Kazdan-Warner Problem on Compact Riemann Surfaces with Smooth Boundary

In this article, we show that (i) any smooth function on compact Riemann surface with non-empty smooth boundary $(M, \partial M, g)$ can be realized as a Gaussian curvature function; (ii) any smooth function on $\partial M$ can be realized as a geodesic curvature function for some metric $\tilde{g} \in [g]$. The essential steps are the existence results of Brezis-Merle type equations $-\Delta_{g} u + Au = K e^{2u} \; {\rm in} \; M$ and $\frac{\partial u}{\partial \nu} + \kappa u = \sigma e^{u} \; {\rm on} \; \partial M$ with given functions $K, \sigma$ and some constants $A, \kappa$. In addition, we rely on the extension of the uniformization theorem given by Osgood, Phillips and Sarnak.