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

TL;DR

This paper establishes global well-posedness for the energy-critical defocusing quintic wave equation with Neumann boundary conditions in three-dimensional smooth domains, using spectral estimates and boundary analysis.

Contribution

It extends global existence results to Neumann boundary conditions, which are more delicate than Dirichlet, by combining spectral projector estimates with boundary problem analysis.

Findings

Proves global well-posedness for the wave equation with Neumann conditions.

Develops new boundary analysis techniques for Neumann problems.

Utilizes spectral projector estimates to handle boundary effects.

Abstract

We prove that the defocusing quintic wave equation, with Neumann boundary conditions, is globally wellposed on $H^1_N(\Omega) \times L^2(\Omega)$ for any smooth (compact) domain $\Omega \subset \mathbb{R}^3$. 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.