Variational Approach to Regularity of Optimal Transport Maps: General Cost Functions

Otto Felix; Prodhomme Maxime; Ried Tobias

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Variational Approach to Regularity of Optimal Transport Maps: General Cost Functions

机译：Variational Approach to Regularity of Optimal Transport Maps: General Cost Functions

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We extend the variational approach to regularity for optimal transport maps initiated by Goldman and the first author to the case of general cost functions. Our main result is an epsilon-regularity result for optimal transport maps between Holder continuous densities slightly more quantitative than the result by De Philippis-Figalli. One of the new contributions is the use of almost-minimality: if the cost is quantitatively close to the Euclidean cost function, a minimizer for the optimal transport problem with general cost is an almost-minimizer for the one with quadratic cost. This further highlights the connection between our variational approach and De Giorgi's strategy for epsilon-regularity of minimal surfaces.

机译：我们扩展规律的变分方法最优交通地图由高盛和第一作者一般费用的情况功能。epsilon-regularity结果最优的交通工具地图持有人之间连续密度定量比德的结果Philippis-Figalli。是使用almost-minimality:如果成本是什么定量接近欧几里得的成本函数,最优运输的最小值问题一般成本是一个almost-minimizer为一个二次成本。突出我们的变分之间的联系方法和De Giorgi的策略epsilon-regularity最小的表面。

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来源
《Annals of PDE. (Partial Differential Equations)》 |2021年第2期|共74页
作者
Otto Felix; Prodhomme Maxime; Ried Tobias;
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作者单位

Max Planck Inst Math Nat Wissensch;

Univ Paul Sabatier;

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原文格式 PDF
正文语种英语
中图分类偏微分方程;
关键词
Optimal transportation; epsilon-regularity; Partial regularity; General cost functions; Almost-minimality; PROOF;

机译：最佳运输;epsilon-正则性;偏正则性;一般成本函数;近乎极简;证明;

Variational Approach to Regularity of Optimal Transport Maps: General Cost Functions

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