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 g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005215&_mathId=si1.gif&_user=111111111&_pii=S0022039615005215&_rdoc=1&_issn=00220396&md5=ec7ef0462b0dea4d435d31304c6e71ff" title="Click to view the MathML source">C^1,αgif" overflow="scroll">C1,α regularity for potential functions in optimal transportation under condition (A3w). These results were obtained by Caffarelli ,  and  for the cost g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005215&_mathId=si3.gif&_user=111111111&_pii=S0022039615005215&_rdoc=1&_issn=00220396&md5=5736091e380c8d753cc23ae0fc6e1955" title="Click to view the MathML source">c(x,y)=|x−y|²gif" overflow="scroll">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 g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005215&_mathId=si1.gif&_user=111111111&_pii=S0022039615005215&_rdoc=1&_issn=00220396&md5=ec7ef0462b0dea4d435d31304c6e71ff" title="Click to view the MathML source">C^1,αgif" overflow="scroll">C1,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 g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005215&_mathId=si4.gif&_user=111111111&_pii=S0022039615005215&_rdoc=1&_issn=00220396&md5=f1319e79c1c859dbdbf75f0b250cda46" title="Click to view the MathML source">2≤n≤4gif" overflow="scroll">2≤n≤4.