Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data

TL;DR

This paper proves global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in the exterior of a ball with radial initial data, and extends the approach to supercritical cases.

Contribution

It establishes the first global well-posedness and scattering results for energy-critical NLS in exterior domains with radial data, and adapts the method to supercritical regimes.

Findings

Proves global well-posedness and scattering for energy-critical NLS exterior of a ball.

Extends the method to energy-supercritical NLS with radial data.

Demonstrates the strategy's applicability to exterior domain problems.

Abstract

We consider the defocusing energy-critical NLS in the exterior of the unit ball in three dimensions. For the initial value problem with Dirichlet boundary condition we prove global well-posedness and scattering with large radial initial data in the Sobolev space $\dot H_0^1$. We also point out that the same strategy can be used to treat the energy-supercritical NLS in the exterior of balls with Dirichlet boundary condition and radial $\dot H_0^1$ initial data.