June  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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