报告人：桂长峰（澳门大学）

邀请人：吴付科

报告时间：2023年10月8日（星期日）10:00-12:00

报告地点：科技楼南楼702室

报告题目：Some New Inequalities in Analysis and Geometry

报告摘要：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 H1, 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 pre- liminary 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，澳门大学数学系讲座教授，数学系主任，澳大发展基金会数学杰出学者，国家级人才项目获得者，博士生导师。1991年在美国明尼苏达大学获博士学位。桂长峰教授曾入选国家级人才计划和海外高层次人才，于2013年当选美国数学会首届会士，获得过IEEE最佳论文奖、加拿大太平洋数学研究所研究成果奖、加拿大数学中心Andrew Aisensdadt奖等荣誉。桂长峰教授现致力于非线性偏微分方程的研究，特别是在Allen-Cahn方程的研究、Moser-Trudinger不等式最佳常数的猜想、De Giorgi 猜想和Gibbons 猜想等方面取得了一系列在国际上有影响的工作，在Ann. of Math., Invent. Math., Comm. Pure Appl. Math.等国际顶级期刊上发表论文80余篇。