Regularity in Monge's mass transfer problem

2013/05/01 by Li, Qi-Rui, Santambrogio, Filippo, Wang, Xu-Jian
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1305.0099

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)=(\eps²+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.

Related

• Obstructions to regularity in the classical Monge problem
• Local regularity result for an optimal transportation problem with rough measures in the plane
• Regularity for a log-concave to log-concave mass transfer problem with near Euclidean cost
• Boundary ε-regularity in optimal transportation
• Canonical duality approach in the approximation of optimal Monge mass transfer mapping