We use PDE methods as developed for the Liouville equation to study the
existence of conformal metrics with prescribed singularities on surfaces with
boundary, the boundary condition being constant geodesic curvature. Our first
result shows that a disk with two corners admits a conformal metric with
constant Gauss curvature and constant geodesic curvature on its boundary if and
only if the two corners have the same angle. In fact, we can classify all the
solutions in a more general situation, that of the 2-sphere cut by two planes.Comment: to appear in Annales de l'IHP-AN