Optimal Mass Transport on Euclidean Spaces

图书简介

Optimal mass transport has emerged in the past three decades as an active field with wide-ranging connections to the calculus of variations, PDEs, and geometric analysis. This graduate-level introduction covers the field’s theoretical foundation and key ideas in applications. By focusing on optimal mass transport problems in a Euclidean setting, the book is able to introduce concepts in a gradual, accessible way with minimal prerequisites, while remaining technically and conceptually complete. Working in a familiar context will help readers build geometric intuition quickly and give them a strong foundation in the subject. This book explores the relation between the Monge and Kantorovich transport problems, solving the former for both the linear transport cost (which is important in geometric applications) and for the quadratic transport cost (which is central in PDE applications), starting from the solution of the latter for arbitrary transport costs.

Preface; Notation; Part I. The Kantorovich Problem: 1. An introduction to the Monge problem; 2. Discrete transport problems; 3. The Kantorovich problem; Part II. Solution of the Monge Problem with Quadratic Cost: the Brenier-McCann Theorem: 4. The Brenier theorem; 5. First order differentiability of convex functions; 6. The Brenier-McCann theorem; 7. Second order differentiability of convex functions; 8. The Monge-Ampère equation for Brenier maps; Part III. Applications to PDE and the Calculus of Variations and the Wasserstein Space: 9. Isoperimetric and Sobolev inequalities in sharp form; 10. Displacement convexity and equilibrium of gases; 11. The Wasserstein distance W2 on P2(Rn); 12. Gradient flows and the minimizing movements scheme; 13. The Fokker-Planck equation in the Wasserstein space; 14. The Euler equations and isochoric projections; 15. Action minimization, Eulerian velocities and Otto’s calculus; Part IV. Solution of the Monge Problem with Linear Cost: the Sudakov Theorem: 16. Optimal transport maps on the real line; 17. Disintegration; 18. Solution to the Monge problem with linear cost; 19. An introduction to the needle decomposition method; Appendix A: Radon measures on Rn and related topics; Appendix B: Bibliographical Notes; Bibliography; Index.

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Optimal Mass Transport on Euclidean Spaces（Cambridge Studies in Advanced Mathematics, 207） 欧基里得空间上的*质量传输 代数学

Optimal Mass Transport on Euclidean Spaces（Cambridge Studies in Advanced Mathematics, 207）

欧基里得空间上的质量传输
代数学*