The effect of curvatures in some functional inequalities

In this book, we show the role played by the scalar and the mean curvatures in the existence of extremals for the optimal constant in the Hardy-Sobolev type inequalities. We investigate Hardy and "pure" Hardy-Sobolev inequalities in Riemanian manifold with singularity an interior point or a higher dimensional manifold. In the case of Hardy inequality, there is no influence of the local geometry of the domain in the existence of extremals. In contrast to the Hardy-Sobolev inquality, the existence of minimizers depend of the topology of the domain in dimension three. In higher dimensional singularities the argument is purely geometric. Our arguments of proof are based on the critical point theory: the minimization methods.