A geometric approach to apriori estimates for optimal transport maps

被引：0

作者：

Brendle, Simon ^{[1
]}

Leger, Flavien ^{[2
]}

McCann, Robert J. ^{[3
]}

Rankin, Cale ^{[4
,5
]}

机构：

[1] Columbia Univ, 2290 Broadway, New York, NY 10027 USA

[2] INRIA Paris, Paris, France

[3] Univ Toronto, Dept Math, Toronto, ON M5S 2E4, Canada

[4] Univ Toronto, Dept Math, Toronto, ON M5S 2E4, Canada

[5] Monash Univ, Sch Math, 9 Rainforest Walk, Melbourne, Vic 3800, Australia

来源：

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK | 2024年 / 2024卷 / 817期

基金：

美国国家科学基金会; 加拿大自然科学与工程研究理事会;

关键词：

BOUNDARY-VALUE PROBLEM; POTENTIAL FUNCTIONS; REGULARITY; EQUATIONS; PRODUCTS; DESIGN;

D O I：

10.1515/crelle-2024-0071

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma-Trudinger-Wang condition is the Pogorelov second-derivative bound. This translates to an apriori interior C 1 C<^>{1} estimate for smooth optimal maps. Here we give a new derivation of this estimate which relies in part on Kim, McCann and Warren's observation that the graph of an optimal map becomes a volume maximizing spacelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost.

引用

页码：251 / 266

页数：16

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