# A variational approach to regularity theory in optimal transportation

## TL;DR

This paper introduces a variational method to analyze the regularity of optimal transport maps, providing quantitative insights and applications to partial regularity and structure predictions in matching problems.

## Contribution

It offers a new variational approach to regularity theory in optimal transportation, including a quantitative linearization of the Monge-Ampère equation and applications to existing regularity results.

## Key findings

- A quantitative linearization of the Monge-Ampère equation around the identity.
- A variational proof of the partial regularity theorem by Figalli and Kim.
- Validation of structure predictions in optimal transport matching problems.

## Abstract

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-Amp{\`e}re 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.