Strict convexity and C1,α regularity of potential functions in optimal transportation under condition A3w

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:260

Article

Chen, Shibing¹  Wang, Xu-Jia¹ 

[1] Australian Natl Univ, Ctr Math & Its Applicat, GPO Box 4, Canberra, ACT 0200, Australia

关键词: Optimal transportation;    Strict convexity;    Regularity;   

DOI  :  10.1016/j.jde.2015.09.047

来源: Elsevier

【 摘 要 】

In this paper we prove the strict c-convexity and the C-1,C-alpha regularity for potential functions in optimal transportation under condition (A3w). These results were obtained by Caffarelli 11,3,41 for the cost c(x, y) = vertical bar x - y vertical bar(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,C-alpha 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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free