Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data

We consider the defocusing energy-critical NLS in the exterior of the unit ball in three dimensions. For the initial value problem with Dirichlet boundary condition we prove global well-posedness and scattering with large radial initial data in the Sobolev space ${\dot{H}}_0^1$. We also point out that the same strategy can be used to treat the energy-supercritical NLS in the exterior of balls with Dirichlet boundary condition and radial ${\dot{H}}_0^1$ initial data.