# Global Strichartz estimates for the Schr\"odinger equation with non zero boundary conditions and applications

## TL;DR

This paper establishes global Strichartz estimates for the Schrödinger equation on a half space with various boundary conditions, enabling analysis of nonlinear problems and scattering in this setting.

## Contribution

It extends local Strichartz estimates to global ones for the Schrödinger equation with nonhomogeneous boundary conditions in any dimension.

## Key findings

- Derived global Strichartz estimates for initial data in H^s and boundary data in ^s
- Solved nonlinear Schrödinger equations using these estimates
- Constructed global asymptotically linear solutions for small data

## Abstract

We consider the Schr\"odinger equation on a half space in any dimension with a class of nonhomogeneous boundary conditions including Dirichlet, Neuman and the so-called transparent boundary conditions. Building upon recent local in time Strichartz estimates (for Dirichlet boundary conditions), we obtain global Strichartz estimates for initial data in $H^s,\ 0\leq s\leq 2$ and boundary data in a natural space $\mathcal{H}^s$. For $s\geq 1/2$, the issue of compatibility conditions requires a thorough analysis of the $\mathcal{H}^s$ space. As an application we solve nonlinear Schr\"odinger equations and construct global asymptotically linear solutions for small data. A discussion is included on the appropriate notion of scattering in this framework, and the optimality of the $\mathcal{H}^s$ space.