Global existence for energy critical waves in 3-d domains

Abstract

We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on H 0 1 ( Ω ) × L 2 ( Ω ) H^1_0(\Omega ) \times L^2( \Omega ) for any smooth (compact) domain Ω ⊂ R 3 \Omega \subset \mathbb {R}^3 . The main ingredient in the proof is an L 5 L^5 spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.