Archive for Rational Mechanics and Analysis

, Volume 219, Issue 2, pp 829–860 | Cite as

BV Estimates in Optimal Transportation and Applications

  • Guido De Philippis
  • Alpár Richárd Mészáros
  • Filippo Santambrogio
  • Bozhidar Velichkov
Article

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Guido De Philippis
    • 1
  • Alpár Richárd Mészáros
    • 2
  • Filippo Santambrogio
    • 2
  • Bozhidar Velichkov
    • 3
  1. 1.UMPA, CNRS and École Normale Supérieure de LyonLyon Cedex 07France
  2. 2.Laboratoire de Mathématiques d’OrsayUniversité Paris-SudOrsay CedexFrance
  3. 3.Laboratoire LJKUniversité J. FourierGrenoble Cedex 9France

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