Abstract

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 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.

个人简介

桂长峰，澳门大学科技学院讲座教授、数学系主任，澳大发展基金会数学杰出学人教授，研究方向为非线性偏微分方程、图像分析和处理，在国际顶级期刊如《Annals of Mathematics》《Inventiones Mathematicae》《Communications on Pure and Applied Mathematics》发表多篇论文。曾获颁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信号处理协会最佳论文奖、中国国家自然科学基金海外合作基金（海外杰青）等奖项。美国数学学会会士、美国西蒙斯会士、美国德州大学圣安东尼奥分校丹．帕尔曼应用数学冠名讲座教授。曾入选国家教育部长江学者讲座教授。