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Journal Articles Journal de Mathématiques Pures et Appliquées Year : 2014

Regularity in Monge's mass transfer problem

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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Dates and versions

hal-00819131 , version 1 (30-04-2013)

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Qi-Rui Li, Filippo Santambrogio, Xu-Jia Wang. Regularity in Monge's mass transfer problem. Journal de Mathématiques Pures et Appliquées, 2014, pp.In Press, Corrected Proof, only online for the moment. ⟨hal-00819131⟩
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