Calculus of Variations and Geometric Measure Theory

I. FragalĂ  - M. S. Gelli - A. Pratelli

Continuity of an optimal transport in Monge problem

created on 28 Jun 2004
modified by fragala on 15 Dec 2005


Published Paper

Inserted: 28 jun 2004
Last Updated: 15 dec 2005

Journal: JMPA
Volume: 84
Number: 9
Pages: 1261-1294
Year: 2005


Given two absolutely continuous probability measures $\nni^\pm$ in $\R^2$, we consider the classical Monge transport problem, with the Euclidean distance as cost function. We prove the existence of a {\it continuous} optimal transport, under the assumptions that (the densities of) $\nni^\pm$ are continuous and strictly positive in the interior part of their supports, and that such supports are convex, compact, and disjoint. We show through several examples that our statement is nearly optimal. Moreover, under the same hypotheses, we also obtain the continuity of the transport density