Calculus of Variations and Geometric Measure Theory

G. Bouchitté - C. Jimenez - R. Mahadevan

A new $L^\infty$ estimate in optimal mass transport

created by rajesh on 11 Sep 2007


Published Paper

Inserted: 11 sep 2007

Journal: Proc. Amer. Math. Soc.
Volume: 135
Number: 11
Pages: 3525-3535
Year: 2007


Let be a bounded Lipschitz regular open subset of $\mathbb{R}^d$ and let $\mu,\nu$ be two probablity measures on $\overline{\Omega}$. It is well known that if $\mu = f dx$ is absolutely continuous, then there exists, for every $p > 1$, a unique transport map $T_p$ pushing forward $\mu$ on $\nu$ which realizes the Monge-Kantorovich distance $W_p(\mu,\nu)$. In this paper, we establish an $L^\infty$ bound for the displacement map $T_px-x$ which depends only on $p$, on the shape of $\Omega$ and on the essential infimum of the density $f$.

Keywords: optimal mass transport, p-Wasserstein distance, p>1, $L^\infty$ estimate