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In this article, we show that (i) any smooth function on compact Riemannsurface with non-empty smooth boundary $ (M, \partial M, g) $ can be realizedas 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 typeequations $ -\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 byOsgood, Phillips and Sarnak.

Kazdan-Warner Problem on Compact Riemann Surfaces with Smooth Boundary