A mean field problem approach for the double curvature prescription problem

2023/09/14 by Luca Battaglia, Battaglia, Luca, Rafael López-Soriano +1
Computer Science · Mathematics · #35J20 #58J32 #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.2309.07735

openalex publication_date 2023/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we establish a new mean field-type formulation to study the problem of prescribing Gaussian and geodesic curvatures on compact surfaces with boundary, which is equivalent to the following Liouville-type PDE with nonlinear Neumann conditions: \-Δu+2K_g=2Ke^u · amp;in Σ
∂_νu+2h_g=2he^\frac u2 · amp;on ∂Σ.. We provide three different existence results in the cases of positive, zero and negative Euler characteristics by means of variational techniques.

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