# American Institute of Mathematical Sciences

2010, 28(2): 559-565. doi: 10.3934/dcds.2010.28.559

## Partial regularity of Brenier solutions of the Monge-Ampère equation

 1 Department of Mathematics, The University of Texas at Austin, Austin TX 78712, United States 2 Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada

Received  December 2009 Revised  April 2010 Published  April 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 [8].
Citation: Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559
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