论文标题
关于ding-jost-li-wang结果的另一个评论
Another remark on a result of Ding-Jost-Li-Wang
论文作者
论文摘要
令$（M，G）$为紧凑的Riemann表面，$ h $是$ M $上的正平滑功能。众所周知，功能性$$ J（u）= \ frac {1} {2} \ int_m | \ nabla u |^2dv_g+8π\ int_m udv_g-8π\ log \ log \ log \ log \ int_mhe^{u} dv_g $$在dv_g $下达到了最低的条件。杨和作者将此结果推广到非负$ h $。 Later, Sun and Zhu (arXiv:2012.12840) showed Ding-Jost-Li-Wang condition is also sufficient for $J$ achieves its minimum when $h$ changes sign, which was reproved later by Wang and Yang (J. Funct. Anal. 282: Paper No. 109449, 2022) and Li and Xu (Calc. Var. 61: Paper No. 143, 2022) respectively using flow approach.本说明的目的是给出阳光和朱的结果的新证明。我们的证明是基于变异方法和最大原理。
Let $(M,g)$ be a compact Riemann surface, $h$ be a positive smooth function on $M$. It is well known the functional $$J(u)=\frac{1}{2}\int_M|\nabla u|^2dv_g+8π\int_M udv_g-8π\log\int_Mhe^{u}dv_g$$ achieves its minimum under Ding-Jost-Li-Wang condition. This result was generalized to nonnegative $h$ by Yang and the author. Later, Sun and Zhu (arXiv:2012.12840) showed Ding-Jost-Li-Wang condition is also sufficient for $J$ achieves its minimum when $h$ changes sign, which was reproved later by Wang and Yang (J. Funct. Anal. 282: Paper No. 109449, 2022) and Li and Xu (Calc. Var. 61: Paper No. 143, 2022) respectively using flow approach. The aim of this note is to give a new proof of Sun and Zhu's result. Our proof is based on the variational method and the maximum principle.
