The Gaussian and geodesic prescription problem via conformal changes of the metric of the disk

Author: David Ruíz

Abstract: The problem of prescribing the Gaussian curvature on compact surfaces is a classic one, and dates back to the works of Berger, Moser, Kazdan and Warner, etc. The case of the sphere receives the name of Nirenberg problem and is particularly delicate.

If the domain has a boundary, the most natural question is to prescribe also the geodesic curvature h(x) of the boundary. This problem reduces to solve a semilinear elliptic PDE under a nonlinear Neumann boundary condition. In this talk we will focus in the case of the standard disk, which is the analogue to the Nirenberg problem in the boundary setting.

First we perform a blow-up analysis for the solutions of this equation. We will show that, if a sequence of solutions blows up, it tends to concentrate around a unique point in the boundary of the disk. We are able to give necessary conditions on such point that, quite interestingly, depend on h(x) in a nonlocal way. This is joint work with A. Jevnikar, R. López-Soriano and M. Medina.

Secondly, we will give existence results. We will show how the blow-up analysis developed before can be used to compute the Leray-Schauder degree associated to the problem.