Calculus of Variations and Geometric Measure Theory

A. Figalli - Y. H. Kim

Partial regularity of Brenier solutions of the Monge-Ampère equation

created by figalli on 11 Mar 2010


Accepted Paper

Inserted: 11 mar 2010

Journal: Discrete Contin. Dyn. Syst. Series A
Year: 2010


Given $\Omega,\Lambda \subset \R^n$ two bounded open sets, and $f$ and $g$ two probability densities concentrated on $\Omega$ and $\Lambda$ respectively, we investigate the regularity of the optimal map $\nabla \varphi$ (the optimality referring to the Euclidean quadratic cost) sending $f$ onto $g$. We show that if $f$ and $g$ are both bounded away from zero and infinity, we can find two open sets $\Omega'\subset \Omega$ and $\Lambda'\subset \Lambda$ such that $f$ and $g$ are concentrated on $\Omega'$ and $\Lambda'$ respectively, and $\nabla\varphi:\Omega' \to \Lambda'$ is a (bi-Hölder) homeomorphism. This generalizes the $2$-dimensional partial regularity result of the first author.