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

2016/10/03 by Yunyan Yang, Yang, Yunyan, Xiaobao Zhu +1 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1610.00774

openalex publication_date 2016/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (M,g) be a compact Riemannian surface without boundary, W^1,2(M) be the usual Sobolev space, J: W^1,2(M)→ ℝ be the functional defined by J(u)=(1)/(2)∫_M|∇ u|²dv_g+8π∫_M udv_g-8πlog∫_Mhe^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≥ 0 and h\not≡ 0, then the above result still holds. Our method is to exclude blow-up points on the zero set of h.

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