Calculus of Variations and Geometric Measure Theory

M. Goldman - F. Otto

A variational proof of partial regularity for optimal transportation maps

created by goldman on 08 Apr 2017
modified on 10 Dec 2018


Accepted Paper

Inserted: 8 apr 2017
Last Updated: 10 dec 2018

Journal: Annales de l'ENS
Year: 2018
Links: constant densities


We provide a new proof of the known partial regularity result for the optimal transportation map (Brenier map) between two sets. Contrary to the existing regularity theory for the Monge-Ampère equation, which is based on the maximum principle, our approach is purely variational. By constructing a competitor on the level of the Eulerian (Benamou-Brenier) formulation, we show that locally, the velocity is close to the gradient of a harmonic function provided the transportation cost is small. We then translate back to the Lagrangian description and perform a Campanato iteration to obtain an epsilon-regularity result.

A simplified version of this paper in the case of constant densities is available at the link above.