The nonlinear Schr\"odinger equation on the half-space

TL;DR

This paper analyzes the initial-boundary value problem for linear and nonlinear Schr"odinger equations on the half-space, deriving new estimates and proving well-posedness using the Fokas method and Bourgain spaces.

Contribution

It introduces a novel approach to solving the Schr"odinger equation on the half-space using the Fokas method, establishing well-posedness with boundary data in Bourgain spaces.

Findings

Derived new linear estimates for the forced linear Schr"odinger problem.

Proved well-posedness of the nonlinear Schr"odinger equation on the half-space.

Established contraction mapping using sharper trilinear estimates.

Abstract

This work studies the initial-boundary value problem for both the linear Schr\"odinger equation and the cubic nonlinear Schr\"odinger equation on the half-space in higher dimensions ($n\ge 2$). First, the forced linear problem is solved on the half-space via the Fokas method and then using the obtained solution formula new and interesting linear estimates are derived with data and forcing in appropriate spaces. Second, the well-posedness of the nonlinear problem on the half-space is proved with initial data in Sobolev spaces $H^s(\mathbb{R}^n_+)$, with $s>\frac{n}{2}-1$, and boundary data in natural Bourgain spaces $\mathcal{B}^s$ that reflect the boundary regularity of the linear problem. The proof method consists of showing that the iteration map defined via the Fokas solution formula is a contraction by establishing sharper trilinear estimates. The presence of the boundary introduces solution spaces that involve temporal Bourgain spaces.