报告题目：Regularity of singular set in optimal transportation
报 告 人：陈世炳（中国科学技术大学 教授）
会议时间：2023年11月10日 15:30开始
会议地点：9号楼 113报告厅
报告摘要：In this talk, we discuss a regularity theory for the optimal transport problem when the target consists of two disjoint convex domains, a fundamental model in which singularities of optimal transport maps arise. When the target is partitioned into two disjoint convex domains, $\Omega^*_i$ for $i=1, 2$, and the densities are bounded from below and above by positive constants, we demonstrate that the singular set of the optimal transport map constitutes a $C^{1,\beta}$ sub-manifold of $\mathbb{R}^n$ for some $\beta \in (0, 1)$. Furthermore, when $\Omega^*_i$ for $i=1, 2$, are assumed to be $C^2$ and uniformly convex, and the densities are both positive and $C^\alpha$ smooth, we establish that the singular set forms an $(n-1)$-dimensional $C^{2,\alpha}$ sub-manifold of $\mathbb{R}^n$. Notably, our results are achieved without requiring any convexity of the source domain, aligning with the celebrated regularity theory of optimal transport maps developed by Caffarelli.
报告人简介：陈世炳，中国科学技术大学数学系特任教授，博士生导师，2018年入选国家级青年人才计划，2022年获得国家杰出青年基金。先后于美国MSRI和澳洲国立大学数学系从事博士后研究。长期致力于Monge-Ampere方程及其在最优传输和几何中的应用。研究成果多次发表在Ann. Math.、Adv. Math.、J. Math. Pures Appl.、 Trans. Amer. Math. Soc.、SIAM J. Math. Anal.、J. Funct. Anal.等国际著名期刊上。
