Maximum principle for singular solutions of nonlinear elliptic equations and its applications
数学专题报告
报告题目(Title)：Maximum principle for singular solutions of nonlinear elliptic equations and its applications
报告人(Speaker)：初保志（Rutgers 大学）
地点(Place)：教二103
时间(Time)：2024年6月24日--6月26日10:00-11:40
邀请人(Inviter)：熊金钢
报告摘要
The maximum principle is fundamental in the study of second order elliptic equations. In this mini course, we will introduce a method to apply the maximum principle to solutions with isolated singularities. Using the Laplace equation as an example, we will illustrate the core ideas behind this method and demonstrate its application by proving the Liouville theorem for positive harmonic functions.
Following this, we will extend these methods to general conformally invariant fully nonlinear elliptic equations. This includes discussing Liouville-type theorems and removable singularities. For instance, we will show how the maximum principle for singular solutions emerges naturally from the method of moving planes, enabling the proof of Liouville-type theorems.
Finally, we will explore applications of Liouville-type theorems for conformally invariant equations, such as local gradient estimates for fully nonlinear elliptic equations involving the Schouten tensor and prescribing symmetric functions of Ricci curvature eigenvalues. Open problems will also be discussed. The material is self-contained with no prior background assumed.
