Optimal Transportation and Applications : Lectures Given at the C.I.M.E. Summer School Held in Martina Franca, Italy, September 2-8, 2001

by Ambrosio, Luigi; Caffarelli, Luis A.; Brenier, Yann; Ambrosio, Luigi; Buttazzo, Giuseppe; Villani, Cedric; Caffarelli, Luis A.; Salsa, S.

• ISBN13: 9783540401926
• ISBN10: 354040192X
• Format: Paperback
• Copyright: 2003-08-01
• Publisher: Springer Verlag

Summary

Leading researchers in the field of Optimal Transportation, with different views and perspectives, contribute to this Summer School volume: Monge-Ampére and Monge-Kantorovich theory, shape optimization and mass transportation are linked, among others, to applications in fluid mechanics granular material physics and statistical mechanics, emphasizing the attractiveness of the subject from both a theoretical and applied point of view. The volume is designed to become a guide to researchers willing to enter into this challenging and useful theory.

Table of Contents

The Monge-Ampere Equation and Optimal Transportation, an elementary review	1	(10)
Luis Caffarelli
Optimal Transportation	1	(1)
The continuous case:	2	(1)
The dual problem:	3	(1)
Existence and Uniqueness:	4	(2)
The potential equation:	6	(1)
Some remarks on the structure of the equation	7	(4)
Optimal Shapes and Masses, and Optimal Transportation Problems	11	(42)
Giuseppe Buttazzo
Luigi De Pascale
Introduction	11	(2)
Some classical problems	13	(10)
The isoperimetric problem	13	(1)
The Newton's problem of optimal aerodynamical profiles	14	(3)
Optimal Dirichlet regions	17	(2)
Optimal mixtures of two conductors	19	(4)
Mass optimization problems	23	(6)
Optimal transportation problems	29	(4)
The optimal mass transportation problem: Monge and Kantorovich formulations	30	(2)
The PDE formulation of the mass transportation problem	32	(1)
Relationships between optimal mass and optimal transportation	33	(2)
Further results and open problems	35	(18)
A vectorial example	35	(2)
A p--Laplacian approximation	37	(1)
Optimization of Dirichlet regions	38	(2)
Optimal transporting distances	40	(4)
References	44	(9)
Optimal transportation, dissipative PDE's and functional inequalities	53	(38)
Cedric Villani
Some motivations	54	(1)
A study of fast trend to equilibrium	55	(9)
A study of slow trend to equilibrium	64	(7)
Estimates in a mean-field limit problem	71	(9)
Otto's differential point of view	80	(11)
References	88	(3)
Extended Monge-Kantorovich Theory	91	(32)
Yann Brenier
Abstract	91	(1)
Generalized geodesics and the Monge-Kantorovich theory	91	(11)
Generalized geodesics	91	(3)
Extension to probability measures	94	(2)
A decomposition result	96	(1)
Relativistic MKT	97	(1)
A relativistic heat equation	98	(2)
Laplace's equation and Moser's lemma revisited	100	(2)
Generalized Harmonic functions	102	(7)
Classical harmonic functions	102	(3)
Open problems	105	(4)
Multiphasic MKT	109	(2)
Generalized extremal surfaces	111	(3)
MKT revisited as a subset of generalized surface theory	113	(1)
Degenerate quadratic cost functions	113	(1)
Generalized extremal surfaces in R5 and Electrodynamics	114	(9)
Recovery of the Maxwell equations	115	(1)
Derivation of a set of nonlinear Maxwell equations	116	(2)
An Euler-Maxwell-type system	118	(2)
References	120	(3)
Existence and stability results in the L1 theory of optimal transportation	123
Luigi Ambrosio
Aldo Pratelli
Introduction	123	(6)
Notation	129	(1)
Duality and optimality conditions	130	(5)
Γ-convergence and Γ-asymptotic expansions	135	(1)
1-dimensional theory	136	(2)
Transport rays and transport set	138	(8)
A stability result	146	(4)
A counterexample	150	(2)
Appendix: disintegration of measures	152
References	158