[1]陈 平.几个最优映射存在唯一性定理的统一证明[J].南京师范大学学报(自然科学版),2015,38(04):82.
 Chen Ping.Several Results About Existence and Uniqueness of Optimal Mapsin Transportation Problems:a Unified Scheme Proof[J].Journal of Nanjing Normal University(Natural Science Edition),2015,38(04):82.
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几个最优映射存在唯一性定理的统一证明()
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《南京师范大学学报》(自然科学版)[ISSN:1001-4616/CN:32-1239/N]

卷:
第38卷
期数:
2015年04期
页码:
82
栏目:
数学
出版日期:
2015-12-30

文章信息/Info

Title:
Several Results About Existence and Uniqueness of Optimal Mapsin Transportation Problems:a Unified Scheme Proof
作者:
陈 平
江苏第二师范学院数学与信息技术学院,江苏 南京 210013
Author(s):
Chen Ping
School of Mathematics and Information Technology,Jiangsu Second Normal University,Nanjing 210013,China
关键词:
凸锥Brenier定理最优运输
Keywords:
convex coneBrenier’s theoremoptimal transportation
分类号:
O174.12
文献标志码:
A
摘要:
基于凸锥的性质以及测度理论,本文给出了几个最优映射存在唯一性定理的统一证明.著名的Brenier定理以及其它几个与光线反射、折射有关的费用函数所对应的最优质量运输问题解的存在唯一性定理可以视为本文主要定理的重要推论.与Brenier定理的原始证明比较而言,本文证明过程简洁明了.
Abstract:
Based on measure theories and convex cones,we give a unified and concise theorem which proves existence and uniqueness of optimal transport maps. Some interested results can be seen as corollaries of this unified theorem,such as the Brenier’s theorem and some Monge’s problems with cost functions coming from far field re?ector problems and refraction problems.

参考文献/References:

[1]AMBROSIO L,GIGLI N. A user’s guide to optimal transport[M]. Berlin Heidelberg:Springer,2013:1-155.
[2]VILLANI C. Optimal transportation,old and new[M]. Berlin Heidelberg:Springer,2008.
[3]BRENIER Y. Polar factorization and monotone rearrangement of vector-valued functions[J]. Communications on pure and applied mathematics,1991,44:375-417.
[4]CHAMPION T,DE PASCALE L. The monge problem in Rd[J]. Duke mathematical journal,2011,157(3):551-572.
[5]JIMENEZ C,SANTAMBROGIO F. Optimal transportation for a quadratic cost with convex constraints and applications[J]. Journal de mathématiques pures et appliquées,2012,98(1):103-113.
[6]CHEN P,JIANG F D,YANG X P. Two dimensional optimal transportation problem for a distance cost with a convex constraint[J]. ESAIM:Control,optimisation and calculus of variations,2013,19(4):1064-1075.
[7]CHEN P,JIANG F D,YANG X P. Optimal transportation in Rn for a distance cost with a convex constraint[J]. Zeitschrift füer angewandte mathematik und physik,2015,66(3):587-606.
[8]DU S Z,LI Q R. Positivity of Ma-Trudinger-Wang curvature on Riemannian surfaces[J]. Calculus of variations and partial differential equations,2014,51(3/4):495-523.
[9]WANG X J. On the design of a reflector antenna Ⅱ[J]. Calculus of variations and partial differential equations,2004,20(3):329-341.
[10]GUTIéRREZ C E,HUANG Q. The refractor problem in reshaping light beams[J]. Archive for rational mechanics and analysis,2009,193(2):423-443.

备注/Memo

备注/Memo:
收稿日期:2015-04-24. 
基金项目:国家自然科学基金青年项目(11401306)、江苏省高校自然科学基金(15KJB110003)、江苏第二师范学院人才培育基金(JSNU2014YB03). 
通讯联系人:陈平,博士,研究方向:偏微分方程与几何分析. E-mail:chenping200507@126.com
更新日期/Last Update: 2015-12-30