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arXiv · 1702.06792

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

Corentin Audiard
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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.

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Corentin Audiard. 2017-02-22. Global Strichartz estimates for the Schr\"odinger equation with non zero boundary conditions and applications. https://arxiv.org/abs/1702.06792

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