On the boundary value problem for the Schrödinger equation : compatibility conditions and global existence

Abstract

We consider linear and nonlinear Schrodinger equations on a domain Omega with nonzero Dirichlet boundary conditions and initial data. We first study the linear boundary value problem with boundary data of optimal regularity (in anisotropic Sobolev spaces) with respect to the initial data. We prove well-posedness under natural compatibility conditions. This is essential for the second part, where we prove the existence and uniqueness of maximal solutions for nonlinear Schrodinger equations. Despite the nonconservation of energy, we also obtain global existence in several (defocusing) cases.