北航数学论坛学术报告

--- 分析与偏微分方程讨论班(2023春季第10讲)

Some New Inequalities in Analysis and Geometry

桂长峰 教授

(澳门大学)

时间：2023年6月12日（周一上午）10：00-11：00

地点：沙河主E404

摘要: The classical Moser-Trudinger inequality is a borderline case of Sobolev inequalities and plays an important role in geometric analysis and PDEs in general. Aubin in 1979 showed that the best constant in the Moser-Trudinger inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space h2 , while Onofri in 1982 discovered an elegant optimal form of Moser-Trudinger inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without mass center constraints. One such inequality, for example, incorporates the mass center deviation (from the origin) into the optimal inequality of Aubin on the sphere, which is for functions with mass centered at the origin. The main ingredient leading to the above inequalities is a novel geometric inequality: Sphere Covering Inequality. Efforts have also been made to show similar inequalities in higher dimensions. Among the preliminary results, we have improved Beckner’s inequality for axially symmetric functions when the dimension n = 4, 6, 8. Many questions remain open. The talk is based on collaborations with Amir Moradifam, Sun-Yung Alice Chang, Yeyao Hu, Weihong Xie, Tuoxin Li, Juncheng Wei, And Zikai Ye.

报告人简介: 桂长峰教授于1991年获得美国明尼苏达大学博士学位。目前是澳门大学教授，德克萨斯大学圣安东尼奥分校(University of Texas at San Antonion)的Dan Parman Endowed Professor。1984年在北京大学获学士学位，1987年在北京大学获硕士学位，1991年在美国明尼苏达大学获得博士学位。曾任纽约大学库郎研究所讲师，加拿大哥伦比亚大学助理教授、副教授，美国康尼迪格大学副教授、教授。2013年评选为美国数学会会士，曾获得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。他主要从事偏微分方程理论研究，特别在Allen-Cahn方程的研究，Moser-Trudinger不等式最佳常数的猜想等方面取得了一系列在国际上有重大影响的工作，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。

邀请人：戴蔚

欢迎大家参加！