Calculus of Variations and Geometric Measure Theory
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N. Gigli

On Hölder continuity in time of the optimal transport map towards measures along a curve

created by gigli on 17 Nov 2008
modified on 09 Feb 2010


Accepted Paper

Inserted: 17 nov 2008
Last Updated: 9 feb 2010

Journal: Proceedings of the Edinburgh Mathematical Society
Year: 2008


We discuss the problem of the regularity in time of the map $t\mapsto T_t\in L^p(R^d,R^d;\sigma)$ where $T_t$ is a transport map (optimal or not) from a reference measure $\sigma$ to a measure $\mu_t$ which lies along an absolutely continuous curve $t\mapsto \mu_t$ in the space $(P_p({R^d}),W_p)$. We prove that in most cases such a map is no more than $p^{-1}$-Hölder continuous.

Keywords: Wasserstein distance, Monge-Ampère equation


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