  We prove that the defocusing quintic wave equation, with Dirichlet boundary
conditions, is globally well posed on $H^1_0(\Omega) \times L^2(\Omega)$ for
any smooth (compact) domain $\Omega \subset \mathbb{R}^3$. The main ingredient
in the proof is an $L^5$ spectral projector estimate, obtained recently by
Smith and Sogge, combined with a precise study of the boundary value problem.
