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

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.


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