The problem of prescribing the Gaussian curvature on compact surfaces is a classic one, and dates back to the works of Berger, Moser, Kazdan & Warner, etc. Our aim is to to prescribe also the geodesic curvature of their boundary. This question gives rise to a Liouville equation under nonlinear Neumann boundary conditions.In this talk we address the case of negative Gaussian curvature. We study the geometric properties of the corresponding energy functional, and deduce the existence of minimum or mountain pass critical points. For that, a compactness result is in order. Here the cancellation between the area and length terms make it possible to have blowing-up solutions with infinite mass. This phenomenon seems to be entirely new in the related literature. For instance, the singular set need not be finite, and the limit profile may have infinite mass. This is joint work with Andrea Malchiodi (SNS Pisa) and Rafael López Soriano (U. Valencia).
org: BATTAGLIA Luca