*Accepted Paper*

**Inserted:** 27 aug 2009

**Last Updated:** 19 jan 2010

**Journal:** Comm. Partial Differential Equations

**Year:** 2009

**Abstract:**

Given two bounded open subsets $\Omega,\Lambda \subset \mathbb{R}^2$, and two densities $f$ and $g$ concentrated on $\Omega$ and $\Lambda$ respectively, we investigate the regularity of the optimal map $\nabla \varphi$ sending $f$ onto $g$. We show that, if $f$ and $g$ are both bounded away from zero and infinity, then we can find two open sets $\Omega'\subset \Omega$ and $\Lambda'\subset \Lambda$ such that $f$ and $g$ are concentrated on $\Omega'$ and $\Lambda'$ respectively, and $\nabla\varphi:\Omega' \to \Lambda'$ is a homeomorphism. Moreover, if $f$ and $g$ are smooth, then $\nabla \varphi$ is a smooth diffeomorphism between $\Omega'$ and $\Lambda'$. Finally, we give a quite precise description of the singular set of $\varphi$, showing that it is a $1$-dimensional manifold of class $C^1$ out of a countable set.

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