*Accepted Paper*

**Inserted:** 4 apr 2013

**Last Updated:** 7 nov 2013

**Journal:** J. Math. Pures et Appl.

**Year:** 2013

**Abstract:**

In this paper, we study the regularity of optimal mappings in Monge's mass transfer problem. Using the approximation to Monge's cost function given by the Euclidean distance $c(x,y)=dist(x,y)$ through the costs $c_{\varepsilon }(x,y)=\sqrt{\varepsilon^{2}+dist(x,y)^{2}}$, we consider the optimal mappings $T_\varepsilon$ for these costs, and we prove that the eigenvalues of the Jacobian matrix $DT_\varepsilon$, which are all positive, are locally uniformly bounded. By an example we prove that $T_\varepsilon$ is in general not uniformly Lipschitz continuous as $\varepsilon\to 0$, even if the mass distributions are positive and smooth, and the domains are $c$-convex.

**Keywords:**
monotone transport, Ma-Trudinger-Wang, Monge-AmpĂ¨re equation

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