Calculus of Variations and Geometric Measure Theory

Q. R. Li - F. Santambrogio - X. J. Wang

Regularity in Monge's mass transfer problem

created by santambro on 04 Apr 2013
modified on 07 Nov 2013


Accepted Paper

Inserted: 4 apr 2013
Last Updated: 7 nov 2013

Journal: J. Math. Pures et Appl.
Year: 2013


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-Ampère equation