Prescribing Gaussian and geodesic curvature on disks

Abstract

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.

18 pages, 1 figure. S. C-B is a student of the INdAM Doctoral Programme in Mathematics and/or Applications Cofunded by Marie Sklodowska-Curie Actions (INdAM-DP-COFUND-2015) and he is supported by the Marie Sklodowska-Curie fellowship of the Istituto Nazionale di Alta Matematica number 713485. D. R. have been supported by the Feder-Mineco Grant MTM2015-68210-P and by J. Andalucia (FQM116)