主讲人：桂长峰 教授 (University of Texas at San Antonio, 中南大学）

主持人：周风 教授

开始时间：2019年11月14日 上午10:30--11:30 

讲座地址：中北校区地理楼 264室

报告人简介：桂长峰教授是美国数学会会士，获得过加拿大太平洋数学研究所研究成果奖， 加拿大数学中心Aisensdadt 奖，IEEE 最佳论文奖，中国国家自然科学基金海外合作基金（海外杰青，与周风教授合作），其研究领域为非线性偏微分方程和应用数学以及图像处理。他在国际一流数学学术期刊发表论文50余篇，其中包括 Annals of Mathematics、Inventiones Mathematicae 等顶级期刊。最近桂长峰教授与合作者共同完成的论文“The sphere covering inequality and its applications”被世界顶级数学期刊Inventiones Mathematicae接受并在线发表。该论文成功解决了美国艺术与科学院院士、美国国家科学院院士、普林斯顿大学数学系Sun-Yung Alice Chang等人于1987年提出的关于Moser-Trudinger不等式最佳常数的猜想。

报告内容简介： In this talk, I will present recent results on axially symmetric solutions of Allen-Cahn equation.

For the existence results, we show in three dimensional Euclidean space the existence of a complete family of axially symmetric solutions with a range of logarithmic growth rates, which may be regarded as the analogue of the family of catenoids and hence called two-end solutions.  Nonexistence of two-end solution with a small growth rate is also shown, which differs from the theory of minimal surfaces.

For the classification of axially symmetric solutions with finite morse index, we show in dimension three that such solutions have finitely many ends. Furthermore, the solution has exactly two ends if its Morse index equals 1. It is also shown that there does not exist such a solution in dimensions between 4 and 10.