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arXiv · 1305.0099

Regularity in Monge's mass transfer problem

Qi-Rui Li·Filippo Santambrogio·Xu-Jian Wang
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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.

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Qi-Rui Li, Filippo Santambrogio, Xu-Jian Wang. 2013-05-01. Regularity in Monge's mass transfer problem. https://arxiv.org/abs/1305.0099

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