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

Abstract

We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class $W_{loc}^{2,p}$ for any $p\ge 1$ 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.

References in corpus (1)

• Continuity and injectivity of optimal maps for non-negatively cross-curved costs