Another remark on a result of Ding-Jost-Li-Wang

2022/12/20 by Xiaobao Zhu, Zhu, Xiaobao · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2212.09943

openalex publication_date 2022/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (M,g) be a compact Riemann surface, h be a positive smooth function on M. It is well known the functional J(u)=(1)/(2)∫_M|∇ u|²dv_g+8π∫_M udv_g-8πlog∫_Mhe^udv_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.

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