Conformal metrics of the disk with prescribed Gaussian and geodesic curvatures

Abstract

This paper is concerned with the existence of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures. Being more specific, given nonnegative smooth functions $K: \overline{\mathbb{D}} \to \mathbb{R}$ and $h: \partial \mathbb{D} \to \mathbb{R}$, we consider the problem of finding a conformal metric realizing $K$ and $h$ as Gaussian and geodesic curvatures, respectively. This is the natural analogue of the classical Nirenberg problem posed on the disk. As we shall see, both curvatures play a role in the existence of solutions. Indeed we are able to give existence results under conditions that involve $K$ and $H$, where $H$ denotes the harmonic extension of $h$. The proof is based on the computation of the Leray-Schauder degree in a compact setting.