Calculus of Variations and Geometric Measure Theory

M. Goldman

A variational approach to regularity theory in optimal transportation

created by goldman on 11 Jul 2019
modified on 10 Sep 2020



Inserted: 11 jul 2019
Last Updated: 10 sep 2020

Journal: Actes du Séminaire Laurent Schwartz
Number: 13
Pages: 14
Year: 2019

ArXiv: 1907.05627 PDF


This paper describes recent results obtained in collaboration with M. Huesmann and F. Otto on the regularity of optimal transport maps. The main result is a quantitative version of the well-known fact that the linearization of the Monge-Ampère equation around the identity is the Poisson equation. We present two applications of this result. The first one is a variational proof of the partial regularity theorem of Figalli and Kim and the second is the rigorous validation of some predictions made by Carraciolo and al. on the structure of the optimal transport maps in matching problems.