Résume | The problem of prescribing the Gaussian curvature K(x) in the conformal class of a 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 has deserved a lot of attention in the literature. In the first part of the talk we will review the known results about compactness and existence of solutions for that problem.
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 focus on the case of the standard disk.
First we will perform a blow-up analysis for the solutions of this equation. We will show that, if a sequence of solutions blow-up, it tends to concentrate around a unique point at the boundary of the disk. We also study the location of such point, which concerns both curvature terms due to the interaction between them. Quite interestingly, such conditions 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 in a compact setting. |