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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.

MSC:

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