  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.
