Prescribing Gaussian curvature on surfaces with conical singularities and geodesic boundary

We study conformal metrics with prescribed Gaussian curvature on surfaces conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive general existence result for at least two components. This seems to be the first this setting. Moreover, allow have both positive negative orders, that is cone angles less grater than $2\pi$.