Nirenberg's problem on domains in the 2-sphere.
type: Item
instance of: scholarly article
title: Nirenberg's problem on domains in the 2-sphere.
zbMATH Open document ID: 1043.53032
DOI: 10.1007/BF02930764
author: Guofang Wang
published in: The Journal of Geometric Analysis
publication date: 2002-08-08T00:00:00Z
review text: Let \(\Sigma\) be a surface with canonical metric \(g_0,\) and let \(K_0\) be the relative Gaussian curvature. The Nirenberg problem consists of finding a conformal metric \(g=e^ug_0\) such that the associated Gaussian curvature equals a given function \(R\colon\Sigma\longrightarrow{\mathbb R}.\) The problem is equivalent to finding a solution \(u\colon\Sigma\longrightarrow{\mathbb R}\) to \[ -\Delta u+K_0=R(x)e^u. \] In this paper, the author considers the case in which \(\Sigma={\mathbb S}^2\) and the above equation is to be satisfied on a domain \(\Omega\subset\Sigma\) with Neumann boundary condition \(\partial u/\partial n=0.\) More precisely, he proves the following theorem: Let \(\Omega\subset{\mathbb S}^2\) be a domain with area \(A(\Omega)\in(2\pi,4\pi),\) and let \(R\colon\Omega\longrightarrow{\mathbb R}\) be a smooth function with \(\inf_{x\in\partial\Omega}R(x)>0.\) If, in addition, \(\Omega\) has at least two boundary components, then there exists a function \(u\colon\Omega \longrightarrow{\mathbb R}\) satisfying the Neumann problem \(-\Delta u+2=R(x)e^u,\) \(\partial u/\partial n\bigl| _{ \partial\Omega}=0\). Equivalently, the Gaussian curvature of \(e^ug_0\) is \(R\). Finally, the author remarks that the above theorem does not hold when \(\Omega\) is simply connected, and that the approach followed in the paper can be applied to the study of the supercritical cases in the problem of prescribing Gaussian curvature on surfaces with conical singularities.
reviewed by: Alberto Parmeggiani
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 58E11
zbMATH DE Number: 1777191
zbMATH Keywords: Nirenberg's problem
zbMATH Keywords: Neumann boundary condition
zbMATH Keywords: supercritical case
MaRDI profile type: Publication
cites work: Q3657016
cites work: Meilleures constantes dans le théorème d'inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire
cites work: Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)euin two dimensions
cites work: Conformal deformations of metric on \({\mathcal S}\) 2
cites work: Prescribing Gaussian curvature on S 2
cites work: ON NIRENBERG'S PROBLEM
cites work: Continuity of solutions of uniformly elliptic equations in \(\mathbb{R}^ 2\)
cites work: Scalar Curvatures on S 2
cites work: Prescribing Gaussian curvatures on surfaces with conical singularities
cites work: Classification of solutions of some nonlinear elliptic equations
cites work: The differential equation \(\Delta u=8\pi-8\pi he^u\) on a compact Riemann surface
cites work: Existence results for mean field equations
cites work: Conformal deformation of metrics on subdomains of surfaces
cites work: Prescribing Gaussian curvature on \(S^ 2\)
cites work: Curvature functions for compact 2-manifolds
cites work: NIRENBERG'S PROBLEM ON THE 2-DIMENSIONAL HEMI-SPHERE
cites work: Harnack type inequality: The method of moving planes
cites work: Q4850193
cites work: Q3348368
cites work: On the shape of least‐energy solutions to a semilinear Neumann problem
cites work: Q5613831
cites work: The evolution of harmonic mappings with free boundaries
cites work: Q3482498
cites work: Prescribing Curvature on Compact Surfaces with Conical Singularities
cites work: Steady state solutions of a reaction-diffusion system modeling chemotaxis
cites work: On a conjecture of Wolansky
cites work: Asymptotic behavior of a nonlinear fourth order eigenvalue problem
cites work: Une remarque sur le comportement asymptotique des solutions de -Δu=λƒ(u)
description: scientific article; zbMATH DE number 1777191