Strict convexity and C^1,α regularity of potential functions in optimal transportation under condition A3w

In this paper we prove the strict c-convexity and the C^1,α regularity for potential functions in optimal transportation under condition (A3w). These results were obtained by Caffarelli [1,3,4] for the cost c(x, y)=|x-y|², by Liu [11], Loeper [15], Trudinger and Wang [20] for costs satisfying the condition (A3). For costs satisfying the condition (A3w), the results have also been proved by Figalli, Kim, and McCann [6], assuming that the initial and target domains are uniformly c-convex, see also [21]; and by Guillen and Kitagawa [8], assuming the cost function satisfies A3w in larger domains. In this paper we prove the strict c-convexity and the C^1,α regularity assuming either the support of source density is compactly contained in a larger domain where the cost function satisfies A3w, or the dimension 2≤n≤4.