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Classification of Solutions of Kantorovich Problems with Strictly Convex Norms(PDF)

《南京师大学报（自然科学版）》[ISSN:1001-4616/CN:32-1239/N]

Issue:
2016年03期
Page:
22-
Research Field:
·特约稿·
Publishing date:

Info

Title:
Classification of Solutions of Kantorovich Problems with Strictly Convex Norms
Author(s):
Chen Ping
School of Mathematics and Information Technology，Jiangsu Second Normal University，Nanjing 210013，China
Keywords:
strictly convex norm; optimal transport plans; classification; Kantorovich problem
PACS:
O174.12
DOI:
10.3969/j.issn.1001-4616.2016.03.004
Abstract:
The paper proposes a classification of solutions of Kantorovich problems with strictly convex norms. We prove a basic property theorem of any optimal plan on transport rays based on strict convexity of the norm. Then we show existence of solutions of the secondary variational problem，whose admissible set is a subset of the collection of all optimal transport plans for a given strictly convex norm. At last，we select different optimal transport plans by solving the secondary variational problem with different integrand functions which are either strictly convex or strictly concave. Furthermore，we prove that those selected optimal transport plans are either ray increasing or ray decreasing，that is we classify optimal transport plans according to ray monotonicity.

References:

［1］ AMBROSIO L，PRATELLI A. Existence and stability results in the L1 theory of optimal transportation［M］//Optimal transportation and applications. Berlin Heidelberg：Springer，2003：123-160.
［2］ 陈平. 次黎曼流形上的极值分解［J］. 安徽师范大学学报（自然科学版），2015，38（6）：533-536.
［3］ VILLANI C. Optimal transportation，old and new［M］. Berlin Heidelberg：Springer，2008.
［4］ SANTAMBROGIO F. Optimal transport for applied mathematicians［M］. Birkauser，NY：Springer，2015.
［5］ 陈平. 几个最优映射存在唯一性定理的统一证明［J］. 南京师大学报（自然科学版），2015，38（4）：82-85.
［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）：1 064-1 075.
［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］ SANTAMBROGIO F. Absolute continuity and summability of optimal transfort densities：simpler proofs and new estimates［J］. Calculus of variations and partial differential equations，2009，36（3）：343-354.
［9］ 张恭庆，郭懋正. 泛函分析讲义（上册）［M］. 北京：北京大学出版社，1990.
［10］ FELDMAN M，MCCANN R J. Uniqueness and transport density in Monge’s mass transportation problem［J］. Calculus of variations，2002，15：81-113.
［11］ CHAMPION T，DE PASCALE L. The monge problem in Rd［J］. Duke mathematical journal，2011，157（3）：551-572.

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Last Update: 2016-09-30