勷勤数学•专家报告

题      目：On a Class of Monge-Amp\`ere Obstacle Problems and Related Results

报  告  人：涂绪山 博士  (邀请人：李进开)

                                             香港科技大学

时      间： 9月5日  15：00-16:00

地     点：数科院西楼114教室

报告人简介:

          涂绪山，2017年和2022年分别获得清华大学学士与博士学位，博士导师为简怀玉教授，现于香港科技大学从事博士后研究，合作导师为金天灵教授。主要研究方向为非线性椭圆偏微分方程及其应用，在Monge-Ampère方程解的正则性理论、分类定理，以及奇点分析与障碍问题等方向做出了系列重要工作，成果发表于Adv. Math.、J. Funct. Anal.等期刊。  

摘      要：

        In this talk, we study a Monge-Amp\`ere obstacle problem, which was initially studied by Savin, and discuss related topics. First, we develop regularity theory for $\det D^2u=u^q$, establish Liouville theorems for its entire solutions. Next, we develop a variational framework for Aleksandrov estimates of convex solutions, identifying both isolated singularity problems and obstacle problems as extremal configurations, and establish refined Alexandrov-Bakelman-Pucci (ABP) estimates under suitable assumptions. Furthermore, we present a global version of these improved ABP estimates and generalize a theorem of Caffarelli and Li. Finally, we may discuss the  Monge-Amp\`ere equations with multiple isolated singularities, with particular emphasis on how obstacle problems influence their structural behavior. This is joint work with Tianling Jin and Jingang Xiong.

        欢迎老师、同学们参加、交流！