Optimal Transportation and Applications : Lectures Given at the C.I.M.E. Summer School Held in Martina Franca, Italy, September 2-8, 2001,9783540401926

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

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ISBN13: 9783540401926
ISBN10: 354040192X
Format: Paperback
Pub. Date: 2003-08-01
Publisher(s): Springer Verlag
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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

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