Mean field equations on a closed Riemannian surface with the action of an isometric group

2018/11/27 by Yunyan Yang, Yang, Yunyan, Xiaobao Zhu +1
Computer Science · Mathematics · #58J05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1811.11036

openalex publication_date 2018/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (Σ,g) be a closed Riemannian surface, G=\σ₁,⋯,σ_N\ be an isometric group acting on it. Denote a positive integer ℓ=inf_x∈ΣI(x), where I(x) is the number of all distinct points of the set \σ₁(x),⋯,σ_N(x)\. A sufficient condition for existence of solutions to the mean field equation Δ_g u=8πℓ((he^u)/(∫_Σhe^udv_g)-\frac1\rm Vol_g(Σ)) is given. This recovers results of Ding-Jost-Li-Wang (Asian J Math 1997) when ℓ=1 or equivalently G=\Id\, where Id is the identity map.

Citations

• Classification of solutions of some nonlinear elliptic equations

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