Regularity Theory for Mean-Field Game Systems

1. Front Matter

   Pages i-xiv

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2. Introduction

   • Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan

   Pages 1-8

3. Explicit Solutions, Special Transformations, and Further Examples

   • Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan

   Pages 9-14

4. Estimates for the Hamilton–Jacobi Equation

   • Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan

   Pages 15-37

5. Estimates for the Transport and Fokker–Planck Equations

   • Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan

   Pages 39-61

6. The Nonlinear Adjoint Method

   • Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan

   Pages 63-76

7. Estimates for MFGs

   • Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan

   Pages 77-95

8. A Priori Bounds for Stationary Models

   • Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan

   Pages 97-103

9. A Priori Bounds for Time-Dependent Models

   • Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan

   Pages 105-109

10. A Priori Bounds for Models with Singularities

    • Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan

    Pages 111-123

11. Non-local Mean-Field Games: Existence

    • Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan

    Pages 125-130

12. Local Mean-Field Games: Existence

    • Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan

    Pages 131-144

13. Back Matter

    Pages 145-156

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Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields.

• Fokker-Planck equation
• Hamilton-Jacobi equation
• Lax-Hopf estimates
• Logarithmic non-linearities
• mean-field games
• stationary mean-field game problems

“This book is concerned with mean field game systems, or MFG systems for short. Such systems describe an infinite number of rational agents in competition. … The book is an accesible introduction to MFG systems, readable by anyone with a basic knowledge of partial differential equations.” (Teemu Lukkari, Mathematical Reviews, October, 2017)

• CEMSE Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

  Diogo A. Gomes, Vardan Voskanyan

• Department of Mathematics, Universidade Federal de Sao Carlos, São Carlos, Brazil

  Edgard A. Pimentel

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