Regularity theory for mean field games systems [electronic resource] / Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan.

Preface; Book Outline; Thanks; Bibliographical Notes; Acknowledgments; Contents; 1 Introduction; 1.1 Derivation of MFG Models; 1.1.1 Optimal Control and Hamilton-Jacobi Equations; 1.1.2 Transport Equation; 1.1.3 Mean-Field Models; 1.1.4 Extensions and Additional Problems; 1.1.5 Uniqueness; 2 Explicit Solutions, Special Transformations, and Further Examples; 2.1 Explicit Solutions; 2.2 The Hopf-Cole Transform; 2.3 Gaussian-Quadratic Solutions; 2.4 Interface Formation; 2.5 Bibliographical Notes; 3 Estimates for the Hamilton-Jacobi Equation; 3.1 Comparison Principle; 3.2 Control Theory Bounds

3.2.1 Optimal Trajectories3.2.2 Dynamic Programming Principle; 3.2.3 Subdifferentials and Superdifferentials of the Value Function; 3.2.4 Regularity of the Value Function; 3.3 Integral Bernstein Estimate; 3.4 Integral Estimates for HJ Equations; 3.5 Gagliardo-Nirenberg Estimates; 3.6 Bibliographical Notes; 4 Estimates for the Transport and Fokker-Planck Equations; 4.1 Mass Conservation and Positivity of Solutions; 4.2 Regularizing Effects of the Fokker-Planck Equation; 4.3 Fokker-Planck Equation with Singular Initial Conditions; 4.4 Iterative Estimates for the Fokker-Planck Equation

4.4.1 Regularity by Estimates on the Divergence of the Drift4.4.2 Polynomial Estimates for the Fokker-Planck Equation, p<∞; 4.4.3 Polynomial Estimates for the Fokker-Planck Equation, p=∞; 4.5 Relative Entropy; 4.6 Weak Solutions; 4.7 Bibliographical Notes; 5 The Nonlinear Adjoint Method; 5.1 Representation of Solutions and Lipschitz Bounds; 5.2 Conserved Quantities; 5.3 The Vanishing Viscosity Convergence Rate; 5.4 Semiconcavity Estimates; 5.5 Lipschitz Regularity for the Heat Equation; 5.6 Irregular Potentials; 5.7 The Hopf-Cole Transform; 5.8 Bibliographical Notes; 6 Estimates for MFGs

6.1 Maximum Principle Bounds6.2 First-Order Estimates; 6.3 Additional Estimates for Solutions of the Fokker-Plank Equation; 6.4 Second-Order Estimates; 6.4.1 Stationary Problems; 6.4.2 Time-Dependent Problems; 6.5 Some Consequences of Second-Order Estimates; 6.6 The Evans Method for the Evans-Aronsson Problem; 6.7 An Energy Conservation Principle; 6.8 Porreta's Cross Estimates; 6.9 Bibliographical Notes; 7 A Priori Bounds for Stationary Models; 7.1 The Bernstein Method; 7.2 A MFG with Congestion; 7.3 Logarithmic Nonlinearity; 7.4 Bibliographical Notes

8 A Priori Bounds for Time-Dependent Models8.1 Subquadratic Hamiltonians; 8.2 Quadratic Hamiltonians; 8.3 Bibliographical Notes; 9 A Priori Bounds for Models with Singularities; 9.1 Logarithmic Nonlinearities; 9.2 Congestion Models: Local Existence; 9.2.1 Estimates for Arbitrary Terminal Time; 9.2.2 Short-Time Estimates; 9.3 Bibliographical Notes; 10 Non-local Mean-Field Games: Existence; 10.1 First-Order, Non-local Mean-Field Games; 10.2 Second-Order, Non-local Mean-Field Games; 10.3 Bibliographical Notes; 11 Local Mean-Field Games: Existence; 11.1 Bootstrapping Regularity