Inserted: 20 apr 2018
Last Updated: 20 jun 2018
Journal: Advanced Nonlinear Studies
In this paper we consider the problem of prescribing the Gaussian and geodesic curvature on a disk and its boundary, respectively, via a conformal change of the metric. This leads us to a Liouville-type equation with a nonlinear Neumann boundary condition. We address the question of existence by setting the problem in a variational framework which seems to be completely new in the literature. We are able to find minimizers under symmetry assumptions.
Keywords: Variational methods, Moser-Trudinger inequality, Prescribed Gaussian curvature problem