Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs (Q2758956)

Monge's optimal transport problem consists in finding, among all maps \(t\) pushing a probability measure \(\mu\) in \({\mathbb R}^n\) into another one \(\nu\) in \({\mathbb R}^n\), the minimizer of the energy NEWLINE\[NEWLINE \int_{{\mathbb R}^n}c(t(x),x)\,d\mu(x). NEWLINE\]NEWLINE Here \(c:{\mathbb R}^n\times{\mathbb R}^n\to [0,+\infty)\) is a given cost function. While there exists an extensive literature on the case when \(c\) is a strictly convex function of the distance between \(t(x)\) and \(x\), the case when \(c\) is the distance between \(t(x)\) and \(x\) has received much less attention. This paper contains a constructive proof of the existence of an optimal map in the case when \(\mu\) and \(\nu\) are absolutely continuous with respect to Lebesgue measure in \({\mathbb R}^n\), and compactly supported. The first rigorous proof of the existence of an optimal transport map goes back to \textit{L. C. Evans} and \textit{W. Gangbo} [Mem. Am. Mat. Soc. 653, 66 p. (1999; Zbl 0920.49004)]. The main new novelty here is that the supports of \(\mu\) and \(\nu\) are allowed to overlap and any regularity assumption on the densities is dropped; at the same time, also costs induced by a \(C^2\) smooth and uniformly convex norm in \({\mathbb R}^n\) are considered. On the technical level, the strategy is to study the limiting behaviour of the dual potentials associated to suitable strictly convex perturbations of the cost to obtain in the limit dual potentials with some additional properties. Then, these dual potentials are used to define the family of transport rays and to make a local change of variables so that they become parallel. In this way, after changing properly the initial and final measures (and this can be done thanks to the co-area formula and a nice Lipschitz estimate on the change of variables), the optimal map is built solving basically a one-dimensional problem. Related but independent result have been obtained by \textit{N. S. Trudinger} and \textit{X.-J. Wang} [Calc. Var. Partial Differential Equations 13, No. 1, 19--31 (2001; Zbl 1010.49030)] for the Euclidean distance and \textit{L. Ambrosio} [Lect. Notes Math. 1812, 1--52 (2003; Zbl 1047.35001)] for the case when the final measure \(\nu\) is not absolutely continuous. See also the later paper by \textit{L. Ambrosio} and \textit{A. Pratelli} [Lect. Notes Math. 1813, 123--160 (2003; Zbl 1065.49026)] for a unified proof of existence and stability in this context.