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

Abstract

We prove that the quintic Schrödinger equation with Dirichlet boundary conditions is locally well posed for $H_0^1(\Omega )$ data on any smooth, non-trapping domain $\Omega \subset {\mathbb{R}}^3$. The key ingredient is a smoothing effect in $L_x^5(L_t^2)$ for the linear equation. We also derive scattering results for the whole range of defocusing sub quintic Schrödinger equations outside a star-shaped domain.