Seminar of young researchers in Mathematics

In this talk we consider two curvature prescription problems on compact riemannian manifolds with boundary via a conformal change of the metric. This leads us to two different semilinear elliptic Partial Differential Equations with nonlinear Neumann boundary conditions. We address the question of existence by methods of Calculus of Variations.

The first one is the problem of prescribing Gaussian and geodesic curvature on the flat disk and its boundary, respectively. Setting the problem in a novel variational framework we are able to find minimizers under symmetry assumptions, when both functions are nonnegative. After that, we study a natural generalization of it to higher dimensions: the prescribed scalar and mean curvature on problem on a compact manifold with boundary.