On prescribing geodesic curvature on \(D^2\)
Let \((D, g_0)\) be the unit disc with the Euclidean metric \(g_0\). Given a continuous function \(k\) on \(S^1=\partial D\), it is an interesting problem whether \(k\) can be the geodesic curvature of some flat metric \(g\) which is pointwise conformal to the metric \(g_0\). If we set \(g=e^{2u} g_0\), where \(u\) is a function defined on \(\overline{D}\), the problem is equivalent to solve the following equation: \[ -\Delta u=0 \quad\text{in}\quad D,\qquad \frac{\partial u}{\partial n}+1=k e^{u} \quad\text{on}\quad \partial D, \tag{1} \] where \(\partial u/\partial n\) is the outer normal derivative of \(u\). The problem (1) may happen to be insolvable for general \(k\). In the paper under review the authors obtain some conditions for solvability of the problem (1) in the case when \(k\) possesses some kinds of symmetries.
scientific article; zbMATH DE number 2132716
scientific article