Curvature prescription problems on manifolds with boundary

Abstract

This thesis addresses the study of two semilinear elliptic problems that arise in Riemannian Geometry. More precisely, we are interested in the prescription of certain geometric quantities on Riemannian manifolds with boundary under conformal changes of the metric, namely, the Gaussian and geodesic curvatures on a compact surface and its boundary, and the scalar and mean curvatures on a manifold of higher dimension. Most of the results available in the literature concern closed manifolds, whereas the boundary cases have been less considered. In that regard, we highlight that the presence of the boundary leads to a wider variety of phenomena, many of which nd no counterpart on the closed versions of these problems. In particular, the variational approach in Chapter 4, and the compactness and existence arguments of Chapter 5 are strictly related to the presence of boundary. Furthermore, the focus of our research concerns the case in which both curvatures are nonconstant, for which there are only a few known results. These problems admit a variational structure, so we will discuss the existence of solutions from the point of view of the Calculus of Variations. Sometimes the energy functionals considered here are bounded from below and a minimizer can be found; in other cases, though, this is not possible, and the use of min-max theory is needed. In the latter situation we are led to the blow-up analysis of solutions of approximated problems. The work developed in this thesis has given rise to two research papers, [31] and [32].