*Accepted Paper*

**Inserted:** 28 nov 2008

**Last Updated:** 2 nov 2010

**Journal:** Arch. Ration. Mech. Anal.

**Year:** 2008

**Abstract:**

Given two densities $f$ and $g$, we consider the problem of
transporting a fraction $m$ of the mass of $f$ onto $g$
minimizing a transportation cost. If the cost per unit of mass is
given by $

x-y

^2$, we will see that uniqueness of solutions holds
for $m$ greater or equal than the common mass.
This extends the result of Caffarelli and McCann,
where the authors consider two densities with disjoint supports.
The free boundaries of the active regions are shown to be
$(n-1)$-rectifiable (provided the supports of $f$ and $g$ have
Lipschitz boundaries), and under some weak regularity assumptions
on the geometry of the supports they are also locally semiconvex.
Moreover, assuming $f$ and $g$ supported on two bounded strictly
convex sets $O$, $L$ in $R^n$, and bounded away from zero and infinity on their respective supports, $C^{0,\alpha}_{\rm loc}$
regularity of the optimal transport map and local $C^1$ regularity
of the free boundaries away from the intersection of $O$ and $L$ are shown. Finally,
the optimal transport map extends to a global homeomorphism
between the active regions.

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