A variational proof of partial regularity for optimal transportation maps
Analysis of PDEs
2017-10-25 v2
Abstract
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-Amp{\`e}re 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 -regularity result.
Cite
@article{arxiv.1704.05339,
title = {A variational proof of partial regularity for optimal transportation maps},
author = {Michael Goldman and F Otto},
journal= {arXiv preprint arXiv:1704.05339},
year = {2017}
}