BV Estimates in Optimal Transportation and Applications

TL;DR

This paper investigates BV regularity in optimal transportation, providing new BV estimates for solutions of nonlinear PDEs and properties of Wasserstein projections, with implications for evolutionary PDEs like crowd motion.

Contribution

It introduces novel BV estimates for solutions in optimal transportation and analyzes Wasserstein projections of BV measures, including their properties and applications.

Findings

BV estimates for nonlinear PDE solutions via optimal transport

Wasserstein projection of BV measures retains bounded variation

Total variation of projection bounded by original measure's variation

Abstract

In this paper we study the BV regularity for solutions of variational problems in Optimal Transportation. As an application we recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. We also prove that the Wasserstein projection of a measure with BV density on the set of measures with density bounded by a given BV function f is of bounded variation as well. In particular, in the case f = 1 (projection onto a set of densities with an L^\infty bound) we precisely prove that the total variation of the projection does not exceed the total variation of the projected measure. This is an estimate which can be iterated, and is therefore very useful in some evolutionary PDEs (crowd motion,. . .). We also establish some properties of the Wasserstein projection which are interesting in their own, and allow for instance to prove uniqueness of such a projection in a very general framework.