A mean field problem approach for the double curvature prescription problem

In this paper we establish a new mean field-type formulation to study the problem of prescribing Gaussian and geodesic curvatures on compact surfaces with boundary, which is equivalent to the following Liouville-type PDE with nonlinear Neumann conditions: $$\{\begin{array}{ll}-\mathrm{\Delta }u+2K_g=2K{e}^u& \text{in }\mathrm{\Sigma }\\ {\mathrm{\partial }}_{\nu }u+2h_g=2h{e}^{\frac{u}{2}}& \text{on }\mathrm{\partial }\mathrm{\Sigma }\mathrm{.}\end{array}$$ We provide three different existence results in the cases of positive, zero and negative Euler characteristics by means of variational techniques.