## A. Figalli - Y. H. Kim - R. J. McCann

# Hölder continuity and injectivity of optimal maps

created by figalli on 01 Mar 2013

modified on 04 Mar 2013

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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 $c$.

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