报告题目：Method of scaling spheres: Liouville theorems in general bounded or unbounded domains, blowing-up analysis on non-C^1 domains and other applications

报告人：戴蔚

报告时间：2023年12月20日（周三）16:00开始

报告地点：腾讯会议（807-874-434）

报告摘要：In this talk, we aim to introduce the method of scaling spheres (MSS) as a unified approach to Liouville theorems on general domains and apply it to establish Liouville theorems on arbitrary unbounded or bounded MSS applicable domains for general ≤n-th order PDEs and integral equations without translation invariance or with singularities. The set of MSS applicable domains includes any unbounded or bounded generalized radially convex domains and any complementary sets of their closures, which is invariant under Kelvin transforms and is the maximal collection of domains such that the MSS works. For instance, R^n, R^n_+, balls, bounded or unbounded cone-like domains, exterior domains, convex domains, star-shaped domains and all the complements of their closures are MSS applicable domains. One should note that, MSS applicable domains is to the MSS what convex domains (at least in one direction) is to the famous method of moving planes. As applications, we derive a priori estimates and hence existence of positive solutions from the boundary Hölder estimates for ≤n-th order elliptic equations by applying the blowing-up argument on domains with blowing-up cone boundary (BCB domains for short). After the blowing-up procedure, the BCB domains allow the limiting shape of the domain to be a cone (half space is a cone). While the classical blowing-up techniques in previous works work on C^1-smooth domains, we are able to apply blowing-up analysis on more general BCB domains on which the boundary Hölder estimates hold (can be guaranteed by uniform exterior cone property etc).

报告人简介：戴蔚，北京航空航天大学数学科学学院教授，基础数学系主任，博士生导师。2012年博士毕业于中国科学院数学与系统科学研究院，曾赴美国UC Berkeley与法国Universite Sorbonne Paris Nord做访问学者。主持国家自然科学基金3项，2022年获国家自然科学基金优秀青年科学基金项目资助。主要研究分数阶与高阶椭圆方程、发展方程及调和分析。相关研究结果发表在Adv. Math.、Trans. AMS、JFA、CVPDE等国际数学权威期刊上。