勷勤数学•专家报告

题      目：Some Recent Results on Nonlinear Problems on the Heisenberg Group

报  告  人： 唐仲伟 教授  (邀请人：钟学秀)

                                          北京师范大学

时      间： 3月21日  09：00-10:00

地     点：数科院西楼112

报告人简介:

        唐仲伟：北京师范大学数学科学学院教授、博士生导师，现任该院党委书记，兼任北京数学会副会长，2004年获中国科学院数学与系统科学研究院博士学位后进入北京师范大学任教，历任讲师、副教授，2016年晋升教授。2007年至2009年作为洪堡学者赴德国吉森大学访问研究。研究方向为偏微分方程与非线性分析，聚焦非线性薛定谔方程及方程组、分式Q-曲率问题。参与复合材料中的偏微分方程等课题，合著《偏微分方程》专著，在Int. Math. Res. Not.、J. Funct. Anal.、Calc. Var. Partial Differential Equations、J. Differential Equation、Nonlinearity、Pacific J. Math.、Sci. China Math.等期刊发表论文70余篇。

摘      要：

       This talk is concerned with nonlinear equations involving sub-elliptic operators on the Heisenberg group.  We first investigate the Bianchi–Egnell stability inequality associated with the sharp Sobolev inequality and establish the existence of a nontrivial minimizer by proving a strict inequality for the optimal stability constant. Second, we study the non-degeneracy of solutions to critical CR-Yamabe type problems on the Heisenberg group, as well as the existence of concentrating solutions to slightly subcritical equations involving the sub-Laplacian on bounded domains. Under suitable assumptions and for sufficiently small parameters, we construct sign-changing solutions with exactly two nodal domains. Finally, we establish the density and multiplicity of positive solutions to the prescribed Webster scalar curvature problem on the standard unit CR sphere. When the curvature function is asymptotically periodic on the Heisenberg group, we also prove the existence of infinitely many positive solutions. This is a joint work with Dr. Jiechen Qiang, Heming Wang, Bingwei Zhang and Yichen Zhang.

          欢迎老师、同学们参加、交流！