走向现代数学学术报告 - 周斌研究员（No. 576）

报告题目：Regularity of the complex Monge-Ampére equation and Moser-Trudinger type inequalities

报告人：周斌 研究员（北京大学）

时间：2023年3月9日 上午9:30

腾讯会议ID：997177365

摘要：A fundamental problem for the complex Monge-Ampére equation is to establish the a priori estimates of solutions when the right hand side f is not smooth. A breakthrough was made by Kolodziej, who obtained the L^\infty-estimate when f is in L^p with p>1. It was later shown that the solution is Hölder continuous when the domain is smooth and strictly pseudo-convex, and the boundary value is Hölder continuous. These results were built upon the pluripotential theory.  In this talk, we will discuss a pure PDE approach to the regularity of the complex Monge-Ampére equation, based on the related Moser-Trudinger type inequalities. More generally, we will also discuss the relations between trace inequalities(Sobolev and Moser-Trudinger type), isocapacitary inequalities and the regularity of the complex Hessian and Monge-Ampére equations when the right hand side is a general positive Borel measure.

报告人简介：周斌，北京大学数学科学学院研究员、博士生导师，2018年获批国家级人才项目，研究方向为几何分析与偏微分方程。相关研究成果发表在Adv. Math, Math. Z., JFA等著名国际数学杂志。