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BV estimates in optimal transportation and applications.

De Philippis, Guido and Mészáros, Alpár Richárd and Santambrogio, Filippo and Velichkov, Bozhidar (2016) 'BV estimates in optimal transportation and applications.', Archive for rational mechanics and analysis., 219 (2). pp. 829-860.


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.

Item Type:Article
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Publisher 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:
Date accepted:02 July 2015
Date deposited:28 February 2020
Date of first online publication:07 September 2016
Date first made open access:28 February 2020

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