Boundary value problem for the mean field equation on a compact Riemann surface

2022/01/05 by Jiayu Li, Li, Jiayu, Linlin Sun +3
Computer Science · Mathematics · #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.2201.01544

openalex publication_date 2022/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (Σ,g) be a compact Riemann surface with smooth boundary ∂Σ, Δ_g be the Laplace-Beltrami operator, and h be a positive smooth function. Using a min-max scheme introduced by Djadli-Malchiodi (2006) and Djadli (2008), we prove that if Σ is non-contractible, then for any ρ∈(8kπ,8(k+1)π) with k∈ℕ^∗, the mean field equation \Δ_g u=ρ(he^u)/(∫_Σ he^udv_g) · amp;\rm in · amp;Σ
u=0 · amp;\rm on · amp;∂Σ. has a solution. This generalizes earlier existence results of Ding-Jost-Li-Wang (1999) and Chen-Lin (2003) in the Euclidean domain. Also we consider the corresponding Neumann boundary value problem. If h is a positive smooth function, then for any ρ∈(4kπ,4(k+1)π) with k∈ℕ^∗, the mean field equation \Δ_g u=ρ((he^u)/(∫_Σ he^udv_g)-(1)/(|Σ|)) · amp;\rm in · amp;Σ
∂ u/∂v=0 · amp;\rm on · amp;∂Σ. has a solution, where v denotes the unit normal outward vector on ∂Σ. Note that in this case we do not require the surface to be non-contractible.

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