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Partial regularity of Brenier solutions of the Monge-Ampère equation

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  • 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 findtwo 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 regularityresult of [8].
    Mathematics Subject Classification: Primary: 35J60; Secondary: 35D10, 49Q20.

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