Calculus of Variations and Geometric Measure Theory

J. Bertrand - A. Pratelli - M. Puel

Kantorovich potentials and continuity of total cost for relativistic cost functions

created by pratelli on 01 Sep 2017
modified on 17 Dec 2019


Published Paper

Inserted: 1 sep 2017
Last Updated: 17 dec 2019

Journal: J. Math. Pures Appl.
Year: 2018


In this paper we consider the optimal mass transport problem for relativistic cost functions, introduced in~BertrandPuel2013 as a generalization of the relativistic heat cost. A typical example of such a cost function is $c_t(x,y)=h(\frac{y-x}t)$, $h$ being a strictly convex function when the variable lies on a given ball, and infinite otherwise. It has been already proved that, for every $t$ larger than some critical time $T>0$, existence and uniqueness of optimal maps hold; nonetheless, the existence of a Kantorovich potential is known only under quite restrictive assumptions. Moreover, the total cost corresponding to time $t$ has been only proved to be a decreasing right-continuous function of $t$. In this paper, we extend the existence of Kantorovich potentials to a much broader setting, and we show that the total cost is a continuous function. To obtain both results the two main crucial steps are a refined ``chain lemma'' and the result that, for $t>T$, the points moving at maximal distance are negligible for the optimal plan.