On the energy critical Schrödinger equation in 3D non-trapping domains

We prove that the quintic Schrödinger equation with Dirichlet boundary conditions is locally well posed for data on any smooth, non-trapping domain Ω⊂R3. The key ingredient is a smoothing effect in for the linear equation. We also derive scattering results for the whole range of defocusing sub quintic Schrödinger equations outside a star-shaped domain.