Conformal metrics with prescribed gaussian and geodesic curvatures

We consider the problem of prescribing the Gaussian and the geodesic curvatures of a compact surface with boundary by a confor- mal deformation of the metric. We derive some existence results using a variational approach, either by minimization of the Euler-Lagrange en- ergy or via min-max methods. One of the main tools in our approach is a blow-up analysis of solutions, which in the present setting can have di- verging volume. To our knowledge, this is the first time in which such an aspect is treated. Key ingredients in our arguments are: a blow-up anal- ysis around a sequence of points different from local maxima; the use of holomorphic domain-variations; and Morse-index estimates.