文摘
In this paper we prove the strict c -convexity and the b=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">C1,α regularity for potential functions in optimal transportation under condition (A3w). These results were obtained by Caffarelli , and for the cost b=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|2, by Liu bbr0110">[11], Loeper bbr0150">[15], Trudinger and Wang bbr0200">[20] for costs satisfying the condition (A3). For costs satisfying the condition (A3w), the results have also been proved by Figalli, Kim, and McCann bbr0060">[6], assuming that the initial and target domains are uniformly c-convex, see also bbr0210">[21]; and by Guillen and Kitagawa bbr0080">[8], assuming the cost function satisfies A3w in larger domains. In this paper we prove the strict c -convexity and the b=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">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 b=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≤4.