Global existence for energy critical waves in 3-D domains: Neumann boundary conditions

We prove that the defocusing quintic wave equation, with Neumann boundary conditions, is globally well-posed on $H^1_N(\Omega) \times L^2( \Omega)$ for any smooth (compact) domain $\Omega \subset {\Bbb R}^3$. The proof relies on one hand on $L^p$ estimates for the spectral projector, and on the other hand on a precise analysis of the boundary value problem, which turns out to be much more delicate than in the case of Dirichlet boundary conditions.