Uniqueness and existence results of mean field equations

讲座时间 Datetime: 2023年4月30日，星期日，16:00-17:30

地点 Venue: 海琴二号457

报告人 Speaker: 胡烨耀 教授

单位 Affiliation: 中南大学

主持人 Host：徐行 副教授

报告摘要 Abstract:

We prove a sharp uniqueness result of a mean field equation on arbitrary flat tori. We first establish the evenly symmetry of the solutions by the newly invented sphere covering inequality which is of independent interest. Then we employ a careful nodal line analysis together with a homotopic argument to show the uniqueness. Beyond the uniqueness threshold, the existence of blow-up solutions in both dimension two and four are also investigated. This is based on joint works with Daniele Bartolucci, Ze Cheng, Guangze Gu, Changfeng Gui, Aleks Jevnikar, Qinfeng Li, Weihong Xie, Wen Yang.

报告人简介：

胡烨耀,中南大学数学与统计学院特聘教授,博士生导师，于2021年入选国家级青年人才计划。主要从事椭圆型偏微分方程与共形几何等方向的研究。近期与合作者一起建立了高维球面上改进的Beckner不等式、彻底解决了著名华人数学家林长寿在2007年提出的平均场方程唯一性猜想。其研究成果在 《J. Funct. Anal.》《Int. Math. Res. Not. IMRN》《SIAM J. Math. Anal.》《Calc. Var. Partial Differential Equations》《中国科学（数学）英文版》《J. Differential Equations》《Nonlinearity》等权威期刊上发表。