Optimal maps in Monge’s mass transport problem. (English. Abridged French version) Zbl 0858.49002

Gangbo, Wilfried; McCann, Robert J.

C. R. Acad. Sci., Paris, Sér. I 321, No. 12, 1653-1658 (1995).

Summary: Choose a cost function \(c({\mathbf x})\geq0\) which is either strictly convex in \(\mathbb{R}^d\), or a strictly concave function of the distance \(|{\mathbf x}|\). Given two nonnegative functions \(f,g\in L^1(\mathbb{R}^d)\) with the same total mass, we assert the existence and uniqueness of a map which is measure-preserving between \(f\) and \(g\), and minimizes the mass transport cost measured against \(c({\mathbf x}-{\mathbf y})\). An analytical proof based on the Euler-Lagrange equation of a dual problem is outlined. It assumes \(f,g\) to be compactly supported, and disjointly supported in the concave case.

MSC:

49J10	Existence theories for free problems in two or more independent variables
49Q20	Variational problems in a geometric measure-theoretic setting

Keywords:

optimal maps; Monge’s mass transport problem; cost function; strictly convex; strictly concave; mass transport cost