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: [Formula: see text] The underlying problem allows the application of straightforward variational techniques. Consequently, we provide three different existence results in the cases of positive, zero and negative Euler characteristics by means of variational techniques.