Conformal metrics with prescribed gaussian and geodesic curvatures

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