Abstract
I will talk about the strict
-convexity
and the
regularity for potential functions in optimal transportation under condition
(A3w). These results were obtained by Caffarelli for the cost function
, 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
-convex, and by
Guillen and Kitagawa, assuming the cost function satisfies (A3w) in larger domains. We prove
the strict -convexity
and the
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
.
This is joint work with Xu-Jia Wang.