd

jD.��=2/j2�.2��/�1=�

C Cı1

� C kVk2L1.Œ0;T�;Lr.Td//C ı1

Z T

0

ZTd

jD.��=2/j2!.2��/�1=�

C Cı1

� C kVk2L1.Œ0;T�;Lr.Td//C Qı kukL1.Td�Œ0;T�/ C C:

(5.24)Hence,

Lip.u/ � C C C kVk2L1.Œ0;T�;Lr.Td// C Qı kukL1.Td�Œ0;T�/ :

ut

5.7 The Hopf–Cole Transform 75

Remark 5.13. The above theorem holds in a more general setting. Namely, it isenough to assume that u solves

�ut C H.x;Du/C V D �u;

where V; as before, is bounded in L1.Œ0;T�;Lr.Td/, HWTd � Rd ! R satisfies for

some constants, c;C > 0,

jDpH.x; p/j2 � Cjpj2 C C; (5.25)

DpH.x; p/p � H.x; p/ � cjpj2; (5.26)

and

jDxH.x; p/j � C C .x/jpjˇ; 0 � ˇ < 2; for some 2 L2r2�ˇ .Td/: (5.27)

5.7 The Hopf–Cole Transform

In this last section, we apply the Hopf–Cole transform and use the results in thepreceding section to get lower bounds on solutions of the Fokker–Planck equation.

Theorem 5.14. Let m W Td � Œ0;T� ! R, m > 0; solve

(mt C div.bm/ D �m;

m.x; 0/ D m0.x/;

where m0 is a given smooth function with m0 > k0 > 0; kDm0k1 � C andkbkL1 � C, k div.b/kLr � C, for some r > d: Assume that we have the boundˇR

Td ln m.x;T/dxˇ � C: Then, m is bounded from below.

Proof. We use the Hopf–Cole transform, w.x; t/ D � ln m.x;T � t/: Then, w solvesthe Hamilton–Jacobi equation,

�wt C jDwj2 C b � Dw � div.b/ D �w; wT D ln m0:

Integrating the equation in t and x and usingˇR

Td ln m.x;T/dxˇ � C; we get

kDwkL2 � C: Finally, we use Theorem 5.12 and Remark 5.13 to concludekD ln mk1 � C; which implies the result. ut



The nonlinear adjoint method was introduced in [91] as a tool to study thevanishing viscosity problem for non-convex Hamiltonians. The earlier work [170]used related ideas to investigate the L1 stability of Hamilton–Jacobi equations.Here, the discussion in Sects. 5.1–5.4 is partially based on [91]. The results onconserved quantities were explored in [50] to develop the Aubry–Mather theoryfor non-convex Hamiltonians. In the context of MFGs, the adjoint method wasused to get Lipschitz regularity for Hamilton–Jacobi equations with Lp potentials[129, 130, 135]. In Sect. 5.6, we follow [135]. The nonlinear adjoint method iscurrently an essential tool for the analysis of Hamilton–Jacobi equations. Someof its applications include stationary Hamilton–Jacobi equations [211], long-timebehavior of Hamilton–Jacobi equations [53, 181], the infinity Laplacian problem[96], convergence of numerical schemes [52], the Aubry–Mather theory for non-convex problems [50], non-convex Hamilton–Jacobi equations [92], obstacle andweakly coupled systems [51]. Some of the techniques in the Aubry–Mather theory[93–95] and its extensions [55, 89, 90, 109–111] are precursors to the adjointmethod. The adjoint equation also appears in optimal transport [29].

Chapter 6Estimates for MFGs

In the absence of special transformations or explicit solutions, the analysis ofMFGs often relies on a priori bounds. Here, we investigate estimates that arecommonly used. We begin by using the maximum principle to obtain one-sidedbounds. Next, we consider energy-type estimates that give additional bounds. Thesetwo techniques extend to a broad class of mean-field game problems. Equallyimportant are the consequences of these bounds when combined with earlier results.In Sect. 6.3, we develop some of these aspects. In the remainder of this chapter, wediscuss other methods that rely on the particular structure of the problems. First,we present a second-order estimate that is used frequently in the periodic setting.Next, we consider a technique that gives Lipschitz bounds for stationary first-orderMFGs. Subsequently, we examine energy conservation principles. Finally, we proveestimates for the Fokker–Planck equation that depend on uniform ellipticity orparabolicity of the MFG system.

One of the problems we consider is the periodic stationary MFG,

(��u C jDuj2

2C V.x/ D F.m/C H

��m � div.mDu/ D 0;(6.1)

where the unknowns are u W Td ! R, m W Td ! R, with m � 0 andR

m D 1, andH 2 R. The other problem we examine is the time-dependent MFG,

(�ut � ��u C jDuj2

2C V.x/ D F.m/

mt � ��m � div.mDu/ D 0;(6.2)

where T > 0, and the unknowns are u W Td � Œ0;T� ! R, m W Td � Œ0;T� ! R, withm � 0, together with the initial-terminal conditions


77

78 6 Estimates for MFGs

(u.x;T/ D uT.x/

m.x; 0/ D m0.x/:(6.3)

Here, m0 > 0 withR

m0 D 1.We suppose that V W Td ! R is a C1 function. The function F that encodes the

interactions between each agent and the mean field is either a real-valued function,F W RC ! R (or F W RC

0 ! R), or a function defined on the space of probabilitymeasures, F W P.Td/ ! C1.Td/. The former is called the local case, and the latteris the non-local case. For local problems, we suppose F.m/ that is C1 in the setm > 0. For some of the estimates, we require that F satisfies the following property:

ZTd

F.m/ � C C 1

2

ZTd

mF.m/: (6.4)

Here, we assume that all functions are C1. In particular, uT ;m0 2 C1.Moreover, we suppose that m and m0 are strictly positive. We say that .u;m;H/or .u;m/ is a classical solution of, respectively, (6.1) or (6.2)–(6.3), if u and m areC1, m > 0, .u;m/ solves (6.1) or (6.2), and, in the time-dependent case, (6.3) holds.

6.1 Maximum Principle Bounds

If F � 0, maximum principle techniques give important bounds for the solutions of(6.1) and (6.2). For stationary problems, the maximum principle implies one-sidedbounds on H. For time-dependent problems, the maximum principle yields lowerbounds for u.

Proposition 6.1. Let u be a classical solution of (6.1). Suppose that F � 0. Then,

H � supTd

V:

Proof. Because u is periodic, it achieves a minimum at a point, x0. At this point,Du.x0/ D 0 and �u � 0. Consequently,

V.x0/ � H C F.m/ � H:

Hence, H � sup V . utIn the case of time-dependent problems, we obtain bounds from below for the

solution, u.

Proposition 6.2. Let u be a classical solution of (6.2) and F � 0: Then, u isbounded from below.

6.2 First-Order Estimates 79

Proof. Since F � 0, we have

�ut � ��u C jDuj22

� �kVkL1.Td�Œ0;T�/:

The function v.x; t/ D �kuTk1 � .T � t/kVkL1.Td�Œ0;T�/ is therefore a subsolution.Hence, by the Comparison Principle given in Proposition 3.1, we have

u.x; t/ � �kuTkL1.Td/ � .T � t/kVkL1.Td�Œ0;T�/: ut

6.2 First-Order Estimates

First-order or energy estimates bound integral norms of classical solutions of first-or second-order MFGs. These estimates are obtained by the multiplier method. Inthe two examples considered here, we multiply the Hamilton–Jacobi equation andthe Fokker–Planck equation by suitable functions of m and u, respectively. Theestimates follow by integration by parts and elementary inequalities. While theproofs are straightforward, these estimates are essential in the theory of MFGs. Inthe next section, we examine some consequences of these results. A more generaltechnique is developed in Sect. 6.8.

Proposition 6.3. There exists a constant, C; such that, for any classical solution,.u;m;H/, of (6.1), we haveZ

Td

jDuj22

.1C m/C 1

2F.m/mdx � C: (6.5)

Proof. Multiply the first equation in (6.1) by .m�1/ and the second equation by �u.Adding the resulting expressions and integrating by parts givesZ

Td

jDuj22

.1C m/C mF.m/dx DZTd

V.m � 1/C F.m/dx: (6.6)

Using (6.4) in the preceding identity gives (6.5). utNext, we obtain a bound for H.

Corollary 6.4. Let .u;m;H/ be a classical solution of (6.1). Suppose that F � 0.Then, there exists a constant, C; not depending on the particular solution, such that

jHj � C:

Proof. By Proposition 6.3, we have jDuj22

2 L1: As a result of (6.4), we have F.m/ 2L1. Therefore, integrating the first equation of (6.1), we obtain the bound for H. utRemark 6.5. An important case not covered by the previous Corollary isF.m/ D ln m. Here, Proposition 6.3 combined with (6.6) provides the bound


ZTd

ln mdx � C:

Because m 7! m ln m is bounded from below, (6.6) also gives the opposite bound.Therefore, by integrating the first equation in (6.1), we conclude that H is bounded.

Now, we focus our attention on the time-dependent problem (6.2) and provebounds to (6.5).

Proposition 6.6. There exists a constant, C > 0; such that, for any classicalsolution, .u;m/, of (6.2), we haveZ

Td

Z T

0

.m C m0/jDuj22

C mF.m/dtdx � C:

Proof. Multiply the first equation in (6.2) by .m � m0/ and the second equation by.uT � u/. Adding the resulting expressions and integrating in T

d � Œ0;T� gives

0 DZTd

Z T

0

Œ.m � m0/.uT � u/t C .uT � u/.m � m0/t� dtdx (6.7)

CZTd

Z T

0

Œ.�.m � m0/�.uT � u/ � �.uT � u/�.m � m0/� dtdx

CZTd

Z T

0

Œ��.m � m0/�uT � �.uT � u/�m0� dtdx

CZTd

Z T

0

�.m � m0/

jDuj22

C u div.mDu/

dtdx

CZTd

Z T

0

Œ�uT div.mDu/C .m � m0/V.x/� dtdx

CZTd

Z T

0

.m0 � m/F.m/dtdx:

Using the boundary conditions, we haveZTd

Z T

0

Œ.m � m0/.uT � u/t C .uT � u/.m � m0/t� dtdx D 0;

and

ZTd

Z T

0

Œ�.m � m0/�.uT � u/ � �.uT � u/�.m � m0/� dtdx D 0:

6.3 Additional Estimates for Solutions of the Fokker–Plank Equation 81

Because m and m0 are probability measures and uT is of class C1, there exists apositive constant, C; such that

ˇˇZTd

Z T

0

Œ��.m � m0/�uT � �uT�m0� dtdx

ˇˇ � C:

Furthermore, for any ı > 0 there exists a constant, C > 0; such that

ˇˇZTd

Z T

0

�u�m0dtdx

ˇˇ � ı

ZTd

Z T

0

jDuj2dtdx C C:

Similarly, we have

ˇˇZTd

Z T

0

�uT div.mDu/dtdx

ˇˇ � ı

ZTd

Z T

0

jDuj2mdtdx C C:

Finally, we get

ˇˇZTd

Z T

0

.m � m0/V.x/dtdx

ˇˇ � C:

Using the preceding identities and estimates in (6.7), selecting ı D 14, and

integrating by parts, we obtain

ZTd

Z T

0

�.m C m0/

jDuj24

C mF.m/

dtdx � C C

ZTd

Z T

0

m0F.m/dtdx:

The statement follows by using (6.4) in the previous estimate. ut

6.3 Additional Estimates for Solutions of the Fokker–PlankEquation

Here, we examine uniformly parabolic MFG. To simplify, we set � D 1 in (6.2). Ourresult combines the first-order estimates with the regularity for the Fokker–Planckequation to get estimates for derivatives of m.

Proposition 6.7. Let .u;m/ solve (6.2). Then, there exists a positive constant, C,independent of the solution such that

Z T

0

ZTd

jD ln mj2 C jDm1=2j2dxdt � C:


Proof. The result follows by combining Proposition 6.6 with Corollaries 4.6and 4.7. utCorollary 6.8. Let .u;m/ solve (6.2). Suppose that F.m/ D m˛ for some ˛ > 0.Then, there exists a positive constant, C, independent of the solution such that

Z T

0

ZTd

jDmjqdxdt � C;

where q D 21C˛2C˛ .

Proof. By Hölder’s inequality, for any s � 0 and 1r C 1

r0

D 1, we have

Z T

0

ZTd

jDmjqdxdt �Z T

0

ZTd

jDmjqr

msrdxdt

�1=r Z T

0

ZTd

msr0

dxdt

�1=r0

:

By Propositions 6.7 and 6.6, we have bounds for, respectively,R T0

RTd

jDmj2m andR T

0

RTd m˛C1. We therefore select

8<ˆ:

qr D 2

sr D 1

sr0 D ˛ C 1:

(6.8)

Solving (6.8) gives q D 21C˛2C˛ . ut

6.4 Second-Order Estimates

Now, we discuss a second-order estimate for mean-field games that gives furtherregularity for the solutions. Remarkably, these second-order estimates are valid forclassical solutions of first-order mean-field games.

6.4.1 Stationary Problems

For stationary problems, the second-order estimates provide a bound on the Hessianof u with respect to the measure, m.

Proposition 6.9. Let .u;m/ be a classical solution of (6.1) with V 2 C2. Assumethat F.m/ is local. Then, there exists a constant, C > 0; that does not depend on thesolution .u;m/ such that

6.4 Second-Order Estimates 83

ZTd

jD2uj2m C F0.m/jDmj2dx � C:

Proof. By applying the operator � to the first equation of (6.1), we obtain

��2u C�V C jD2uj2 C Du � D�u � div.F0.m/Dm/ D 0:

Integrating with respect to m and using

ZTtd

��2u C Du � D�u

�mdx D �

ZTd�u .��m � div.Dum// dx D 0;

we get

ZTd

F0.m/jDmj2dx CZTd

jD2uj2mdx �ZTd

j�Vjmdx � C:

ut

6.4.2 Time-Dependent Problems

The next proposition is the counterpart of Proposition 6.9 for time-dependent MFGs.

Proposition 6.10. Let .u;m/ be a classical solution of (6.2) with m0; uT ;V 2 C2.Assume that F.m/ is local. Then, there exists a constant, C > 0; that is independentof the solution, .u;m/; such that

Z T

0

ZTd

jD2uj2m C F0.m/jDmj2dxdt � C C Cku.�; 0/kL1.Td/:

Proof. Applying the operator � to the first equation of (6.2), we have

�.�u/t � ��2u C�V C jD2uj2 C Du � D�u � div.F0.m/Dm/ D 0:

Multiplying by m, integrating in x and t, using integration by parts and the equationfor m, we get

Z T

0

ZTd

jD2uj2m C F0.m/jDmj2dxdt

D �Z T

0

ZTd�Vmdxdt C

ZTd

m.x;T/�u.x;T/dx �ZTd�m.x; 0/u.x; 0/dx:


Consequently,

Z T

0

ZTd

jD2uj2m C F0.m/jDmj2dxdt � C C Cku.�; 0/kL1.Td/:

utRemark 6.11. If F � 0, the boundedness of ku.�; 0/kL1.Td/ follows from the lowerbounds in Proposition 6.2 combined with the bound

ZTd

u.x; 0/dx � C

from Proposition 6.6.

6.5 Some Consequences of Second-Order Estimates

Next, we continue our study of the parabolic case with � D 1 and combinethe previous estimates with the regularity results for the Fokker–Planck equationobtained earlier.

Proposition 6.12. Let .u;m/ solve (6.2)–(6.3) with � D 1. Suppose that F.m/ Dm˛ for some ˛ > 0. Then, D.mDu/ 2 L1.Td � Œ0;T�/.Proof. We have

Dxi.mDxj u/ D Dxi mDxj u C mD2xixj

u:

Next,

jDxi mDxj uj � jDmjm1=2

m1=2jDuj � jDmj22m

C mjDuj22

:

The expression on the right-hand side is in L1.Td � Œ0;T�/ due to Propositions 6.7and 6.6. Finally, we have

jmD2xixj

uj � m

2C mjD2uj2

2:

Because m is a probability measure and because of Proposition 6.10, the right-handside in the previous bound is also integrable. ut

According to the preceding estimate mt ��m 2 L1.

6.6 The Evans Method for the Evans–Aronsson Problem 85

6.6 The Evans Method for the Evans–Aronsson Problem

The Evans–Aronsson problem consists of minimizing the integral functional,

ZTd

ejDuj

2

2 CV.x/dx; (6.9)

among all functions, u 2 W1;1.Td/. A smooth enough minimizer solves the Euler–Lagrange equation,

� div

e

jDuj

2

2 CV.x/Du

�D 0: (6.10)

Because the functional (6.9) is convex, any solution to the Euler–Lagrange equationis a minimizer. Here, we prove a priori Lipschitz bounds for any solution, u, of(6.10). Because of this estimate, the methods we develop later give the existence ofa solution for (6.10) and, consequently, of a minimizer for (6.9).

Remarkably, (6.10) can be written as an MFG. For that, we set

m D ejDuj

2

2 CV.x/�H;

where H is chosen such thatRTd m D 1. Thus,

( jDuj22

C V.x/ D ln m C H

� div.mDu/ D 0:(6.11)

Because the function m 7! m ln m is bounded from below, Proposition 6.3 givesDu 2 L2. In addition, by Remark 6.5, we have that H is bounded. In the nextproposition, we prove our main result, the Lipschitz regularity for u.

Proposition 6.13. Let .u;m;H/ solve (6.11). Then, there exists a constant, C > 0,independent of the solution such that kDukL1 � C.

Proof. We begin by multiplying the second equation in (6.11) by div.mpDu/. Afterthat, we integrate on T

d, integrate by parts, and apply the identity

Xi;j

ZTd.muxi/xi.m

puxj/xj DX

i;j

ZTd.muxi/xj.m

puxj/xi

to get

ZTd

mpC1Xi;j

jDxixj uj2 C pmp�1jDm � Duj2 C .p C 1/mpD2xixj

uDxi uDxj m D 0:

(6.12)


Next, we differentiate the first equation in (6.11) and get

Xi

Dxi uD2xixj

u C Dxj V D Dxj m

m:

Next, we multiply the preceding identity by mpmxj to conclude that

ZTd

mpX

i;j

D2xixj

uDxi uDxj m DZTd

mp�1jDmj2 �ZTd

mpX

j

Dxj mDxj V:

Combining the prior identity with (6.12) givesZTd

mpC1X

i;j

jDxixj uj2 C pmpjDm � Duj2 C .p C 1/

ZTd

mp�1jDmj2

D .p C 1/

ZTd

mpX

j

Dxj mDxj V � p C 1

2

ZTd

mp�1jDmj2 C C.p C 1/

ZTd

mpC1

for some positive constant, C; independent of p. Accordingly, we have the estimate

ZTd

mp�1jDmj2 � CZTd

mpC1: (6.13)

By Sobolev’s theorem, we have

�ZTd

m2�.pC1/

2

12� � C

�ZTd

mpC1 CZTd

jDmpC12 j2

12

� C.1C jpj/�Z

TdmpC1

12

:

Thus,

kmkL2�.pC1/

2

� C.1C jpj/ 2pC1 kmkLpC1 : (6.14)

To finish the proof, we use Moser’s iteration method. First, we define thesequence pn D �n for some 1 < � < 2�

2. Because m � 0 and

Rm D 1, we

have kmkp0 D 1. Suppose that

1

�D ˛ C 1 � ˛

2�=2:

Then,

kmkpnC1� kmk˛pn

kmk1�˛2�pn=2:

6.7 An Energy Conservation Principle 87

By (6.14), we obtain

kmkpnC1� Cjpnj.1�˛/ 2pn kmk˛pn

kmk.1�˛/pn� Cjpnj.1�˛/ 2pn kmkpn :

By induction, we get

kmkpnC1� C‚n;

where ln‚n � PnjD1

CCln pj

pj. Because the previous series is convergent, we have

kmkLq � C for all 1 � q < 1; that is, C is independent of q. Hence, m 2 L1. Thus,the first equation in (6.11) gives Du 2 L1. ut

6.7 An Energy Conservation Principle

Here, we give a conservation of energy principle for time-dependent MFGs. Thisenergy conservation principle is essential to the study of the long-time limit ofMFGs.

Let ˆ W R ! R be an increasing function. We consider the time-dependentmean-field game

(�ut C H.x;Du/ D ��u Cˆ0.m/;mt � div.DpH.x;Du/m/ D ��m:

(6.15)

Then, we have

Proposition 6.14. Let .u;m/ solve (6.15). Then,

d

dt

ZTd

Hm �ˆ.m/C �DuDm D 0:

Proof. We have

d

dt

ZTd

Hm �ˆ.m/C �DuDm

DZTd.H �ˆ0.m//mt C DpHDxutm C �.DutDm C DuDmt/

DZTd.H �ˆ0.m/ � ut � ��u/mt D 0: ut


While the energy conservation provides strong estimates for solutions to (6.15),its application is somewhat restricted as it depends on the specific form of theequations. For example, no energy conservation principle is known for generalMFGs.

6.8 Porreta’s Cross Estimates

We end this chapter with a class of estimates that involve the solution .u;m/ to (6.2).These estimates build upon the idea that �.u/ and .m/ are approximate solutionsto (6.2). Therefore, using a multiplier method, we gain control over several integralquantities. These estimates give compactness for approximate solutions of (6.2). Webegin our discussion with an auxiliary identity.

Lemma 6.15. Let � D 1 and V D 0. Assume that .u;m/ solves (6.2)–(6.3) andsuppose that �; 2 C2. Then,

Z T

0

ZTd

h�.�.uT// � DuD.�.uT//

C F.m/�0.u/C��00.u/C �0.u/

2

jDuj2

i. .m/ � .m0//dxdt

DZ T

0

ZTd

hdiv.. .m0/Du/C�. .m0//:

� 00.m/jDmj2 � . .m/ � 0.m/m/�ui.�.u/ � �.uT//dxdt:

Proof. We fix two convex increasing functions, � and . To begin with, we multiplythe Hamilton–Jacobi equation in (6.2) by �0.u/ to get

� .�.u//t ��.�.u//C DuD.�.u// D F.m/�0.u/C��00.u/C �0.u/

2

jDuj2:

(6.16)

Next, we multiply the Fokker–Planck equation by 0.m/ and get

. .m//t � div. .m/Du/ ��. .m// D � 00.m/jDmj2 � . .m/ � 0.m/m/�u:(6.17)

Now, to cancel the boundary conditions, we rewrite (6.16) and (6.17) as

� .�.u/ � �.uT//t ��.�.u/ � �.uT//C DuD.�.u/ � �.uT// (6.18)

D �.�.uT// � DuD.�.uT//C F.m/�0.u/C��00.u/C �0.u/

2

jDuj2

6.8 Porreta’s Cross Estimates 89

and

. .m/ � .m0//t � div .. .m/ � .m0//Du/ ��. .m/ � .m0// (6.19)

D div. .m0/Du/C�. .m0// � 00.m/jDmj2 � . .m/ � 0.m/m/�u:

Next, we multiply (6.18) by .m/� .m0/ and (6.19) by �.u/��.uT/. Finally, wesubtract the resulting expressions and integrate in T

d � Œ0;T�. These operations givethe desired identity. utCorollary 6.16. Let � D 1 and V D 0. Assume that m0 is of class C1 and boundedfrom below, m0 � 0 > 0, uT > 1 is C2, F is non-decreasing and non-negative.Suppose that uT � 1. Then, for any solution .u;m/ to (6.2)–(6.3) and r � 1, wehave

Z T

0

ZTd

mF.m/ur�1dxdt CZ T

0

ZTd

ur�1jDuj2.m C m0/dxdt �

CZ T

0

ZTd

ur�1dxdt C cr;

where the constants cr;C depend only on r;F; km0kC1 and kuTkC2 .

Remark 6.17. Proposition 6.6 is a particular case of the previous result.

Remark 6.18. The condition uT � 1 simplifies the statement because u � 1.A similar result holds if uT is bounded from below. Because F is non-negative, uis bounded from below. The same technique gives an estimate for .u C k/r for someconstant, k.

Proof. Let �.u/ D ur and .m/ D m in Lemma 6.15. Then,Z T

0

ZTd

��.�.uT//� DuD.�.uT//C F.m/rur�1 C

��00.u/C �0.u/

2

jDuj2

.m � m0/dxdt

DZ T

0

ZTdŒdiv.m0Du/C�.m0/� .�.u/� �.uT//dxdt

DZ T

0

ZTd

��0.u/m0jDuj2 C m0DuD .�.uT//� D.m0/D .�.u//� m0� .�.uT// dxdt:

After some cancellations, we gather

Z T

0

ZTd

F.m/rur�1.m � m0/dxdt CZ T

0

ZTd

�0.u/2

jDuj2.m C m0/dxdt

DZ T

0

ZTd�00.u/.m � m0/jDuj2 C mDuD .�.uT// � D.m0/D .�.u//

� m�.�.uT// dxdt:


Since F is non-decreasing and non-negative, we have

F.m/m0 � 1

2mF.m/1m�2m0 C m0F.2m0/1m�2m0 � 1

2mF.m/C C.F; km0k1/:

Thus,

Z T

0

ZTd

1

2mF.m/ur�1dxdt C 1

2

Z T

0

ZTd

ur�1jDuj2.m C m0/dxdt

� .r � 1/Z T

0

ZTd

ur�2.m � m0/jDuj2dxdt �Z T

0

ZTd

ur�1D.m0/Dudxdt

C 1

r

Z T

0

ZTd.mDuD .�.uT// � m�.�.uT/// dxdt C C.F; km0k1

/

Z T

0

ZTd

ur�1dxdt:

For the first three terms on the right-hand side of the above equality, we have

.r � 1/Z T

0

ZTd

ur�2.m � m0/jDuj2dxdt � ı

Z T

0

ZTd.m C m0/u

r�1jDuj2dxdt

C cı.r/Z T

0

ZTd.m C m0/jDuj2dxdt;

with cı.1/ D 0,

�Z T

0

ZTd

ur�1D.m0/Dudxdt � ı

Z T

0

ZTd

ur�1jDuj2dxdt

C C.ı; km0kC1 /

Z T

0

ZTd

ur�1dxdt;

and

Z T

0

ZTdŒmDuD .�.uT// � m�.�.uT//� dxdt � C.ı; kuTkC2 /C ı

Z T

0

ZTd

mjDuj2dxdt:

Hence,

Z T

0

ZTd


0

ZTd

ur�1jDuj2.m C m0/dxdt �

cr

Z T

0

ZTd.m C m0/jDuj2dxdt C C.F; km0kC1 ; kuTkC2 /

1C

Z T

0

ZTd

ur�1dxdt

�:

To end the proof, we apply Proposition 6.6. ut


Corollary 6.19. Let � D 1 and V D 0; assume that m0 is in C1 and bounded frombelow, m0 � 0 > 0, uT is C2, F is non-decreasing and non-negative. Then, for anysolution .u;m/ to (6.2)–(6.3) and r � 1, we have

ZTd.u.x; t//rdx C

Z T

t

ZTd


t

ZTd

ur�1jDuj2.m C 1/dxdt � Cr:

Proof. Integrating (6.16) for �.u/ D ur, we get

ZTd.u.x; t//rdx D

ZTd

urTdx C

Z T

t

ZTd

rF.m/ur�1dxdsCZ T

t

ZTd

h�r.r � 1/ur�2 � r

2ur�1i jDuj2dxds:

Thus, from Corollary 6.16 and F.m/ � F.1/C mF.m/, we get

ZTd.u.x; t//rdx � C C Cr

Z T

t

ZTd

urdxds:

By Gronwall’s inequality,

supŒ0;T�

ZTd.u.x; t//rdx � Cr:

Corollary 6.16 concludes the proof. utNext, we organize the identity Lemma 6.15 in a more convenient form.

Lemma 6.20. Let � D 1 and V D 0. Let .u;m/ solve (6.2)–(6.3). Then,

Z T

0

ZTd

F.m/�0.u/ .m/dxdt CZ T

0

ZTd

�0.u/2

jDuj2. .m/C .m0//dxdt

CZ T

0

ZTd 00.m/jDmj2�.u/dxdt C

Z T

0

ZTd

m 00.m/DmDu�.u/dxdt

DZ T

0

ZTd

F.m/�0.u/ .m0/dxdtCZ T

0

ZTd

��00.u/. .m/ � .m0//C . .m/ � 0.m/m/�0.u/

� jDuj2dxdt

ZTd. .m0/ � .mT// �.uT/dx C

Z T

0

ZTd�. .m0//�.u/dxdt:


Proof. From Lemma 6.15, after integrating by parts, we have

Z T

0

ZTd


0

ZTd

�0.u/2


CZ T

0

ZTd

� 00.m/jDmj2 C . .m/ � 0.m/m/�u

�.�.u/ � �.uT//dxdt

DZ T

0

ZTd

F.m/�0.u/ .m0/dxdt CZ T

0

ZTd�00.u/jDuj2. .m/ � .m0//dxdt

CZ T

0

ZTd.� div. .m/Du/ ��. .m/// �.uT/C�. .m0//�.u/dxdt:

Next, using Eq. (6.17), we obtain

Z T

0

ZTd


0

ZTd

�0.u/2


CZ T

0

ZTd

� 00.m/jDmj2 C . .m/ � 0.m/m/�u

��.u/dxdt

DZ T

0

ZTd

F.m/�0.u/ .m0/dxdt CZ T

0

ZTd�00.u/jDuj2. .m/ � .m0//dxdt

CZ T

0

ZTdŒ.� .m//t �.uT/C�. .m0//�.u/� dxdt:

Furthermore,

Z T

0

ZTd. .m/ � 0.m/m/�.u/�udxdt D

Z T

0

ZTd

m 00.m/DmDu�.u/dxdt

�Z T

0

ZTd. .m/ � 0.m/m/�0.u/jDuj2dxdt:

Hence, we have the claim. utCorollary 6.21. Let � D 1 and V D 0. Assume that m0 is of class C1 and boundedfrom below, m0 � 0 > 0, and that uT is of class C2. In addition, suppose that Fis non-decreasing and non-negative. Then, there exist �; � > 0 such that, for anysolution .u;m/ of (6.2)–(6.3), we have

Z T

0

ZTd

F.m/e�um1C� C e�ujDuj2.m1C� C 1/C m��1jDmj2e�udxdt � C:


Proof. Let �.u/ D ��1e�u and .m/ D m1C� . Then, from the identity inLemma 6.20, we get

Z T

0

ZTd

F.m/e�um1C�dxdt CZ T

0

ZTd

e�u

2jDuj2.m1C� C m1C�

0 /dxdtC

��1�.1C �/

Z T

0

ZTd

m��1jDmj2e�udxdt C ��1�.1C �/

Z T

0

ZTd

m�DmDue�udxdt

DZ T

0

ZTd

F.m/e�um1C�0 dxdt C

Z T

0

ZTd

�.� � �/m1C� � �m1C�

0

�e�ujDuj2dxdtC

��1ZTd

�m1C�0 � m1C�

T

�e�uT dx �

Z T

0

ZTd��1e�u�.m1C�

0 /dxdt:

(6.20)Next, we take � < � < 1

4. Then, using

F.m/m1C�0 � ıF.m/m1C� C C.ı;F;m0/

and weighted Cauchy inequalities, we can absorb the first two terms of the right-hand side into the first two terms of the left-hand side. Accordingly, we have

1

2

Z T

0

ZTd

F.m/e�um1C�dxdt C 1

4

Z T

0

ZTd

e�ujDuj2.m1C� C m1C�0 /dxdtC

��1�.1C �/

Z T

0

ZTd

m��1jDmj2e�udxdt C ��1�.1C �/

Z T

0

ZTd

m�DmDue�udxdt �

��1

ZTd

�m1C�0 � m1C�

T

�e�uT dx C C

Z T

0

ZTd

e�udxdt:

Choosing a small enough � such that ��1�.1C �/ < 1; we can further absorb thefourth term on the left-hand side of the above inequality:

Z T

0

ZTd

F.m/e�um1C�dxdt CZ T

0

ZTd

e�ujDuj2.m1C� C m1C�0 /dxdtC

��1�.1C �/

Z T

0

ZTd

m��1jDmj2e�udxdt � C C CZ T

0

ZTd

e�udxdt:

To finish the proof, we need to estimateR T0

RTd e�udxdt.

First, we consider the case when � D 0. Accordingly, the preceding equationgives

Z T

0

ZTd

F.m/e�umdxdt � C C CZ T

0

ZTd

e�udxdt: (6.21)


On the other hand, integrating (6.16), we obtain

ZTd

e�u.x;t/dx �ZTd

e�uT dx C �

�C 1

2

�Z T

t

ZTd

jDuj2e�udxds

D �

Z T

t

ZTd

F.m/e�udxds: (6.22)

Combining the foregoing estimates, we get

ZTd

e�u.x;t/dx CZ T

0

ZTd

jD.e �2 u/j2dxdt � C C C

Z T

0

ZTd

e�udxds: (6.23)

Let v D e�2 u.x;t/. By the Gagliardo–Nirenberg inequality, we have

Z T

0

ZTdv2C4=ddxdt � sup

Œ0;T�

ZTd

jv.x; t/j2� 2

dZ T

0

ZTd

jDvj2dxdt

�:

Thus,

Z T

0

ZTd

e�.1C2=d/udxdt � C

1C

Z T

0

ZTd

e�udxdt

�1C2=d!:

Using Hölder’s inequality

Z T

0

ZTd

e�udxdt

�1C 2d

� C.Te�N/1C 2d C Cjfu > Ngj 2

dC2

Z T

0

ZTd

e�.1C2=d/udxdt:

Since jfu > Ngj � kuk1N � C

N by Corollary 6.19, choosing a large enough N, gives

Z T

0

ZTd

e�udxdt � C:

To end the proof, we observe that the case � > 0 is immediate because we have

Z T

0

ZTd

e�udxdt � C:

ut



The estimates in Sects. 6.1–6.4 appeared first in [164–166]. A version of the second-order estimates was introduced in [90] for stationary problems. In the context of theAubry–Mather theory, similar bounds appeared in [93, 110, 111]. The discussionin Sect. 6.6 is a simplified version of the argument in [89]. The results in Sects. 6.5and 6.8 are taken from [195] (also see [196]). The energy conservation identity wasused in [66] and [67] in the study of the long-time convergence of mean-field games.

Other bounds for stationary MFGs that rely only on elementary methods can befound in [127, 129] and [213]. The methods considered here can be generalized tomany other cases including obstacle-type problems [116], weakly coupled MFGs[115], and multi-population models [77].

Chapter 7A Priori Bounds for Stationary Models

We draw upon our earlier results to study stationary MFGs. Here, we illustratevarious techniques in three models. First, we use the Bernstein estimates givenin Theorem 3.11, to obtain Sobolev estimates for the value function. Next, weconsider a congestion problem and show, through a remarkable identity, that m > 0.Finally, we examine an MFG with a logarithmic nonlinearity. This model presentssubstantial challenges since the logarithm is not bounded from below. However, aclever integration by parts argument gives the necessary bounds for its study.

7.1 The Bernstein Method

We fix a C2 potential, V W Td ! R, and look for a solution, .u;m;H/, of the MFG

8<ˆ:


C V.x/ D NH C m˛;

��m � div .mDu/ D 0;Rudx D 0;

Rmdx D 1;

(7.1)

with u;m W Td ! R and H 2 R. Here, we establish a preliminary result, namely thatu 2 W1;q for any q > 1. For that, we combine the integral Bernstein estimates inChap. 3 with the first-order estimates in Chap. 6. In Chap. 11, we use these estimatesto prove the existence of a classical solution of (7.1).

Theorem 7.1. Let .u;m;H/ solve (7.1) and 0 < ˛ < 1d�1 . Suppose that u;m 2

C2.Td/. Then, for every q > 1; there exists a constant, Cq > 0; that depends onlyon kVk

L1C1˛ .Td/

, such that kDukLq.Td/ � Cq.


97

98 7 A Priori Bounds for Stationary Models

Proof. To prove the claim, we use Theorem 3.11 with V replaced by V.x/ � m˛ .By Proposition 6.3 and Corollary 6.4, we have

j NHj � C; km˛kL1C

1˛ .Td/

� C:

Because d < 1C 1˛

, we have that for p large �p < 1C 1˛

. Thus, Theorem 3.11 gives

kDukLq.Td/ � Cq for every q > 1: ut

7.2 A MFG with Congestion

The second problem we examine in this chapter is the MFG with congestiongiven by

(u ��u C jDuj2

2m˛ C V.x/ D 0

m ��m � div�m1�˛Du

� D 1;(7.2)

where u;m 2 C2.Td/ and m � 0. In addition, we suppose that V 2 C.Td/ and0 < ˛ < 1. We use the particular structure of (7.2) to prove that 1

m is bounded.First, we state an auxiliary Lemma:

Lemma 7.2. There exists a constant, C WD C.kVk1/ � 0; such that, for anyclassical solution, .u;m/; of (7.2), we have

kukL1.Td/ � C: (7.3)

Furthermore, any solution, m; to the second equation in (7.2) is a probabilitydensity; that is, m � 0 on T

d and kmkL1.Td/ D 1.

Proof. To get the L1 bound, we evaluate the first equation in (7.2) at a point ofmaximum of u (resp., minimum). At that point Du D 0, �u � 0 (resp., � 0) andV is bounded on T

d. Thus, (7.3) follows. If we argue as in Proposition 4.3, then,m is non-negative. Furthermore, it has a total mass of 1 by integrating the secondequation in (7.2). ut

In the next proposition, we improve the previous lemma and prove that m isstrictly positive.

Proposition 7.3. There exists a constant, C WD C.kVk1/ � 0; such that for anyclassical solution, .u;m/; of (7.2), we have

�� 1m��

L1.Td/

� C:

7.2 A MFG with Congestion 99

Proof. Let r > ˛. For the proof, we first establish the identity:ZTd

1

.r C 1 � ˛/mrC1�˛ dx CZTd

jDuj22rmrC˛ dx C

ZTd

jDmj2mrC2�˛ dx

DZTd

�� V

rmr� u

rmrC 1

.r C 1 � ˛/mr�˛

dx:

(7.4)

To prove the above, we subtract the second equation of (7.2) divided by .r C 1 �˛/mrC1�˛ from the first equation of (7.2) divided by rmr. Then,Z

Td

hu ��u C jDuj2

2m˛C V

i� 1

rmrdx (7.5)

�ZTd

hm ��m � div

�m1�˛Du

�i � 1

.r C 1 � ˛/mrC1�˛ dx

D �ZTd

1

.r C 1 � ˛/mrC1�˛ dx:

Next, we observe that ZTd

�u

rmrdx D

ZTd

Du � Dm

mrC1 dx

and

ZTd

div�m1�˛Du

�.r C 1 � ˛/mrC1�˛ dx D

ZTd

Du � Dm

mrC1 dx:

Hence,

ZTd

�u

rmrdx �

ZTd

div�m1�˛Du

�.r C 1 � ˛/mrC1�˛ dx D 0: (7.6)

Therefore, (7.5) is reduced to (7.4).Now, we note that Lemma 7.2 combined with (7.4) gives

ZTd

1

.r C 1 � ˛/mrC1�˛ dx CZTd

jDuj22rmrC˛ dx C

ZTd

jDmj2mrC2�˛ dx

�ZTd

C

rmrdx C

ZTd

C

.r � ˛/mr�˛ dx:

By Young’s inequality for ˛ 2 Œ0; 1/, we have

C

rmr� 1

4.r C 1 � ˛/mrC1�˛ C C1r


and

C

.r � ˛/mr�˛ � 1

4.r � ˛/mrC1�˛ C C2r

with

C1r WD .1 � ˛/4 r

1�˛ CrC1�˛1�˛

r.r C 1 � ˛/ ; C2r WD 4r�˛CrC1�˛.r � ˛/r�˛�1

.r C 1 � ˛/rC1�˛ :

Therefore,

1

r C 1 � ˛ZTd

1

mr�˛C1 � 2.C1r C C2

r /:

Thus, we get

�� 1m��

LrC1�˛.Td/

�h2.r C 1 � ˛/.C1

r C C2r /i 1

rC1�˛ DW C˛.r/:

Finally, we check that, for any r0 > ˛; there exists C˛ for which

C˛.r/ � C˛; for all r 2 Œr0;1/: ut

7.3 Logarithmic Nonlinearity

Because ln m is not bounded from below, MFGs with logarithmic nonlinearitiespresent substantial challenges. Here, we collect some estimates to overcome theseissues. For .u;m;H/, u;m W Td ! R and H 2 R, we consider the problem

8<ˆ:


C V.x/ D NH C ln m;

��m � div .mDu/ D 0;Rudx D 0;

Rmdx D 1:

(7.7)

Proposition 7.4. Let .u;m/ solve (7.7). Then, there exists a constant, C > 0; thatdepends only on kVkL1.Td/ and j NHj, such that k ln mkH1.Td/ � C:

Proof. Integrating the first equation in (7.7), we get

ZTd

jDuj22

dx D �ZTd

Vdx C NH CZ

ln mdx:

7.3 Logarithmic Nonlinearity 101

Using Jensen’s inequality, we have

0 �ZTd

ln mdx � �C CZTd

jDuj22

dx � �C:

Therefore,

ZTd

jDuj22

dx;

ˇˇZTd

ln mdx

ˇˇ � C:

Multiplying the second equation in (7.7) by 1m , integrating by parts and using the

above, we get

ZTd

jD ln mj2dx � CZTd

jDuj2dx � C: (7.8)

Finally, the foregoing bound and the Poincaré inequality give

ZTd

j ln mj2dx � C

"ZTd

ln mdx

�2CZTd

jD ln mj2dx

#� C:

utRemark 7.5. The estimate (7.8) is a stationary version of the result in Proposi-tion 4.5.

The previous proposition can be improved as follows.

Proposition 7.6. Let .u;m;H/ solve (7.7). Then, for every 1 � p < 1, there existsa constant, Cp > 0; that does not depend on the solution, such that

kj ln mjpkH1.Td/ � Cp:

Proof. We prove by induction that fk D j ln mj kC12 2 H1.Td/ for any k 2 N. The

case k D 1 is given by Proposition 7.4. Let l � 1 and suppose that kfkkH1.Td/ � Cl

for all k � l. Then, we have

kDfkk2L2 DZTd

j ln mjk�1

m2jDmj2dx � C2

l

and

kfkk2L2 DZTd

j ln mjkC1dx � C2l :


Next, we show that flC1 2 H1.Td/: Let Fl.z/ D R z1

j ln yjly2

dy. Multiplying the secondequation of (7.7) by Fl.m/ and integrating by parts, we get

ZTd

j ln mjlm2

jDmj2dx D �ZTd

j ln mjlm

DmDudx � 1

2

ZTd

j ln mjlm2

jDmj2dx

C 1

2

ZTd

j ln mjljDuj2dx:

Thus,

ZTd

j ln mjlm2

jDmj2dx �ZTd

j ln mjljDuj2dx: (7.9)

From the first equation of (7.7), we infer that

jDuj22

� C C j ln mj C�u:

Therefore, multiplying by j ln mjl and integrating gives

ZTd

j ln mjljDuj2dx � CZTd

j ln mjldx C CZTd

j ln mjlC1dx CZTd�uj ln mjldx:

Integrating by parts the last term yields

ZTd


j ln mjldx C CZTd

j ln mjlC1dx

�ZTd

Duj ln mjl�1Dm

msgn .ln m/dx

from the last term. This integration by parts is valid because, for any smooth functionf ; the identity D.jf jp/ D pjf jp�2 sgn.f /Df holds both a.e. and in the sense ofdistributions. Accordingly,

ZTd


j ln mjldx C CZTd

j ln mjlC1dx

C CZTd

j ln mjl�1jDuj2dx C CZTd

j ln mjl�1 jDmj2m2

dx

� C C CZTd

j ln mjlC1dx

C CZTdŒ�j ln mjl C C.�/�jDuj2dx C C

ZTd


dx;


which yields

ZTd


j ln mjlC1dx C CZTd


dx C C: (7.10)

Combining (7.9) and (7.10), we get

ZTd

j ln mjlm2

jDmj2dx � CZTd

j ln mjlC1dx C CZTd


dx C C:

Consequently,

kDflC1k2L2 � Ckflk2L2 C CkDflk2L2 C C � ClC1:

Since j ln mjlC1 D f 2l 2 L1, we have flC1 D j ln mj lC12 2 L1. By Poincaré’s

inequality,

kflC1k2L2 � kflC1k2L1 C CkDflC1k2L2 � ClC1I

this concludes the proof. ut


The paper [164] introduced the first a priori estimates for stationary MFGs.Subsequently, several other estimates were developed in [127, 129] and [130]. Theintegral Bernstein method was introduced in [173] in the context of Hamilton–Jacobi equations. It was then used in an MFG in [77]. Our presentation follows[194], where this method is explored for more refined estimates.

For MFGs with congestion, the bound for 1m considered here was obtained in

[114]. Because this estimate depends in a crucial way on a cancellation between theHamilton–Jacobi equation and the Fokker–Planck equation, general Hamiltonianscannot be studied with this method. In this case, the existence of solutions remainsan open problem.

First-order MFGs with logarithmic nonlinearities were studied in the context ofthe Aubry–Mather theory in [90]. Then, a one-dimensional problem was inves-tigated in [125]. Second-order MFGs with logarithmic nonlinearities were firststudied in [130]. These results were subsequently improved in [194].

Chapter 8A Priori Bounds for Time-Dependent Models

We continue our study of the regularity of MFGs by considering the time-dependentproblem

(�ut C 1

�jDuj� D �u C m˛ in T

d � Œ0;T�;mt � div.jDuj��1 m/ D �m in T

d � Œ0;T�;(8.1)

where 1 < � � 2 and ˛ > 0. For � < 2; we are in the subquadratic case; for� D 2 the quadratic case. In the first instance, the non-linearity jDuj� acts as aperturbation of the heat equation and the main regularity tool is the Gagliardo–Nirenberg inequality. In the second instance, the Hopf–Cole transformation givesan explicit way to study (8.1). However, this transformation cannot be used tosuperquadratic problems. As a consequence, here, we use a technique that extendsfor superquadratic problems, � > 2, based on the nonlinear adjoint method. In thenext chapter, we investigate two time-dependent problems with singularities—thelogarithmic nonlinearity and the congestion problem—for which different methodsare required.

To get bounds for (8.1), we combine the estimates for Fokker–Planck equationswith estimates for Hamilton–Jacobi equations according to the strategy that weoutline next. First, we fix two function spaces, X and Y , which are typicallyLebesgue or Sobolev spaces. Next, we use the regularity of the Fokker–Planckequation to show that

km˛kX � C1 C C1 kDuk�1Y ;

where C1 > 0 is a constant and �1 > 0. Subsequently, we apply bounds for theHamilton–Jacobi equation to get an estimate of the form:

kDukY � C2 C C2 km˛k�2X ;


105

106 8 A Priori Bounds for Time-Dependent Models

where C2 > 0 and �2 > 0. Finally, we combine the prior inequalities to obtain

kDukY � C C C kDuk�1�2Y : (8.2)

If �1�2 < 1, we have an estimate for Du in Y . The choice of the spaces, X and Y;and of the corresponding estimates depends on � and is distinct for subquadraticand quadratic problems.

8.1 Subquadratic Hamiltonians

To study subquadratic Hamiltonians, we combine the polynomial estimates forFokker–Planck equations in Proposition 4.17, the integral bounds for Hamilton–Jacobi equations in Proposition 3.15, and the bounds given by the Gagliardo–Nirenberg inequality in Theorem 3.23. Because our goal is to illustrate the mainideas in the simplest possible way, the next result is far from optimal. We refer thereader to Remark 8.2 and to the bibliographical notes at the end of the chapter forsharper results.

Theorem 8.1. Let .u;m/ solve (8.1) with 1 < � < 2. Suppose that

0 < ˛ <2 � �

2d.� � 1/ :

Then, for any 1 < s < 1; there exists a constant, Cs; such that

kmkLs.Td�Œ0;T�/ C kDukLs.Td�Œ0;T�/ � Cs:

Proof. By Theorem 3.23 with r D p D ��1�

Nr, we have

kDukL.��1/Nr.Td�Œ0;T�/ � Ckm˛kL.��1/Nr=� .Td�Œ0;T�/ C Ckuk2=.2��/L1.Td�Œ0;T�/; (8.3)

provided that

.� � 1/�

Nr > 1: (8.4)

Proposition 3.15, with p D ��1�

Nr, gives

kukL1.Td�Œ0;T�/ � C C Ckm˛kL.��1/Nr=� .Td�Œ0;T�/ D C C Ckmk˛L˛.��1/Nr=� .Td�Œ0;T�/(8.5)

8.1 Subquadratic Hamiltonians 107

if

.� � 1/�

Nr > d

2C 1: (8.6)

Finally, Proposition 4.17 with r D p D Nr2, ˇ0 D 1; and n D 1 implies that

kmk�L� .Td�Œ0;T�/ � C C CkDuk.��1/NrL.��1/Nr.Td�Œ0;T�/; (8.7)

provided that Remark 4.16 holds, and

Nr2>

d

2and

Nr2>

NrNr � d

; (8.8)

or, equivalently,

Nr > d C 2; (8.9)

and conditions (4.23) and (4.26) are met such that

� D 2.Nr=2 � 1/d

: (8.10)

Suppose that � satisfies

� � ˛� � 1�

Nr: (8.11)

Then, we get

kDukL.��1/Nr.Td�Œ0;T�/ � C C CkDuk.��1/Nr˛=�L.��1/Nr.Td�Œ0;T�/ C CkDuk2.��1/Nr˛=.�.2��//

L.��1/Nr.Td�Œ0;T�/ :

(8.12)

Therefore, if

2.� � 1/Nr˛�.2 � �/ < 1; (8.13)

we have

kDukL.��1/Nr.Td�Œ0;T�/ � C:

Elementary computations show that for 0 < ˛ <2��

2d.��1/ , there exist Nr > 1 and� > 1 such that (8.6), (8.9), (8.10), (8.11), and (8.13) hold simultaneously. Then,the iterative estimates in Proposition 4.17 give m 2 Ls for any 1 < s < 1. Thus, byProposition 3.15, we have Du 2 Ls for any 1 < s < 1. ut

108 8 A Priori Bounds for Time-Dependent Models

Remark 8.2. The preceding result can be improved by taking ˇ0 > 1. For that, wecan use the first-order estimates in Sect. 6.2 of Chap. 6 or the second-order estimatesin Sect. 6.4 of the same chapter combined with the iterative estimates in Sect. 4.4.1of Chap. 4. We refer the reader to the bibliographical notes for additional results.

8.2 Quadratic Hamiltonians

Here, we consider � D 2 in (8.1) and get bounds for the L1 norm of Du.

Theorem 8.3. Let .u;m/ solve (8.1) with � D 2. Suppose that d > 2 and

0 < ˛ <3

4d� 1

2d2:

Then, there exists a constant, C, that does not depend on the solution, such thatkDukL1.Td�Œ0;T�/ � C.

Proof. From Theorem 5.12, we have

kDukL1 � C C Ckm˛k2L1.Œ0;T�;LNr.Td//C CkukL1.Td�Œ0;T�/; (8.14)

provided that

Nr > d: (8.15)

Next, by Remark 3.16, we have

kukL1.Td�Œ0;T�/ � C C Ckm˛kL1.Œ0;T�;LNr.Td// � C C Ckmk˛L1.Œ0;T�;L˛Nr.Td//;

provided that Nr > d2, which holds by (8.15). Finally, Corollary 4.21 with ˇ0 D 1 and

n D 0 gives

kmk�L1.Œ0;T�;L� .Td//� C C C kDuk2NrL1.Td�Œ0;T�/ ; (8.16)

with

Nr D 2C d.� � 1/2

and � > 1: (8.17)

Suppose that

� � ˛Nr: (8.18)


Combining the previous estimates, we get

kDukL1.Td�Œ0;T�/ � C C Ckmk2˛L1.Œ0;T�;L˛Nr.Td//� C C CkDuk 4˛Nr

�

L1.Td�Œ0;T�/:

Thus, if

4˛Nr < �; (8.19)

we get Du 2 L1.Td � Œ0;T�/.Elementary computations show that there exist r > 1 and � > 1 satisfying (8.15),

(8.17), (8.18), and (8.19) if ˛ < 34d � 1

2d2for d > 2. ut

Remark 8.4. As in the preceding section, the previous result can be improvedby selecting ˇ0 > 1. Moreover, the methods in this section can be used forsuperquadratic Hamiltonians; that is, � > 2. We refer the reader to the bibliograph-ical notes for additional results.


The techniques used in this chapter were developed in the thesis [193] and in thepapers [134] and [135] for subquadratic and superquadratic problems, respectively.For unbounded domains, see [118]. An application of these methods to the forward–forward MFG problem can be found in [119].

Chapter 9A Priori Bounds for Models with Singularities

Here, we discuss two problems—an MFG with a logarithmic nonlinearity and anMFG with congestion effects. Stationary versions of these two problems wereconsidered in Chap. 7. However, the techniques for time-dependent problems aresubstantially different from the ones used in the stationary case.

9.1 Logarithmic Nonlinearities

We begin our study of MFGs with singularities by examining the system(�ut.x; t/ C 1

2jDu.x; t/j� D �u.x; t/ C ln m.x; t/; .x; t/ 2 T

d � Œ0;T�;mt.x; t/ � div.mjDuj��2Du/ D �m.x; t/; .x; t/ 2 T

d � Œ0;T�;(9.1)

with initial-terminal boundary conditions(u.x;T/ D uT.x/; x 2 T

d;

m.x; 0/ D m0.x/; x 2 Td;

(9.2)

where T > 0 is a fixed terminal instant. The Hamiltonian associated with (9.1) is

jpj��

� ln m:

The corresponding Lagrangian given by the Legendre transform [see (1.7)] is thus

jvj� 0

� 0 C ln m;


111

112 9 A Priori Bounds for Models with Singularities

with 1�

C 1� 0

D 1. Hence, areas of low density are highly desirable from the pointof view of agents. This effect should therefore force the density, m; to be boundedfrom below. Here, however, the primary mechanism for regularity is given by thediffusion that overcomes the nonlinearity, jpj� ; if � is close to 1 and prevents low-density regions.

Lemma 9.1. Let .u;m/ be a classical solution of (9.1)–(9.2) with m > 0. Then,

d

dt

�ln

ZTd

1

mdx

�� C

��jDuj2��1

L1.Td/C C:

Proof. Proposition 4.5 ensures that, for some constants, C > 0 and c > 0, we have

d

dt

ZTd

1

mdt � C

ZTd

jDuj2.��1/

mdx � c

ZTd

jDmj2m3

dx

� C kDuk2.��1/L1.Td/

ZTd

1

mdx:

Hence, we conclude that

d

dt

�ln

ZTd

1

mdx

�� C kDuk2.��1/

L1.Td/:

utLemma 9.2. Assume that m W Td ! R

C is integrable withRTd m D 1. Then, for

p > 1, there exists a constant, C > 0; such that

ZTd

j ln mjpdx � C C CZ

m�1

ln1

m

�p

dx:

Proof. For p > 1, we have

ZTd

j ln mjpdx DZ

m�1

ln1

m

�p

dx CZ

m>1.ln m/pdx:

Because ln z is sublinear for z > 1, we get

ln m � Cımı;

for every ı > 0, provided that m > 1. Hence, we infer that

ZTd

j ln mjpdx �Z

m�1

ln1

m

�p

dx C C

and conclude the proof. ut

9.1 Logarithmic Nonlinearities 113

Next, we present an auxiliary lemma:

Lemma 9.3. There exists 0 < A D A.p/, such that .ln z/p is a concave function ofz for z > 1

A .

Proof. By elementary computations, we have

Œ.ln z/p�00 D p .ln z/p�2

z2Œp � 1 � ln z� :

For z > ep�1, we get

Œ.ln z/p�00 < 0:

Hence, by setting A D A.p/ WD e1�p, we prove the lemma. utLemma 9.4. Let .u;m/ solve (9.1)–(9.2). Then, there exists a constant, C > 0; suchthat Z

m�1

ln

1

m.x; �/

�p

dx � C C C kDuk2p.��1/L1.Td�Œ0;T�/ :

Proof. First, we observe that

Zm�1

ln1

m

�p

dx DZ

A�m�1

ln1

m

�p

dx

CZ

m<A

ln1

m

�p

dx;

for every 0 < A < 1. By choosing A as in Lemma 9.3, we get

ZA�m�1

ln1

m

�p

dx � C maxA�m�1

ˇˇln 1m

ˇˇp

� C:

Now, we set ‰.z/ WD .ln z/p for z � 1A and extend ‰ continuously and linearly for

z < 1A . We can do this so that ‰ is globally concave and increasing.

Therefore, Jensen’s inequality implies that

1

jfm � AgjZ

m�A

ln1

m

�p

dx D 1

jfm � AgjZ

m�A‰

1

m

�dx

� ‰

1

jfm � AgjZ

m<A

1

m

�:


Because ‰ is increasing, we have that

1

jfm � AgjZ

m�A

ln1

m

�p

dx � ‰

1

jfm � AgjZTd

1

m

�:

In the sequel, we consider two cases: either

1

jfm � AgjZTd

1

m<1

A;

or

1

jfm � AgjZTd

1

m>1

A:

The former inequality yields

Zm�A

ln1

m

�p

dx � ‰

1

A

�;

whereas the latter implies that

‰

1

jfm � AgjZTd

1

m

�D�

ln

1

jfm � Agj�

C ln

ZTd

1

m

�p

:

Because jfm � Agj � 1, we conclude that

Zm�A

ln1

m

�p

dx � ‰

1

A

�C Cp

�ln

ZTd

1

m

�p

C Cp jfm � Agj�

ln

1

jfm � Agj�p

:

Since

1

jfm � Agj � 1;

we have

ln

1

jfm � Agj�

� Cı

1

jfm � Agj�ı;

9.2 Congestion Models: Local Existence 115

for every ı > 0. Set ı WD 1p ; then,

Zm�A

ln1

m

�p

dx � C C C

�ln

ZTd

1

m

�p

C Cjfm � Agjjfm � Agj ;

i.e.,

Zm�A

ln1

m

�p

dx � C C C

�ln

ZTd

1

m

�p

I

this, in light of Lemma 9.1, concludes the proof. utWe close this section with the main theorem that gives a bound for ln m in terms

of norms of Du.

Theorem 9.5. Let .u;m/ solve (9.1)–(9.2). Then, for every p > 1,

kln mkL1.0;TILp.Td// � C C CkDuk2.��1/L1.Td�Œ0;T�/:

Proof. The claim results from combining Lemmas 9.2 and 9.4 to conclude that

ZTd

j ln mjpdx � C C C��jDuj2��p.��1/

L1.Td�Œ0;T�/ :

ut

9.2 Congestion Models: Local Existence

Here, we consider the MFG congestion model given by

8<:

�ut ��u C m˛

�

� jDujm˛

�� D 0;

mt ��m � div

Dum˛

� jDujm˛

��2m

�D 0;

(9.3)

together with the initial-terminal conditions

u.x;T/ D uT.x/; m.x; 0/ D m0.x/: (9.4)

As before, we work in the spatially periodic setting. The unknowns in (9.3) areuWTd � Œ0;T� ! R and mWTd � Œ0;T� ! R

C. We assume that uT ;m0WTd ! R

are given C1 functions with m0 > 0. The terminal cost, uT WTd ! R, is globally


bounded with bounded derivatives of all orders. The initial distribution, m0; is a C1probability density:

RTd m0.x/dx D 1. Finally, we suppose that 1 � � < 2 and, as

previously, 0 < ˛ < 1.The specific form of the Hamiltonian in (9.3) is motivated by the following

consideration. With congestion, agents face difficulties in moving at high speed inhigh-density areas. Hence, it is natural to consider the Lagrangian,

L.x; v;m/ D m˛ jvj� 0

� 0 ;

with 1�

C 1� 0

D 1. The corresponding Hamiltonian is

H.x; p;m/ D m˛

�

jpjm˛

��:

9.2.1 Estimates for Arbitrary Terminal Time

We begin our study of (9.3) by proving estimates that are valid for all terminal times,T > 0. Unfortunately, these estimates are not strong enough to prove the existenceof solutions. Therefore, in the next section, we consider the short-time problem.There, we examine a local-in-time estimate that gives a bound for 1

m provided thatT is small enough.

The next two results are straightforward consequences of the comparisonprinciple in Proposition 3.1.

Proposition 9.6. For any C1 solution, .u;m/; of (9.3)–(9.4), we have

km.�; t/kL1.Td/ D 1;

and u � �kuTkL1.Td/, for all 0 � t � T.

Proof. Proposition 4.1 givesRTd m.x; t/dx D 1 for all t � 0. Moreover, by

Proposition 4.3, we have m � 0 for all t � 0.The lower bound on u results from the comparison principle in Proposition 3.1

with v D min uT . utProposition 9.7. For any C1 solution, .u;m/; of (9.3)–(9.4), u � kuTkL1.Td/.

Proof. Because m˛

�

� jDujm˛

�� 0, we have

�ut ��u � 0:

Then, Proposition 3.1 gives u � max uT . ut


The two preceding propositions give the following corollary.

Corollary 9.8. For any C1 solution .u;m/ of (9.3)–(9.4), kukL1.Td/ �kuTkL1.Td/.

Proposition 9.9. There exists a constant, C WD C.kuTkL1.Td/;T/; such that, forany C1 solution, .u;m/, of (9.3)–(9.4), we have

Z T

0

ZTd

jDuj�m N dxdt � C; (9.5)

where

N D .� � 1/˛ < 1: (9.6)

Proof. We integrate the first equation in (9.3) with respect to x and t. Then, we usethe bounds on u from the previous Corollary to get

Z T

t

ZTd

m˛

�

jDujm˛

��dxds D

ZTd

u.x;T/dx �ZTd

u.x; 0/dx � C;

Thus, (9.5) follows. utProposition 9.10. There exists a constant, C WD C.kuTkL1.Td/;T/; such that, forany C1 solution, .u;m/; of (9.3)–(9.4), we have

Z T

0

ZTd

jDuj�m1� N dxdt � C;

where N is given by (9.6).

Proof. We multiply the first equation in (9.3) by m and subtract the second equationmultiplied by u. Then, integration by parts yields

Z t

0

ZTd

m1C˛"

Du

m˛� Du

m˛

jDujm˛

��2� 1

�

jDujm˛

��#dxds D

ZTd

u.x; 0/m0.x/dx

�ZTd

u.x; t/m.x; t/dx � 2kukL1.Td�Œ0;T�/:

The claim in the statement follows from Corollary 9.8 and the identity p � pjpj��2 �1�jpj� D 1

� 0

jpj� : utProposition 9.11. There exist constants, cr;Cr WD Cr.˛;T/ > 0; that havepolynomial growth in r, such that, for any C1 solution, .u;m/; of (9.3)–(9.4) andr > 1,


ZTd

1

mr.x; t/dx C cr

Z t

0

ZTd

ˇˇD 1

mr=2

ˇˇ2

dxds CZ t

0

ZTd

jDuj�mrC N dxds

� Cr C Cr

Z t

0

ZTd

1

mqdxds

for all 0 � t � T, where N is given by (9.6) and

q D r C 2 N2 � � : (9.7)

Proof. By adding a constant to uT , we can assume, without loss of generality, thatu � �1. We fix r > 1, multiply the first equation in (9.3) by 1

mr and add it to thesecond equation multiplied by r u

mrC1 . After integrating by parts, we obtain

�ZTd

� u

mr

�tdx �

ZTd

r.r C 1/ujDmj2mrC2 dx C

ZTd

m˛1�

� jDujm˛

�� C r Dum˛ � Du

m˛

� jDujm˛

��2

mr dx

�ZTd

r.r C 1/u Du

m˛

� jDujm˛

��2 � Dm

mrC1 dx D 0:

We integrate the preceding identity in t and use �u � 1, juj � C; to get

ZTd

1

mr.x; t/dx C r.r C 1/

Z t

0

ZTd

jDmj2mrC2 dxds C

1

�C r

�Z t

0

ZTd

jDuj�mrC N dxds �

r.r C 1/CZ t

0

ZTd

jDuj��1 jDmjmrC NC1 dxds C C

ZTd

1

mr.x; 0/dx:

The required estimate follows from the inequality

jDuj��1 jDmjmrC NC1 � �

jDuj�mrC N C �

jDmj2mrC2 C C�

1

mq;

where q is given by (9.7). ut

9.2.2 Short-Time Estimates

In this section, we establish estimates for C1 solutions of (9.3)–(9.4) for smallvalues of T . The key idea is to use the estimate in Proposition 9.11 to control thegrowth of 1

m . Because q > r in (9.7), we can achieve bounds only for small T . Webegin with the following bound on 1

m .


Theorem 9.12. There exist r0 > 0, a time, t1.r/ > 0; and constants, C DC.r; �; ˛/ > 0 and ı > 0, such that, for any C1 solution, .u;m/, of (9.3)–(9.4)and r � r0, Z

Td

1

mr.x; t/dx � C

�1C 1

.t1 � t/ı

; 8t < t1:

Proof. Let 2� D 2dd�2 be the Sobolev conjugate exponent to 2. We choose a

sufficiently large r0 such that

2�

2r D dr

d � 2 > q D r C 2 N2 � �

for r � r0. Let � > 0 be such that

2�

2r�C r.1 � �/ D qI

that is,

� D N .d � 2/.2 � �/r < 1

for r � r0. We set N� D 2�

2� D Nd

.2��/r . If r0 is large enough, we have N� < 1 and

ˇ D 1 � �1 � N� > 1

for all r � r0. Then, using Hölder’s and Young’s inequalities, we get

ZTd

1

mqdx �

ZTd

1

m2�

2 rdx

�� ZTd

1

mrdx

�1��

D"Z

Td

1

m2�

2 rdx

�2=2�

#N� "ZTd

1

mrdx

�ˇ#1�N�

� " N�Z

Td

1

m2�

2 rdx

�2=2�

C 1

"�.1 � N�/

ZTd

1

mrdx

�ˇ

for any " > 0 and some exponent � > 0. From Sobolev’s inequality,

ZTd

jDmj2mrC2 dx D 4

r2

ZTd

ˇˇD

1

mr=2

�ˇˇ2

dx � c4

r2

ZTd

1

m2�

2 rdx

�2=2�

� 4

r2

ZTd

1

mrdx:


By combining Proposition 9.11 and the above inequalities with the estimate

ZTd

1

mrdx � "

ZTd

1

mrdx

�ˇC C"; 8" > 0;

we get

ZTd

1

mr.x; t/dx � C C C

Z t

0

ZTd

1

mrdx

�ˇdt; 8t 2 Œ0;T�: (9.8)

Let h.t/ D RTd

1m.x;t/r dx and H.t/ D

tR0

hˇ.s/ds. Then, the previous inequality reads

h.t/ � Cr;�;T C Cr;�;TH.t/:

Thus,

PH.t/ D hˇ.t/ � Cr;˛;�;T.1C H.t//ˇ: (9.9)

Integrating (9.9) and taking into account that H.0/ D 0, we get

.1C H.t//1�ˇ � 1 � .ˇ � 1/Cr;�;T t:

Accordingly,

H.t/ � 1�1 � .ˇ � 1/Cr;�;T t

� 1ˇ�1

for all t < t1.r/ WD 1

.ˇ � 1/Cr;�;T:

Consequently,

ZTd

1

m.x; t/rdx D h.t/ � Cr;�;T C Cr;�;TH.t/ � C C C

.t1 � t/1

ˇ�1

; t < t1:

utCorollary 9.13. Let r0 and t1.r/ be as in Theorem 9.12. For r > r0, let t � t1.r/ �t1. Then, there exist constants, Cr and ır; such that, for any C1 solution, .u;m/; of(9.3)–(9.4),

Z t

0

ZTd

ˇˇD 1

mr=2

ˇˇ2

dxdt � Cr C Cr

.t1 � t/ır; 8 t < t1: (9.10)


Iterating the estimates from Proposition 9.11, we get uniform bounds in r, as weprove next.

Proposition 9.14. There exist r1 > 0 and constants, Ct D Ct.r; �; ˛/ > 0 andˇr > 1, such that, for any C1 solution, .u;m/; to (9.3)–(9.4) and r � r1,

�� 1m��

L1.Œ0;t��Td/

� Ct

1C

�� 1m��ˇr

L1.Œ0;t�;Lr.Td//

!:

Remark 9.15. We observe that the previous result is not a local result. If weestablish bounds for 1

m in L1.Œ0; t�;Lr.Td// for some t > 0, we get 1m 2 L1

.Œ0; t� � Td/.

Proof. For r > 1, choose �n > 0 such that

rnC1 C ı D .1 � �n/rn C �n

2�

2rnC1;

where ı D 2 N2�� ; that is, �n D 1� 1

r C ı

rnC1

2�

2 � 1r

> 0. Set �n D 2�

2�n and ˇn D 1��n

1��n. Then,

there exists r1 > 1 such that, for any r � r1 and any n � 1, we have �n < 1. We fixa time, t. As in the previous proposition, using a weighted Hölder’s inequality, wehave

ZTd

1

mrnC1Cı dx �"Z

Td

1

m2�

2 rnC1dx

�2=2�

#�n"Z

Td

1

mrn dx

�ˇn#1��n

� "�n

ZTd

1

m2�

2 rnC1dx

�2=2�

C 1

"�.1 � �n/

ZTd

1

mrn dx

�ˇn

;

where " > 0 and � > 0 is a suitable exponent. Next, Proposition 9.11 and Sobolev’sinequality imply that

ZTd

1

mrnC1.x; t/

dx CZ t

0

ZTd

1

m2�

2 rnC1.x; s/

dx

!2=2�

ds

� CrnC1 C CrnC1

Z t

0

ZTd

1

mrnC1Cı.x; s/dxds:

From these two inequalities, we conclude that

ZTd

1

mrnC1.x; t/

dx � CrnC1 C CrnC1

Z t

0

ZTd

1

mrn.x; s/

dx

�ˇn

ds:


Define

An.t/ D maxŒ0;t�

ZTd

1

mrn.x; �/dx:

From the above estimate,

1C AnC1.t/ � maxf1; tgCn.1C An.t//ˇn ;

where Cn D O.rnk/ for some k > 1: Proceeding inductively, we get

.1C AnC1.t//1

ˇ1 �:::�ˇn � CPn

iD11

ˇ1 �:::�ˇit r

PniD1

ikˇ1 �:::�ˇi .1C A1/:

Note that

ˇn D 1 � �n

1 � �nD r

1C . 2

�

2r � 1/ı

rnC1. 2�

2� 1 � 2�

2ırn /

!WD r.1C qn/:

Because

limn!1 rn . 2

�

2r � 1/ı

rnC1. 2�

2� 1 � 2�

2ırn /

D2�

2ı

2�

2� 1 ;

we have qn D O.r�n/ > 0, the seriesP1

iD1 ikˇ1�:::�ˇi

;P1

iD1 1ˇ1�:::�ˇi

, and the infinite

productQ1

iD1.1C qi/ converges. From this, we get

�� 1m��

L1.Œ0;t�;LrnC1.Td//

� Ct

1C

�� 1m��ˇr

L1.Œ0;t�;Lr.Td//

!;

for some constants, Ct > 0 and ˇr D Q1iD1.1C qi/ > 1; that do not depend on the

solution. By letting n ! 1, we obtain the result. utThe results of Theorem 9.12, Proposition 9.14, and Corollary 9.13 prove the

following:

Theorem 9.16. There exist a time, T0 > 0; and a constant, C D C.�; ˛/ > 0; suchthat, for any C1 solution, .u;m/; of (9.3)–(9.4), we have

�� 1m��

L1.Œ0;T0��Td/

� C:



Time-dependent MFGs with logarithmic nonlinearity were first studied in [117].MFGs with congestion were introduced in [174]. The existence of classical solutionsfor stationary MFGs with quadratic Hamiltonians was proven in [114]. Theexistence of solutions for time-dependent problems is known only for the short-time problem. Weak solutions of congestion problems were investigated in [136].Here, in our approach to the congestion problem, we follow [123].

Chapter 10Non-local Mean-Field Games: Existence

MFGs where the Hamilton–Jacobi equation depends on the distribution of playersin a non-local way make up an important group of problems. In many examples, thisdependence is given by regularizing convolution operators. We split the discussionof non-local problems into two cases. First, we consider first-order MFGs. Here,semiconcavity bounds and the optimal control characterization of the Hamilton–Jacobi equation are the main tools. Next, we examine second-order MFGs. Here,the regularizing effects of parabolic equations and the L2 stability of the Fokker–Planck equation are the main ingredients of the proof.

10.1 First-Order, Non-local Mean-Field Games

We denote by P1.Rd/ the set of Borel probability measures in Rd with a finite first

moment. The 1-Wasserstein distance between two probability measures, �1 and �2;with finite first moments is

d1.�1; �2/ D inf

ZRd�Rd

jx � yjd.x; y/;

where the infimum is taken over the set, ….�1; �2/; of all probability measures,; in R

d � Rd whose first marginal is �1 and whose second marginal is �2. The

1-Wasserstein distance makes P1.Rd/ a metric space.We define the norm, k � kC2 , as

kgkC2 D supx2Rd

�jg.x/j C jDxg.x/j C jD2xxg.x/j� ;

for any g 2 C2.Rd/.


125

126 10 Non-local Mean-Field Games: Existence

Fix F W Rd �P1.Rd/ ! R, and suppose that the map m 7! F.�;m/, m 2 P1.Rd/,is continuous from P1 to C2.Rd/. Next, select initial and terminal conditions,m0 2 P1.Rd/ and uT 2 C2.Rd/. We consider the MFG(

�ut C jDuj22

D F.x;m/;

mt � div.mDu/ D 0;(10.1)

with the initial-terminal conditions(u.x;T/ D uT.x/;

m.x; 0/ D m0.x/:(10.2)

Because first-order Hamilton–Jacobi equations may fail to have C1 solutions, welook for a solution, .u;m/, with u W R

d � Œ0;T� ! R, u bounded and locallyLipschitz, and m 2 C.Œ0;T�;P1.Rd//.

Fix m 2 C.Œ0;T�;P1.Rd//. In (10.1), the Hamiltonian, H.x; p/ D jpj22

� F.x;m/;is convex in p. For that reason, we say that u is a viscosity solution of the firstequation in (10.1) if

u.x; t/ D infx

Z T

t

� jPx.s/j22

C F.x.s/;m.�; s//

ds C uT.x.T//;

where the infimum is taken over all absolutely continuous trajectories, x; withx.t/ D x. Though this is not the usual definition of a viscosity solution, it isequivalent to the usual one in this case. We refer the reader to the end of Chap. 3for bibliographical references.

We say that .u;m/ solves (10.1)–(10.2) if u is a viscosity solution of the firstequation in (10.1), m 2 C.Œ0;T�;P1.Rd// is a solution in the sense of distributionsof the second equation, and (10.2) holds.

Theorem 10.1. Assume that m0 is absolutely continuous with respect to theLebesgue measure and that there exists a constant, C > 0; such that

supm2P1.Rd/

kF.�;m/kC2 � C (10.3)

and

supx2Rd

kF.x;m/ � F.x; Nm/kC1 � Cd1.m; Nm/; 8m; Nm 2 P1.Rd/: (10.4)

Then, there is a solution, .u;m/; of (10.1)–(10.2) such that u is a Lipschitzcontinuous and semiconcave viscosity solution of the Hamilton–Jacobi equation,

�ut C jDuj22

D F.x;m/;

10.1 First-Order, Non-local Mean-Field Games 127

and m 2 C.Œ0;T�;P1.Rd// is a weak solution of the transport equation

mt � div.mDu/ D 0:

Proof. We use a fixed-point argument. Let ‰ W P1.Rd/ 7! C.Rd/ be as follows: form1 2 P1.Rd/, we define ‰.m1/ as the solution of the optimal control problem

u1.x; t/ D infx

TZt

jPx.s/j22

C F.x.s/;m1.s//ds C uT.x.T//; (10.5)

where the infimum is taken over all absolutely continuous trajectories, x, withx.t/ D x. Then, u1 is the viscosity solution to

� u1t CˇDu1

ˇ22

D F.x;m1/; (10.6)

with the terminal condition u1.x;T/ D uT.x/.We proceed with the analysis of the operator, ‰. Fix m1 2 P1.Rd/ and let

u1 D ‰.m1/. By Proposition 3.8, u1 is uniformly bounded, Lipschitz, and locallyuniformly semiconcave on Œ0;T/:

Though viscosity solutions may fail to be differentiable, by semiconcavity, theyare differentiable almost everywhere. The Hamiltonian corresponding to (10.6) isjpj22

� F.x;m1/. The corresponding Hamiltonian dynamics is

(Px D �p;

Pp D �DxF.x;m1/;(10.7)

Consequently, if x 2 Rd is a point of differentiability of u1.x; 0/, the solution of

(10.7) with the initial conditions

x.0/ D x; p.0/ D Dxu1.x; 0/;

is an optimal trajectory for (10.5). Moreover, u1 is differentiable at .x.t/; t/, withp.t/ D Dxu1.x.t/; t/ for 0 < t < T .

Let .ˆ1.x; t; s/; ˆ2.x; t; s// D .x.s/;p.s// be the flow defined for almost every.x; t/ through (10.7), satisfying

8<ˆ:@sˆ

1.x; t; s/ D �ˆ2.x; t; s/;@sˆ

2.x; t; s/ D �DxF.ˆ1.x; t; s/;m1/;

ˆ1.x; t; t/ D x; ˆ2.x; t; t/ D Du1.x; t/:

(10.8)


Note that ˆ2.x; t; s/ D Du1.ˆ1.x; t; s/; s/. Furthermore, ˆ1 satisfies

jˆ1.x; t; s0/ �ˆ1.x; t; s/j � Cjs � s0j:Taking into account that m1.0/ is absolutely continuous, we define

m2.t/ D ˆ1.�; 0; t/]m1.0/:

Finally, we define the operator, UW C.Œ0;T�;P1/ ! C.Œ0;T�;P1/; by U.m1/ D m2:

Next, we check that U is continuous. Consider a sequence, m1n; in C.Œ0;T�;P1/.

Suppose that m1n ! m1 in C.Œ0;T�;P1/, for some m1 2 P1. The property (10.4)

ensures the stability of the control problem (10.5); thus, u1n ! u1 a.e. Becauseu1n and u1 are Lipschitz continuous and semiconcave, we have Du1n ! Du1 a.e.The latter convergence combined with (10.4) and the stability of the ODE (10.7)give kˆn.x; 0; �/ � ˆ.x; 0; �/kC.Œ0;T�/ ! 0 for a.e. x, where ˆn D .ˆ1n; ˆ

2n/ and

ˆ D .ˆ1;ˆ2/. Since

d1.m2n.s/;m

2.s// �ZRd

jˆ1n.x; 0; s/ �ˆ1.x; 0; s/jdm0.x/;

the Dominated Convergence Theorem implies that m2n ! m2 in C.Œ0;T�;P1/.

Hence, U is continuous. Consequently, by Schauder’s Fixed-point Theorem, U hasa fixed point, m. Moreover, u D ‰.m/ is a viscosity solution to

�ut C jDuj22

D F.x;m/:

Thus, m D U.m/ D ˆ1.�; 0; t/]m.0/, where Dsˆ1.x; 0; s/ D �ˆ2.x; 0; s/ D

�Dxu.ˆ1.x; 0; s/; s/. By the argument outlined in Sect. 1.1.2, m is a weak solution of

mt � div.Dum/ D 0: ut

10.2 Second-Order, Non-local Mean-Field Games

To study non-local, second-order MFGs, we use a fixed-point argument. In contrastto first-order equations, second-order equations have strong regularizing properties.Hence, our proof uses a distinct argument from the first-order case. This argumentrelies on the L2 stability of the Fokker–Planck equation.

Theorem 10.2. Suppose that

1. F W Rd � L2.Td/ ! R is uniformly bounded,

2. There exists a constant, C > 0; such that

jF.x;m1/ � F.x;m2/j � Ckm1 � m2kL2.Td/;

10.2 Second-Order, Non-local Mean-Field Games 129

8m1; m2 2 L2.Td/; and

kF.�;m/kC2 ; kuTkC2 � C; 8m 2 L2.Td/:

3. m0 is continuous,R

m0 D 1, m0 � 0.

Then, there exists a solution , .u;m/; of

8<ˆ:

�ut C jDuj22

D F.x;m/C�u;

mt � div.mDu/ D �m;

m.x; 0/ D m0.x/; u.x;T/ D uT.x/;

(10.9)

with m 2 C.Œ0;T�;L2.Td// and u 2 C1;2.Œ0;T� � Rd/.

Proof. Let m 2 C.Œ0;T�;L2.Td//. Then, there exists a unique solution, U 2 C1;2

.Œ0;T� � Td/, of

�Ut C jDUj22

D �U C F.x;m/; U.x;T/ D uT.x/:

Furthermore, U is globally bounded, Lipschitz continuous, and locally semiconcaveon Œ0;T/ with uniform bounds depending on the assumptions on F and uT . Next, wedefine the map, ˆ W C.Œ0;T�;L2.Td// ! C.Œ0;T�;L2.Td//; as ˆ.m/ D Qm, where Qmsolves the Fokker–Planck equation:

Qmt � div.DU Qm/ D � Qm; Qm.x; 0/ D m0.x/:

Lemma 10.3. The map ˆ W C.Œ0;T�;L2.Td// ! C.Œ0;T�;L2.Td// is continuous.

Proof. Let mn ! m in C.Œ0;T�;L2.Td//. By the continuity of F and the stability ofsolutions of Hamilton–Jacobi equations, we conclude that Un ! U uniformly onŒ0;T� � R

d. Next, the uniform semiconcavity of U and Un gives DUn ! DU a.e.We denote wn D Qmn � Qm. Then,

wnt � div.DUnwn/ � div . Qm.DUn � DU// D �wn; wn.x; 0/ D 0:

Multiplying the preceding PDE by wn and integrating on Rd, we get

d

dtkwn.�; t/k2L2.Rd/

D 2

ZRd

��jDwnj2 � wnDwn � DUn � QmDwn � .DUn � DU/�

dx:

Therefore, using Cauchy’s inequality,

d

dtkwn.�; t/k2L2.Rd/

� CkDUnk21kwn.�; t/k2L2.Rd/C Ck Qm.DUn � DU/k2L2.Rd/

:


Thus, by Gronwall’s inequality,

kwn.�; t/k2L2.Rd/� Ck Qm.DUn � DU/k2L2.Œ0;t��Rd/

:

Further, by the Dominated Convergence Theorem, the right-hand side in the priorestimate converges to 0. Thus, wn ! 0 in C.Œ0;T�;L2.Td//I that is,ˆ.mn/ ! ˆ.m/.utProposition 10.4. The set K D ˆ.C.Œ0;T�;L2.Td/// is compact.

Proof. It is enough to prove that, for any sequence, mn 2 C.Œ0;T�;L2.Td//, wehave that ˆ.mn/ has a convergent subsequence. Because the sequence Un D ˆ.mn/

is equibounded and equiLipschitz, the Arzelá–Ascoli theorem gives that fUng hasa convergent subsequence. Moreover, because Un is uniformly semiconcave, DUn

converges a.e. Therefore, arguing as in the previous lemma, we conclude that,through the same subsequence, ˆ.mn/ converges in C.Œ0;T�;L2.Td//. ut

Finally, using Schauder’s fixed-point theorem, there exists m 2 K, withˆ.m/ D m, and so

(�ut C jDuj2

2D F.m/C�u; u.x;T/ D uT.x/

mt � div.mDu/ D �m; m.x; 0/ D m0.x/: ut


The use of fixed-point methods to prove the existence of solutions was first discussedin [174]. The proof of Theorem 10.1 is a variation of the fixed-point argumentfrom [61]. In the previous reference, the proof uses the quadratic structure of theHamiltonian. Here, we adapt the ideas from [124], and our proof extends to a wideclass of Hamiltonians and avoids measurable selection arguments. The book [212]is a standard reference for optimal transport and Wasserstein distance. The proof ofTheorem 10.2 uses only PDE arguments and does not require the use of stochasticdifferential equations as in [61].

Chapter 11Local Mean-Field Games: Existence

In this last chapter, we address the existence problem for local mean-field games.First, we illustrate the bootstrapping technique and put together the previousestimates. Thanks to this technique, we show that solutions of stationary MFGsare bounded a priori in all Sobolev spaces. This is an essential step for the twoexistence methods developed next. The first method is a regularization procedure inwhich we perturb the original local MFG into a non-local problem. By the resultsof the preceding chapter, this non-local problem admits a solution. Then, a limitingprocedure gives the existence of a solution. The second method we consider is thecontinuation method. The key idea is to deform the original MFG into a problemthat can be solved explicitly. Then, a topological argument shows that it is possibleto deform the solution of the latter MFG into the former. This argument uses both theearlier bounds and an infinite dimensional version of the implicit function theorem.

11.1 Bootstrapping Regularity

Next, we develop a way to prove the regularity of solutions of stationary MFGscalled the bootstrapping method. This method is based on the observation that ifthe solutions of MFGs are regular enough, the equations give further regularityimmediately. Moreover, this process can be iterated indefinitely. Here, we illustratethis idea in the stationary case. The time-dependent case is similar.

For illustration, we consider (7.1) with ˛ as in Theorem 7.1. For every q > 1,Theorem 7.1 ensures that

kDukLq.Td/ � Cq;

for some constant, Cq. Next, we show that this regularity for u immediately impliesthat u and m are a priori bounded in all Sobolev spaces and, hence, are a prioribounded in C1.


131

132 11 Local Mean-Field Games: Existence

Proposition 11.1. Let .u;m;H/ solve (7.1) with u and m in C1.Td/, and let m > 0.Then, there exists a constant, C > 0; such that

kln mkW1;q.Td/ � C:

Proof. Because of Theorem 7.1, standard elliptic regularity theory applied to thefirst equation in (7.1) yields

kukW2;q.Td/ � Cq

for every q > 1. Therefore, Morrey’s Embedding Theorem implies that u 2 C1;ˇ

.Td/ for some ˇ 2 .0; 1/. Next, set w D �2 ln m. Straightforward computationsshow that w satisfies

��w C 1

2jDwj2 � Du � Dw C 2 div.Du/ D 0:

Thus, Theorem 3.11 gives

kDwkL2d.pC1/

d�2 .Td/� Cp

C C kDu � Dwk

L2d.1Cp/

dC2p .Td/

C k div.Du/kL2d.1Cp/

dC2p .Td/

�:

Since Dw 2 L2 and 2d.pC1/d�2 >

2d.1Cp/dC2p , we get Dw 2 Lq for any q � 1 by

using an interpolation argument in the preceding estimate. Hence, ln m is a Höldercontinuous function. Because

RTd mdx D 1, m is bounded from above and from

below. Consequently, k ln mkLq.Td/ is a priori bounded by some universal constantthat depends only on q. utProposition 11.2. Let .u;m;H/ solve (7.1) with u and m in C1.Td/, and let m > 0.For any k � 1 and q > 1, there exists a constant, Ck;q > 0, such that

��Dku��

Lq.Td/;��Dkm

��Lq.Td/

� Ck;q:

Proof. The preceding results give

kukW2;a.Td/ � Ca

for every 1 < a < 1. Also, Proposition 11.1 gives

kln mkW1;a.Td/ � C

for any 1 < a < 1. By differentiating the first equation in (7.1), we obtain

� Dx�u D Dxg.m/ � D2uDu: (11.1)

Finally, we observe that the right-hand side of (11.1) is bounded in La.Td/. Thus,

kukW3;a.Td/ � C3;a;

11.2 Regularization Methods 133

which leads to

kmkW2;a.Td/ � C2;a:

The proof proceeds by iterating this procedure up to order k. utRemark 11.3. Because bootstrapping arguments are very similar, we do not discusshere the other stationary models nor the time-dependent cases and refer the readerto the bibliography.

11.2 Regularization Methods

Frequently, to investigate the existence of solutions of a partial differential equation,we introduce a regularized version of that PDE. Usually, this new problem is wellunderstood or easier to study, and the existence of solutions is straightforward. Then,a limiting procedure together with a compactness argument gives a solution to theoriginal problem.

Here, we illustrate the regularization method in the time-dependent MFG givenby (8.1). We introduce the regularized non-linearity,

g�.m/ WD � � g . � � m/ ;

where � is a symmetric standard mollifier and the convolution in the previousdefinition is in the variable x only. The regularized system is

(�u�t C 1

�jDu�j� D �u� C g�.m�/; in T

d � Œ0;T�;m�

t � div.jDu�j��1 m�/ D �m�; in Td � Œ0;T�;

(11.2)

with the initial-terminal boundary conditions

u�.x;T/ D uT.x/ and m�.x; 0/ D m0.x/:

Remarkably, (11.2) satisfies the same a priori estimates as (8.1). In particular, u�

and m� are bounded in all Sobolev space with � independent bounds. Hence, up to asubsequence, .u�;m�/ ! .u;m/, and .u;m/ solves (8.1). Moreover, .u;m/ inheritsthe regularity of the limiting sequence. Thus, .u;m/ is in any Sobolev space and isof class C1.Td � Œ0;T�/.

For example, here, we examine the first-order estimates and the second-orderestimates from Propositions 6.6 and 6.10, respectively. For g.m/ D m˛ , Proposi-tion 6.6 means that there exists C > 0 not depending on � such thatZ T

0

ZTd.m� C m0/

jDu�j22

C m�� . � � m�/˛

�dxdt � C:


Because � is symmetric, we obtainZ T

0

ZTd.m� C m0/

jDu�j22

C . � � m�/˛C1 dxdt � C:

Therefore, in the case of polynomial nonlinearities, the mean-field coupling for theregularized problem, g�.m�/, is in L˛C1.Td/, uniformly in �.

A similar reasoning applies to the second-order estimates, namely, Proposi-tion 6.10. We haveZ

TdjD2u�j2m C ˛. � � m�/˛�1jD. � � m�/j2dx � C:

To get the previous estimate, we follow the proof of Proposition 6.10. The onlydifference is the term

˛

Z T

0

ZTd

div� � � �. � � m�/˛�1D. � � m�/

��m�dxdt (11.3)

that we address as follows. Integrating by parts in (11.3), we get

˛

Z T

0

ZTd

div� � � .. � � m�/˛�1D. � � m�/

�m�dxdt

D �˛Z T

0

ZTd. � � m�/˛�1jD. � � m�/j2dxdt;

using the symmetry of � . The previous computation thus ensures the second-orderestimates in Proposition 6.10 hold for the regularized problem.

11.3 Continuation Method: Stationary Problems

The regularization methods examined earlier depend on the particular structure ofthe problem. In some cases, it may be difficult to construct a regularized problemwith strong enough bounds. An alternative is the continuation method. Here, weillustrate it by proving the existence of smooth solutions of

8<ˆ:

��u C jDuj22

C V.x/ D H C g.m/;

��m � div .Dum/ D 0;RTd u D 0;

RTd m D 1:

(11.4)

As usual, in the previous equation, the unknowns are the smooth functions, uWTd !R; mWTd ! R

C; and the constant, H 2 R.

11.3 Continuation Method: Stationary Problems 135

First, for 0 � � � 1, we consider the family of problems

8<ˆ:

��m� � div.Du�m�/ D 0;

�u� � jDu�j22

� �V C H� C g.m�/ D 0;RTd u� D 0;

RTd m� D 1:

(11.5)

For � D 1, (11.5) is (11.4).Next, we set

PHk.Td;R/ D

f 2 Hk.Td;R/ WZTd

f dx D 0

�

and consider the Hilbert space, Fk D PHk.Td;R/ � Hk.Td;R/ � R; with the norm

kwk2Fk D k k2PHk.Td ;R/C kf k2Hk.Td ;R/

C jhj2

for w D . ; f ; h/ 2 Fk: By Sobolev’s Embedding Theorem for k > d2, we have

Hk.Td;R/ C0;ˇ.Td;R/ for some ˇ 2 .0; 1/. Thanks to this embedding, we definethe space, HkC.Td;R/; for k > d

2as the set of (everywhere) positive functions in

Hk.Td;R/: For any k > d2, we let

FkC D PHk.Td;R/ � HkC.Td;R/ � R:

Finally, we recall that a classical solution to (11.5) is a tuple, .u�;m�;H�/ 2 Tk�0

FkC.

Theorem 11.4. Assume that g; V 2 C1.Td/ with g0.z/ > 0 for z 2 .0;C1/ andthat we have the a priori estimate for any solution of (11.5):

jHj C�� 1m�

��L1.Td/

C ku�kWk;p.Td/ C km�kWk;p.Td/ � Ck;p:

Then, there exists a classical solution to (11.4).

Proof. For large enough k; we define EWR � FkC ! Fk�2 by

E.�; u;m;H/ D

0B@ ��m � div.Dum/

�u � jDuj22

� �V C H C g.m/� R

Td m C 1

1CA :

Then, (11.5) is equivalent to E.�; v�/ D 0; where v� D .u�;m�;H�/: The partialderivative of E in the second variable at the point v� D .u�;m�;H�/,

L� D D2E.�; v�/W Fk ! Fk�2;


is

L�.w/.x/ D0@ ��f .x/ � div.Du�f .x/C m�D /� .x/ � Du�D C g0.m�.x//f .x/C h

� RTd f

1A ;

where w D . ; f ; h/ 2 Fk. In principle, L� is only defined as a linear map on Fk fora large enough k. However, by inspection of the coefficients, it is easy to see that itadmits a unique extension to Fk for any k > 1:

We define the set

ƒ WD f� j 0 � � � 1; (11.5) has a classical solution .u�;m�;H�/ g:

Note that 0 2 ƒ as .u0;m0;H0/ � .0; 1;�g.1// is a solution to (11.5) for � D 0:

Our goal is to prove ƒ D Œ0; 1�: The a priori bounds in the statement mean that ƒis a closed set. To prove that ƒ is open, we show that L� is invertible and apply theimplicit function theorem. To prove invertibility, we use arguments related to theones in the proof of the Lax–Milgram theorem and the structure of L�. Let F D F1.For w1;w2 2 F with smooth components, we define

B�Œw1;w2� DZTd

w2 � L�.w1/:

For smooth w1;w2, integration by parts gives

B�Œw1;w2� DZTdŒm�D 1 � D 2 C f1Du�D 2 � f2Du�D 1

C g0.m�/f1f2 C Df1D 2 � Df2D 1 C h1f2 � h2f1�:

(11.6)

This last expression is well defined on F �F: Thus, it defines a bilinear form B�W F �F ! R.

Claim 11.5. B� is bounded, i.e.,

jB�Œw1;w2�j � Ckw1kFkw2kF:

To prove the claim, we use Holder’s inequality on each summand.

Claim 11.6. There exists a linear bounded mapping, AW F ! F; such thatB�Œw1;w2� D .Aw1;w2/F.

This claim follows from Claim 11.5 and the Riesz Representation Theorem.

Claim 11.7. There exists a positive constant, c; such that kAwkF � ckwkF for allw 2 F:

If the previous claim were false, then there would exist a sequence, wn 2 F; withkwnkF D 1 such that Awn ! 0: Let wn D . n; fn; hn/: Then,

11.3 Continuation Method: Stationary Problems 137

ZTd

m�jD nj2 C g0.m�/f2n D B�Œwn;wn� ! 0: (11.7)

By combining the a priori estimates on 1m�

with the fact that g is strictly increasing

and smooth, we have g0.m�/ > ı > 0. Then, (11.7) implies that n ! 0 in PH10 and

fn ! 0 in L2. Taking Lwn D .fn � Rfn; 0; 0/ 2 F, we getZ

TdŒjDfnj2 C m�D n � Dfn C fnDu�Dfn� D BŒwn; Lwn� D .Awn; Lwn/;

Therefore,

1

2kDfnk2L2.Td/

� C�kD nk2L2.Td/

C kfnk2L2.Td/

�� .Awn; Lwn/ ! 0;

where the constant, C; depends only on u�. Because D n; fn ! 0 in L2, we havefn ! 0 in H1.Td/: Finally, we take Mw D .0; 1; 0/. Accordingly, we get

ZTdŒ�Du�D n C g0.m�/fn�C hn D BŒwn; Mw� D .Awn; Mw/ ! 0:

Because D n; fn ! 0 in L2, we have hn ! 0. Hence, kwnkF ! 0, which contradictskwnkF D 1.

Claim 11.8. R.A/ is closed in F.

This claim follows from the preceding one.

Claim 11.9. R.A/ D F.

By contradiction, suppose that R.A/ ¤ F. Then, because R.A/ is closed in F, thereexists a vector, w ¤ 0; with w?R.A/. Let w D . ; f ; h/. Then,

0 D .Aw;w/ D B�Œw;w� �ZTd� jD j2 C ıjf j2:

Therefore, D 0 and f D 0. Next, we choose Nw D .0; 1; 0/. Similarly, we haveh D B�Œ Nw;w� D .A Nw;w/ D 0. Thus, w D 0, and, consequently, R.A/ D F.

Claim 11.10. For any w0 2 F0, there exists a unique w 2 F such that B�Œw; Qw� D.w0; Qw/F0 for all Qw 2 F: Consequently, w is the unique weak solution of the equationL�.w/ D w0. Moreover, w 2 F2 and L�.w/ D w0 in the sense of F2:

Consider the functional Qw 7! .w0; Qw/F0 on F. By the Riesz RepresentationTheorem, there exists ! 2 F such that .w0; Qw/F0 D .!; Qw/F. Taking w D A�1!,we get

BŒw; Qw� D .Aw; Qw/F D .!; Qw/F D .w0; Qw/F0 :


Therefore, f is a weak solution to

��f � div.m�D C fDu�/ D 0

and is a weak solution to

� � Du�D C g0.m�/f C h D f0:

Standard results from the regularity theory for elliptic equations combined withbootstrapping arguments give w D . ; f ; h/ 2 F2. Thus, L�.w/ D w0:

Consequently, L� is a bijective operator from F2 to F0. Then, L� is injective asan operator from Fk to Fk�2 for any k � 2: To prove that it is also surjective, takeany w0 2 Fk�2. Then, there exists w 2 F2 such that L�.w/ D w0. Returning againto elliptic regularity and bootstrapping arguments, we conclude that w 2 Fk. Hence,L�W Fk ! Fk�2 is surjective and, therefore, also bijective.

Claim 11.11. L� is an isomorphism from Fk to Fk�2 for any k � 2:

Because L�W Fk ! Fk�2 is bijective, we just need to check that it is also bounded.The boundedness follows directly from bounds on u� and m� and the smoothness ofV and g.

Claim 11.12. The set ƒ is open.

We choose k > d=2 C 1 so that Hk�1.Td;R/ is an algebra. For each �0 2 ƒ, thepartial derivative, L D D2E.�0; v�0/W Fk ! Fk�2, is an isometry. By the ImplicitFunction Theorem, there exists a unique solution v� 2 FkC to E.�; v�/ D 0; in someneighborhood, U; of �0. Since Hk�1.Td;R/ is an algebra, bootstrapping argumentsyield that u� and m� are smooth. Therefore, v� is a classical solution to (11.4).Hence, U ƒ, which in turn proves that ƒ is open.

We have proven thatƒ is both open and closed; hence,ƒ D Œ0; 1�. This argumentends the proof of the theorem. ut

11.4 Continuation Method: Time-Dependent Problems

The continuation method can also be used for time-dependent problems. Here, weexamine the model 8<

ˆ:�ut ��u C jDuj2

2D V.x;m/;

mt ��m � div.Du m/ D 0;

u.x;T/ D uT.x/; m.x; 0/ D m0.x/;

(11.8)

11.4 Continuation Method: Time-Dependent Problems 139

where uT WTd ! R; m0WTd ! RC and VWTd � Œ0;C1/ ! R are given smooth

functions, with V.x; z/ non-decreasing in z. As usual, the unknowns are the smoothfunctions uWTd � Œ0;T� ! R and mWTd � Œ0;T� ! R

C.We introduce the problem

8<ˆ:

�ut ��u C jDuj22

D .1 � �/V.x;m/C � arctan.m/;

mt ��m � div.Du m/ D 0;

u.x;T/ D .1 � �/uT WD ‰�; m.x; 0/ D .1 � �/m0 C � WD m0;�;

(11.9)

where 0 � � � 1. For convenience, we set

V�.x;m/ D .1 � �/V.x;m/C � arctan.m/:

We assume that the following a priori bounds hold for any solution, .u�;m�/; of(11.9).�� 1m

��L1.Td�Œ0;T�//

C kDkt Dl

xm�kLp.Td�Œ0;T�/ C kDkt Dl

xu�kLp.Td�Œ0;T�/ � Ck;l;p

(11.10)for any k; l 2 N and 1 � p < 1.

Theorem 11.13. Assume that the a priori bound (11.10) holds for any solution,.u�;m�/; of (11.9) and that uT ;m0;V are as above. Then, there exists a smoothsolution, .u;m/; of (11.13).

When � D 1, (11.9) has a unique solution, namely u D 4.T � t/, m D 1. As in

the preceding section, our goal is to prove that the set, ƒ; of values 0 � � � 1 forwhich (11.9) admits a solution is relatively open and closed. Therefore, ƒ D Œ0; 1�

and, thus, (11.13) has a solution.For k � �1, we set

Fk.Œ0;T�;Td/ D \2k1Ck2DkHk1 .Œ0;T�;Hk2 .Td//;

where the intersection is taken over all integers, k1 � 0; k2 � �1. The spaceFk.Œ0;T�;Td/ is a Banach space endowed with the norm

kf kFk.Œ0;T�;Td/ DX

2k1Ck2Dk

kf kHk1 .Œ0;T�;Hk2 .Td// :

Moreover, there exists Qkd, depending only on the dimension d, such that, for k � Qkd,Fk�2 is an algebra. Let k � Qkd and consider the operator

M�W Fk.Œ0;T�;Td/ � Fk.Œ0;T�;Td/ ! Fk�2.Œ0;T�;Td/ � Fk�2.Œ0;T�;Td/

� Hk�1.Td/ � Hk�1.Td/


given by

M�

�um

D

26664

mt ��m � div.Dum/

ut C�u � jDuj22

C V�.x;m/m.x; 0/ � m0;�.x/u.x;T/ �‰�.x/

37775 :

Then, (11.9) is equivalent to

M�

�um

D 0; (11.11)

and (11.13) then reads as M0

�um

D 0: Moreover, as we remarked before,

M1

�um

D 0 has only the trivial solution u D

4.T � t/ and m D 1. We consider

the linearized operator, L, given by

L��v

f

D lim

"!0

M�

�u C "v

m C "f

� M�

�um

"

D

26666666664

ft ��f � div ŒDuf C mDv�

vt C�v � DuDv C DzV�f

f .x; 0/

v.x;T/

37777777775:

Note that L� W Fk.Œ0;T�;Td/ � Fk.Œ0;T�;Td/ ! Fk�2.Œ0;T�;Td/ �Fk�2.Œ0;T�;Td/ � Hk�1.Td/ � Hk�1.Td/ is a bounded linear operator for all largeenough k. However, if u and m are C1 solutions to (11.9), then L� admits a uniqueextension as a bounded linear operator L� W Fk.Œ0;T�;Td/ � Fk.Œ0;T�;Td/ !Fk�2.Œ0;T�;Td/ � Fk�2.Œ0;T�;Td/ � Hk�1.Td/ � Hk�1.Td/, for all k � 1.

The form h�; �i denotes the scalar product on L2.Td/. To apply the inverse functiontheorem, we need to prove that the linear operator, L�, is invertible. For this, webegin by showing that the equation L�w D W has a unique weak solution in thesense of the following definition.

Definition 11.14. For h; g 2 L2.Œ0;T�;L2.Td//;A;B 2 L2.Td/, set

W.x; t/ D

2664

h.x; t/g.x; t/A.x/B.x/

3775 :


A function, w D�v

f

, with

v; f 2 L2.Œ0;T�;H1.Td// and vt; ft 2 L2.Œ0;T�;H�1.Td//; that is v; f 2 F1.Œ0;T�;Td/;

(11.12)

is called a weak solution of L�w D W if:

1. for any Nv; Nf 2 H1.Td/ and for a.e. t; 0 � t � T0; we have

(hft; Nf i C ˝

Df C Du�f C mDv;DNf ˛ D hh; Nf ihvt; Nvi � hDv;D Nvi � hDu� � Dv � DzV�f ; Nvi D hg; Nvi: (11.13)

2. f .x; 0/ D A.x/; v.x;T/ D B.x/.

Remark 11.15. Note that (11.12) implies that v; f 2 C.Œ0;T�;L2.Td// (see, e.g.,[88], Section 5.9.2, Theorem 3). Therefore, the traces f .x; 0/ and v.x;T/ are welldefined.

Lemma 11.16 (Uniqueness of Weak Solutions). Let .u�;m�/ be a C1 solutionto (11.9) and the a priori bounds (11.10) hold. Then, there exists at most one weaksolution to the equation L�w D W in the sense of Definition 11.14.

Proof. Since the equation L�w D W is linear, it is enough to prove that L�w D 0

has only the trivial solution w D 0. For this, we take Nf D v; Nv D f in (11.13).Adding both equations and integrating in time, we obtain

Z T0

0

ZTd

m�jDvj2 C DzV�f 2dxdt D 0:

Thus, we get f D 0;Dv D 0. Consequently, v � v.t/. Next, by looking at thesecond equation in (11.13), for Nv D v.t/, we obtain

d

dthv; vi D 0:

Using the boundary conditions for v, we conclude that v D 0, therefore, thatw D 0. ut

To prove the existence of weak solutions, we use the Galerkin approximationmethod. We consider a sequence of C1 functions, ek D ek.x/; k 2 N; such thatf ek g1

kD1 is an orthogonal basis of H1.Td/ and an orthonormal basis of L2.Td/.We construct a sequence of finite dimensional approximations to weak solutions of(11.9) as follows. Let vN ; fN W Œ0;T� ! H1.Td/ be given by

fN.t/ DNX

kD1Ak

N.t/ek; vN.t/ DNX

kD1Bk

N.t/ek:


Next, we show that we can select the coefficients, AkN ;B

kN ; such that

(hf 0

N ; eki C hDfN C fNDu� C m�DvN ;Deki D hh; eki;hv0

N ; eki � hDvN ;Deki � hDu� � DvN � DzV�fN ; eki D hg; eki;(11.14)

and

AkN.0/ D hA; eki; Bk

N.T/ D hB; eki; k D 1; 2; : : : ;N: (11.15)

First, we observe that (11.14) is equivalent to:

( PAkN CPN

lD1 hDel C elDu�;Deki AlN CPN

lD1hm� � Del;DekiBlN D hh; eki;

PBkN �PN

lD1hDel C Du� � Del;DekiBlN �PN

lD1 helDzV�; eki AlN D hg; eki:

(11.16)

Second, because (11.16) is a linear system of ODEs, the only difficulty in provingthe existence of solutions concerns the boundary conditions (11.15). Existence isnot immediate because half of the boundary conditions are given at the initial time,whereas the other half are given at the terminal time. From the standard theory ofordinary differential equations, the initial value problem for (11.16) (that is, withAk

N.0/ and BkN.0/ prescribed) has a unique solution. Hence, to prove the existence

of solutions to (11.16), it is enough to show the existence of solutions for thecorresponding homogeneous problem:

( PAkN CPN

lD1 hDel C elDu�;Deki AlN CPN

lD1hm � Del;DekiBlN D 0;

PBkN �PN

lD1hDel C Du� � Del;DekiBlN �PN

lD1 helDzV�; eki AlN D 0;

(11.17)with arbitrary QAk

N.0/ and QBkN.T/, 1 � k � N. Indeed, any solution to (11.16)–

(11.15), .A;B/ can be written as a sum of a particular solution to (11.16), . NA; NB/, forinstance, with

NAkN.0/ D 0; NBk

N.0/ D 0; k D 1; 2; : : : ;N;

and a solution, . QA; QB/, to (11.17) with suitable initial and terminal conditions suchthat (11.15) holds for .A;B/ D . NA C QA; NB C QB/.

Finally, we regard the solution of the initial value problem for the homogeneoussystem corresponding to (11.16) as a linear operator on R

2N :

.AN.0/;BN.0// 7! .AN.0/;BN.T//: (11.18)

We need to prove that this mapping is surjective. Because (11.18) is a linear mappingfrom R

2N to R2N , surjectivity is equivalent to injectivity. Therefore, it suffices to

prove that the homogeneous system of ODEs corresponding to (11.16), subject toinitial-terminal conditions, AN.0/ D BN.T/ D 0; has only the trivial solution AN DBN � 0. Let fN ; vN solve (11.14) with h D g � 0;A D B � 0. From (11.14),


we obtain (11.13) for f D Nv D fN ; v D Nf D vN . Using the same argument as inLemma 11.16, we conclude that fN D vN � 0.

Next, we provide energy estimates for these approximations to ensure the weakconvergence of approximate solutions through some subsequence.

Lemma 11.17. Let .u�;m�/ be a C1 solution to (11.9) and let the a priori boundsin (11.10) hold. Then, there exists a constant, C; such that, for any C1 solution.u�;m�/ to (11.9), we have

max0�t�T

k.fN ; vN/k.L2.Td//2 C k.fN ; vN/k.L2.0;TIH1.Td///2 C k.f 0N ; v

0N/k.L2.0;TIH�1.Td///2

� C�khkL2.0;TIL2.Td// C kgkL2.0;TIL2.Td// C kAkL2.Td/ C kBkL2.Td/

�:

Lemma 11.18 (Existence of Weak Solutions). Let .u�;m�/ be a C1 solution to(11.9) and let the a priori bounds (11.10) hold. Then, there exists a weak solution ofL�w D W in the sense of (11.13).

Proof. By the energy estimates, there exist subsequences of vN ; fN and functionsv; f 2 L2.0;T0I H1.Td//; with v0 D vt; f 0 D ft 2 L2.0;T0I H�1.Td//; such that

(vN * v; fN * f ; weakly in L2.0;T0I H1.Td//

v0N * v0; f 0

N * f 0; weakly in L2.0;T0I H�1.Td//:

For fixed N0, let Nv; Nf 2 spanfek W 1 � k � N0g with k NvkL2.0;T0IH1.Td//; kNf kL2.0;T0IH1.Td//

� 1. According to the definition of vN ; fN , (11.13) holds with v D vN and f D fNfor every N � N0. Weak convergence then implies (11.13) for v; f and anyNv; Nf 2 spanfek W 1 � k � N0g. The above convergence implies that vN * v; fN * falso in C.0;T0I L2.Td//. Therefore, the initial and terminal conditions on f ; v holdas well. Since [N�1 spanfek W 1 � k � Ng is dense in L2.0;T0I H1.Td//, the proofis complete. utLemma 11.19 (Higher Regularity). Let .u�;m�/ be a C1 solution to (11.9) andlet the a priori bounds in (11.10) hold. Assume that A;B 2 HkC1.Td/; h; g 2F2k.Œ0;T�;Td/ and let W D Œh; g;A;B�t. Then, for any weak solution, w D Œf ; v�t; ofL�w D W, we have v; f 2 F2kC2.Œ0;T�;Td/.

The proof follows from the following result on the regularizing properties of theheat equation and a bootstrap argument.

Lemma 11.20. Let Qh 2 Hk1 .Œ0;T�;Hk2 .Td//, Qg 2 H2k1Ck2C1.Td/ for somek1; k2 � 0, and let Qu 2 F1.Œ0;T�;Td/ be a weak solution of the heat equation(

Qut ��Qu D QhQu.x; 0/ D Qg.x/:

Then, Qu 2 Hk1 .Œ0;T�;Hk2C2.Td// \ Hk1C1.Œ0;T�;Hk2 .Td//:


Proof of Theorem 11.13. The bounds (11.10) and the Arzela–Ascoli Theoremimply that ƒ is a closed subset of the interval Œ0; 1�. We prove that it is also open.Let �0 2 ƒ. Using (11.10), we see that the operator

L�0 W F2k � F2k ! F2k�2 � F2k�2 � H2k�1 � H2k�1

is bounded for every k � 1. Using Lemmas 11.16, 11.18, and 11.19, we concludethat L�0 is bijective. It is thus also invertible. We choose a large enough k andl D b 2k

3c such that Hl.Œ0;T�;Hl.Td// is an algebra. By the inverse function

theorem, there is a neighborhood U of �0 where the equation M�

�um

D 0 has

a unique solution, .u�;m�/; in F2k.Œ0;T�;Td/ � F2k.Œ0;T�;Td/. Then, u�;m� 2Hl.Œ0;T�;Hl.Td//. The inverse function theorem implies that the mapping � 7!.u�;m�/ is continuous. Hence, we can assume that in the neighborhood, U,m� is bounded away from zero. This observation together with the fact thatHl.Œ0;T�;Hl.Td// is an algebra allows us to use the regularity theory and a bootstrapargument to conclude that .u�;m�/ are C1. Accordingly, U ƒ. Consequently, wehave proved that ƒ is an open set in Œ0; 1�. Because 1 2 ƒ, we know that ƒ ¤ ;.Therefore, ƒ D Œ0; 1�. In particular, 0 2 ƒ. ut


The book [171] gives a systematic account on regularization and compactnessmethods in partial differential equations. A comprehensive account of the main ideasbehind bootstrapping methods can be found in [205]. In MFGs, the regularizationexamined in Sect. 11.2 was introduced in [134] and [135]. The constructionof solutions for time-dependent MFGs in [117] also relies on a regularizationargument. In the context of weak solutions, other regularizations were proposed in[196] and [195]. The continuation method is a well-known technique for elliptic andparabolic equations. In elliptic equations, the continuation argument is usually setup in C2;˛ spaces using Schauder estimates; see, for example, [107]. In contrast, forMFGs, it is more convenient to work with Sobolev spaces. A proof of the implicitand inverse function theorems in Banach spaces can be found in [84]. For theapplication of the continuation methods to stationary MFGs, see [114, 129, 130].For time-dependent MFGs, this method has been used in [123]. For the Galerkinmethod, see, for example, [88].

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Index

AAction functional, 3

BBernstein estimate, 25Bootstrapping method, 131

CComparison principle, 16Congestion MFGs, 98, 115Continuation Method, 134Cross estimates, 88

DDynamic Programming Principle, 19

EEnergy conservation, 65, 87Energy estimates, 79Euler-Lagrange equation, 18Evans-Aronsson problem, 85

FFirst-order estimates, 79

GGaussian-quadratic solutions, 11

HHamilton-Jacobi equation, 3Hamiltonian dynamics, 18, 65Hopf-Cole transform, 10

LLasry-Lions monotonicity condition, 7Legendre Transform, 3Logarithmic nonlinearity, 100, 111

MMass conservation, 39

OOptimal control, 2Optimal trajectory, 17

PPoisson bracket, 66Porretta Method, 88Positivity, 39

QQuadratic Hamiltonian, 108

RRegularization method, 133

© Springer International Publishing Switzerland 2016D.A. Gomes et al., Regularity Theory for Mean-Field Game Systems,SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-38934-9

155

156 Index

SSecond-order estimates, 82Semiconcavity, 23Subdifferential, 21Subquadratic Hamiltonian, 106Superdifferential, 20Superquadratic Hamiltonian, 109

TTransport equation, 4

UUniqueness, 7

VValue function, 3, 17Verification theorem, 4

WWasserstein distance, 125

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Regularity Theory for Mean-Field Game Systems

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