Nirenberg's problem on domains in the 2-sphere.
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.
scientific article; zbMATH DE number 1777191
scientific article