Strict convexity and $C^{1,\alpha }$–regularity of potential functions in optimal transportation under condition (A3w)

Abstract

I will talk about the strict $c$-convexity and the $C^{1,\alpha }$ regularity for potential functions in optimal transportation under condition (A3w). These results were obtained by Caffarelli for the cost function $c\left(x,y\right)=|x-y{|}^2$, by Liu, Loeper, Trudinger and Wang for cost functions satisfying the condition (A3). For cost functions satisfying the condition (A3w), the results have also been proved by Figalli, Kim, and McCann assuming that the initial and target domains are uniformly $c$-convex, and by Guillen and Kitagawa, assuming the cost function satisfies (A3w) in larger domains. We prove the strict $c$-convexity and the $C^{1,\alpha }$ regularity assuming either the support of the source density is compactly contained in a larger domain where the cost function satisfies (A3w), or, in dimension $2\le n\le 4$.

This is joint work with Xu-Jia Wang.