TY - JOUR

T1 - BV Estimates in Optimal Transportation and Applications

AU - De Philippis, Guido

AU - Mészáros, Alpár Richárd

AU - Santambrogio, Filippo

AU - Velichkov, Bozhidar

N1 - Funding Information:
The authors would like to thank te referee for a careful reading of the manuscript and for her/his comments. The second and third author gratefully acknowledge the support of the ANR Project ANR-12-MONU-0013 ISOTACE.
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2016/2/1

Y1 - 2016/2/1

N2 - 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.

AB - 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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U2 - 10.1007/s00205-015-0909-3

DO - 10.1007/s00205-015-0909-3

M3 - Article

AN - SCOPUS:84952631431

VL - 219

SP - 829

EP - 860

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -