  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_\eps(x,y)=(\eps^2+dist(x,y)^2)^{1/2}, we consider the optimal mappings T_\eps
for these costs, and we prove that the eigenvalues of the Jacobian matrix
DT_\eps, which are all positive, are locally uniformly bounded. By an example
we prove that T_\eps is in general not uniformly Lipschitz continuous as
\eps-0, even if the mass distributions are positive and smooth, and the domains
are c-convex.
