Regularity in Monge's mass transfer problem

TL;DR

This paper investigates the regularity properties of optimal transport mappings in Monge's problem, demonstrating bounds on eigenvalues of approximating solutions and highlighting limitations in Lipschitz continuity.

Contribution

It introduces a new approximation approach to Monge's cost function and analyzes the eigenvalues of the Jacobian of the resulting optimal mappings, revealing their boundedness and regularity issues.

Findings

Eigenvalues of Jacobian matrices are locally uniformly bounded.

T_ ext{eps} mappings are not uniformly Lipschitz continuous as ext{eps} approaches zero.

Regularity of optimal mappings can fail even with smooth, positive mass distributions.

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.