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

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• 作者：Shibing Chen ; ^{Shibing.Chen@anu.edu.au" class="auth_mail" title="E-mail the corresponding author} ; Xu-Jia Wang ^{Xu-Jia.Wang@anu.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J60 ; 35B45 ; 49Q20 ; 28C99
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：260
• 期：2
• 页码：1954-1974
• 全文大小：359 K

文摘

In this paper we prove the strict c  -convexity and the C^1,α$C^{1,\alpha }$ regularity for potential functions in optimal transportation under condition (A3w). These results were obtained by Caffarelli ,  and  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$C^{1,\alpha }$ 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$2\le n\le 4$.