# Hölder continuity and injectivity of optimal maps

created by figalli on 01 Mar 2013
modified on 04 Mar 2013

[BibTeX]

Accepted Paper

Inserted: 1 mar 2013
Last Updated: 4 mar 2013

Journal: Arch. Ration. Mech. Anal.
Year: 2013

Abstract:

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from $x$ to $y$ is given by a smooth function $c(x,y)$. If the source density $f^+(x)$ is bounded away from zero and infinity in an open region $U' \subset R^n$, and the target density $f^-(y)$ is bounded away from zero and infinity on its support $V \subset R^n$, which is strongly $c$-convex with respect to $U'$, and the transportation cost $c$ satisfies the condition of Trudinger and Wang, we deduce local Hölder continuity and injectivity of the optimal map inside $U'$ (so that the associated potential $u$ belongs to $C^{1,\alpha}_{loc}(U')$). Here the exponent $\alpha>0$ depends only on the dimension and the bounds on the densities, but not on $c$.