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# On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS

Published:2014-02-27
Printed: Oct 2014
• Dong Li,
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2
• Guixiang Xu,
Institute of Applied Physics and Computational Mathematics, Beijing, China, 100088,
• Xiaoyi Zhang,
Department of Mathematics, University of Iowa, Iowa City, IA, USA, 52242
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## Abstract

We consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under the radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator $e^{it\Delta_D}$ and give a robust algorithm to prove sharp $L^1 \rightarrow L^{\infty}$ dispersive estimates. We showcase the analysis in dimensions $n=5,7$. As an application, we obtain global well-posedness and scattering for defocusing energy-critical NLS on $\Omega=\mathbb{R}^n\backslash \overline{B(0,1)}$ with Dirichlet boundary condition and radial data in these dimensions.
 Keywords: Dirichlet Schrödinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy critical
 MSC Classifications: 35P25 - Scattering theory [See also 47A40] 35Q55 - NLS-like equations (nonlinear Schrodinger) [See also 37K10] 47J35 - Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25]

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