Epigraph

pp v-vi

Dedication

pp vii-viii

Contents

pp ix-xii

Preface

pp xiii-xviii

Notation

pp xix-xx

Part I - The Kantorovich Problem

pp 1-2

1 - An Introduction to the Monge Problem

pp 3-13

2 - Discrete Transport Problems

pp 14-34

3 - The Kantorovich Problem

pp 35-56

Part II - Solution of the Monge Problem with Quadratic Cost: The Brenier–McCann Theorem

pp 57-58

4 - The Brenier Theorem

pp 59-66

5 - First Order Differentiability of Convex Functions

pp 67-72

6 - The Brenier–McCann Theorem

pp 73-76

7 - Second Order Differentiability of Convex Functions

pp 77-85

8 - The Monge–Ampère Equation for Brenier Maps

pp 86-94

Part III - Applications to PDE and the Calculus of Variations and the Wasserstein Space

pp 95-96

9 - Isoperimetric and Sobolev Inequalities in Sharp Form

pp 97-103

10 - Displacement Convexity and Equilibrium of Gases

pp 104-117

11 - The Wasserstein Distance W2 on P2(Rn)

pp 118-128

12 - Gradient Flows and the Minimizing Movements Scheme

pp 129-138

13 - The Fokker–Planck Equation in the Wasserstein Space

pp 139-163

14 - The Euler Equations and Isochoric Projections

pp 164-173

15 - Action Minimization, Eulerian Velocities, and Otto's Calculus

pp 174-192

Part IV - Solution of the Monge Problem with Linear Cost: The Sudakov Theorem

pp 193-194

16 - Optimal Transport Maps on the Real Line

pp 195-203

17 - Disintegration

pp 204-215

18 - Solution to the Monge Problem with Linear Cost

pp 216-246

19 - An Introduction to the Needle Decomposition Method

pp 247-267

Appendix A - Radon Measures on Rn and Related Topics

pp 268-280