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 wellposed on $H_N^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$ for any smooth (compact) domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$. The proof relies on one hand on $L^p$ estimates for the spectral projector by Smith and Sogge, 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.