BV Estimates in Optimal Transportation and Applications

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

Abstract

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.

Citation

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. https://doi.org/10.1007/s00205-015-0909-3

• 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
• DOI: https://doi.org/10.1007/s00205-015-0909-3
• Public URL: https://durham-repository.worktribe.com/output/1320039
• Related Public URLs: https://arxiv.org/abs/1503.06389