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BV Estimates in Optimal Transportation and Applications

De Philippis, Guido; Mészáros, Alpár Richárd; Santambrogio, Filippo; Velichkov, Bozhidar

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Guido De Philippis

Filippo Santambrogio

Bozhidar Velichkov


In this paper we study the BV regularity for solutions of certain variational problems in Optimal Transportation. We 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 and we also provide a precise estimate of its BV norm. Of particular interest is the case f = 1, corresponding to a projection onto a set of densities with an L ∞ bound, where we prove that the total variation decreases by projection. This estimate and, in particular, its iterations have a natural application to some evolutionary PDEs as, for example, the ones describing a crowd motion. In fact, as an application of our results, we obtain BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. We also establish some properties of the Wasserstein projection which are interesting in their own right, and allow, for instance, for the proof of the uniqueness of such a projection in a very general framework.


De Philippis, G., Mészáros, A. R., Santambrogio, F., & Velichkov, B. (2016). BV Estimates in Optimal Transportation and Applications. Archive for Rational Mechanics and Analysis, 219(2), 829-860.

Journal Article Type Article
Acceptance Date Jul 2, 2015
Online Publication Date Sep 7, 2016
Publication Date Feb 28, 2016
Deposit Date Oct 1, 2019
Publicly Available Date Feb 28, 2020
Journal Archive for Rational Mechanics and Analysis
Print ISSN 0003-9527
Electronic ISSN 1432-0673
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 219
Issue 2
Pages 829-860
Related Public URLs


Accepted Journal Article (407 Kb)

Copyright Statement
This is a post-peer-review, pre-copyedit version of an article published in Archive for rational mechanics and analysis. The final authenticated version is available online at:

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