Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem

Abstract

We survey recent wellposedness and illposedness results for the Cauchy problem for nonlinear Schrödinger equations in inhomogeneous media such as Riemannian manifolds or domains of the Euclidean space, trying to emphasize the influence of the geometry. The main tools are multilinear Strichartz estimates for the Schrödinger group.