The nonlinear Schrödinger equation on the half-space

Abstract

This work studies the initial-boundary value problem for both the linear Schrödinger equation and the cubic nonlinear Schrödinger 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.

39 pages, 4 figures