Regularity in Monge's mass transfer problem
Analysis of PDEs
2013-05-02 v1
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_\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.
Cite
@article{arxiv.1305.0099,
title = {Regularity in Monge's mass transfer problem},
author = {Qi-Rui Li and Filippo Santambrogio and Xu-Jian Wang},
journal= {arXiv preprint arXiv:1305.0099},
year = {2013}
}