Calculus of Variations and Geometric Measure Theory

A. Pratelli

Equivalence between some definitions for the optimal mass transport problem and for the transport density on manifolds

created on 18 Mar 2003
modified by pratelli on 04 Oct 2005


Published Paper

Inserted: 18 mar 2003
Last Updated: 4 oct 2005

Journal: Ann. Mat. Pura Appl.
Volume: 184
Number: 2
Pages: 215-238
Year: 2005


In this paper we consider three problems, which are related to the classical Monge's optimal mass transport problem and which are known to be equivalent when the ambient space is an open, convex and bounded subset of $\R^n$; to these problems there correspond different definitions of particular measures (often called {\it transport densities}), which are also known to be equivalent. Here we will generalize the setting of these problems and the resulting definitions of transport densities to the case of a Riemannian manifold endowed with a finslerian semidistance, and we will see that the equivalences still hold.