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

Published online by Cambridge University Press:  20 November 2018

Dong Li ,
Guixiang Xu  and
Xiaoyi Zhang
Dong Li
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2. e-mail: mpdongli@gmail.com
Guixiang Xu
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing, China, 100088. e-mail: xu guixiang@iapcm.ac.cn
Xiaoyi Zhang
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA, USA, 52242 and Chinese Academy of Science, Beijing, China. e-mail: zh.xiaoyi@gmail.com
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Abstract

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We consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under 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 ${{L}^{1}}\,\to \,{{L}^{\infty }}$ 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 $\text{NLS}$ on $\Omega \,=\,{{\mathbb{R}}^{n}}\backslash \overline{B\left( 0,\,1 \right)}$ with Dirichlet boundary condition and radial data in these dimensions.

Keywords

35P2535Q5547J35Dirichlet Schrödinger propagatordispersive estimateDirichlet boundary conditionscattering theoryenergy critical
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 5 , 01 October 2014 , pp. 1110 - 1142
Copyright
Copyright © Canadian Mathematical Society 2014

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