This paper is concerned with the compactness of metrics of the disk with
prescribed Gaussian and geodesic curvatures. We consider a blowing-up sequence
of metrics and give a precise description of its asymptotic behavior. In
particular, the metrics blow-up at a unique point on the boundary and we are
able to give necessary conditions on its location. It turns out that such
conditions depend locally on the Gaussian curvatures but they depend on the
geodesic curvatures in a nonlocal way. This is a novelty with respect to the
classical Nirenberg problem where the blow-up conditions are local, and this
new aspect is driven by the boundary condition.Comment: 31 pages, 1 figur