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Evolution equations on non flat waveguides
- Publication Year :
- 2010
-
Abstract
- We investigate the dispersive properties of evolution equations on waveguides with a non flat shape. More precisely we consider an operator $H=-\Delta_{x}-\Delta_{y}+V(x,y)$ with Dirichled boundary condition on an unbounded domain $\Omega$, and we introduce the notion of a \emph{repulsive waveguide} along the direction of the first group of variables $x$. If $\Omega$ is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation $Hu-\lambda u=f$. As consequences we prove smoothing estimates for the Schr\"odinger and wave equations associated to $H$, and Strichartz estimates for the Schr\"odinger equation. Additionally, we deduce that the operator $H$ does not admit eigenvalues.<br />Comment: 22 pages, 4 figures
- Subjects :
- Mathematics - Analysis of PDEs
Mathematical Physics
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1010.0817
- Document Type :
- Working Paper