On Asymptotic Behaviour and W2,p Regularity of Potentials in Optimal Transportation

被引：5

作者：

Liu, Jiakun ^{[1
]}

Trudinger, Neil S. ^{[2
]}

Wang, Xu-Jia ^{[2
]}

机构：

[1] Univ Wollongong, Sch Math & Appl Stat, Inst Math & Its Applicat, Wollongong, NSW 2522, Australia

[2] Australian Natl Univ, Ctr Math & Its Applicat, Canberra, ACT 0200, Australia

来源：

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS | 2015年 / 215卷 / 03期

基金：

澳大利亚研究理事会;

关键词：

MONGE-AMPERE EQUATION; STRICT CONVEXITY; INTERIOR; CONTINUITY; MAPS;

D O I：

10.1007/s00205-014-0797-y

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper we study local properties of cost and potential functions in optimal transportation. We prove that in a proper normalization process, the cost function is uniformly smooth and converges locally smoothly to a quadratic cost x center dot y, while the potential function converges to a quadratic function. As applications we obtain the interior W (2, p) estimates and sharp C (1, alpha) estimates for the potentials, which satisfy a Monge-AmpSre type equation. The W (2, p) estimate was previously proved by Caffarelli for the quadratic transport cost and the associated standard Monge-AmpSre equation.

引用

页码：867 / 905

页数：39

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