A Remark on a Result of Ding-Jost-Li-Wang
主 题: A Remark on a Result of Ding-Jost-Li-Wang
报告人: Xiaobao Zhu (Renmin University of China)
时 间: 2016-04-12 10:10 - 2016-04-12 12:00
地 点: Room 29, Quan Zhai, BICMR
Let $(M,g)$ be a compact Riemannian surface without boundary, $W^{1,2}(M)$ be the usual Sobolev space,$J: W^{1,2}(M)\ra \mathbb{R}$ be the functional defined by $$J(u)=\f{1}{2}\int_M|\nabla u|^2dv_g+8\pi \int_M udv_g-8\pi\log\int_M he^udv_g,$$where $h$ is a positive smooth function on $M$. In an inspiring work (Asian J. Math., vol. 1, pp. 230-248, 1997),Ding, Jost, Li and Wang obtained a sufficient condition under which $J$ achieves its minimum.In this note, we prove that if the smooth function $h$ satisfies $h\geq 0$ and $h\not\equiv 0$, thenthe above result still holds. Our method is to exclude blow-up points on the zero set of $h$. This is a joint work with Prof. Yunyan Yang.
