On the energy critical Schrodinger equation in 3D non-trapping domains

TL;DR

This paper establishes local well-posedness and scattering results for the energy-critical and sub-critical Schrödinger equations in three-dimensional non-trapping domains, extending understanding of boundary effects on solution behavior.

Contribution

It proves local well-posedness for the energy-critical quintic Schrödinger equation with Dirichlet boundary conditions in non-trapping domains and derives scattering results for sub-quintic cases.

Findings

Local well-posedness in non-trapping domains

Smoothing effect in L^5_x L^2_t for the linear equation

Scattering results for defocusing sub-quintic Schrödinger equations

Abstract

We prove that the quintic Schrodinger equation with Dirichlet boundary conditions is locally well posed for H^{1}_{0} data on any smooth, non-trapping domain of R^3. The key ingredient is a smoothing effect in L^{5}_{x}L^{2}_{t} for the linear equation. We also derive scattering results for the whole range of defocusing sub-quintic Schrodinger equations outside star-shaped domains.