讲座题目：Improved Beckner's inequality for axially symmetric functions on $\mathbb{S}^4$ 主讲人🧏🏿♀️：桂长峰 教授 主持人：叶东 教授 开始时间：2020-11-13 08:30:00 讲座地址：数学楼102报告厅、腾讯会议 ID：197 139 648 主办单位：数学科学学院
报告人简介🧓🏿： 桂长峰教授，德克萨斯大学圣安东尼奥分校。于1984年本科毕业北京大学♣︎，1987年在北京大学获得硕士学位💂🏿♂️，1991年在美国明尼苏达大学获博士学位。曾任纽约大学库郎研究所讲师，加拿大哥伦比亚大学助教🧑🏿🚒、副教授👩🏽🦳，美国康尼迪格大学副教授👩🏿🚒、教授🧑🏻💻、德克萨斯大学圣安东尼奥分校教授𓀀。他主要从事偏微分方程理论研究，特别是对Allen-Cahn方程的研究上取得了一系列在国际上有影响的工作，在国际一流数学学术期刊发表论文70余篇，其中包括 Annals of Mathematics、Inventiones Mathematicae 等顶级期刊🙅🏿♀️🧑🏽🦱。同时他在图像处理方面也有很好的工作，他与合作者撰写的论文Distance Regularized Level Set Evolution and Its Application to Image Segmentation 获得了2015年IEEE SIGNAL PROCESSING SOCIETY颁发的最佳论文奖♠️。
报告内容🏌️♂️： We show that axially symmetric solutions on $\mathbb{S}^4$ to a constant $Q$-curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter $\alpha$ in front of the Paneitz operator belongs to $[\frac{473 + \sqrt{209329}}{1800}\approx0.517, 1)$. This is in contrast to the case $\alpha=1$, where a family of non constant solutions exist, known as the standard bubbles. The phenomenon resembles the Gaussian curvature equation on $ \mathbb{S}^2$ in connection to the Moser-Trudinger-Onofri inequality. As a consequence, we prove an improved Beckner's inequality on $\mathbb{S}^4$ for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when $\alpha=\frac15$ by exploiting Pohozaev-type identities, and prove existence of a non-constant axially symmetric solution for $\alpha \in (\frac15, \frac12)$ via a bifurcation method. This is a joint work with Yeyao Hu and Weihong Xie from the Central South University. |
