## 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