Regularity in Monge's mass transfer problem

In this paper, we study the regularity of optimal mappings in Monge's mass transfer problem. Using the approximation to Monge's cost function c(x, y)=|x-y| through the costs cε(x,y)=ε2+|x-y|2, we consider the optimal mappings T_ε for these costs, and we prove that the eigenvalues of the Jacobian matrix DT_ε, which are all positive, are locally uniformly bounded. By an example we prove that T_ε is in general not uniformly Lipschitz continuous as ε→0, even if the mass distributions are positive and smooth, and the domains are c-convex.