Calculus of Variations and Geometric Measure Theory

S. Chen - A. Figalli

Partial $W^{2,p}$ regularity for optimal transport maps

created by figalli on 02 Mar 2017


Accepted Paper

Inserted: 2 mar 2017
Last Updated: 2 mar 2017

Journal: J. Funct. Anal.
Year: 2017


We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class $W^{2,p}_{loc}$ for any $p \geq 1$ outside of a closed singular set of measure zero. We also establish global $W^{2,p}$ estimates when the cost is a small perturbation of the quadratic cost. The latter result is new even when the cost is exactly the quadratic cost.