On vanishing near corners of transmission eigenfunctions
type: Item
instance of: scholarly article
title: On vanishing near corners of transmission eigenfunctions
zbMATH Open document ID: 1387.35437
author: Eemeli Blåsten
author: Hongyu Liu
published in: Journal of Functional Analysis
publication date: 2017-09-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1701.07957
review text: The paper investigates the following (interior) transmission eigenvalue problem for nontrivial \(v, w\in L^2(\Omega)\), \[ \begin{cases} (\Delta+k^2)v=0\quad &\text{in}\;\;\Omega,\\ (\Delta+k^2(1+V))w=0\quad &\text{in}\;\;\Omega,\\ w-v\in H^2_0(\Omega),\;\;||v||_{L^2(\Omega)}&=1, \end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(n\geq2\), \(k\in \mathbb{R}_+\) and \(V\in L^\infty(\Omega)\) is a potential function. The transmission eigenvalue problem is a type of non elliptic and non self-adjoint problem. The existing results are mainly concerned with the spectral properties of the transmission eigenvalues, including the existence, discreteness and infiniteness, and Weyl laws. In this paper the vanishing properties of interior transmission eigenfunctions is investigated. It is proved that \(v\) and \(w\) vanish near a corner point on \(\partial \Omega\) in a generic situation where the corner possesses an interior angle less than \(\pi\) and the potential function \(V\) does not vanish at the corner point. This is proved by an indirect approach, connecting to the wave scattering theory. Note that the vanishing behavior carries geometric information of the support of the involved potential function \(V\). Implications of the obtained resuls to inverse scattering theory and invisibility are also discussed at end of the paper.
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 81V80
zbMATH DE Number: 6783144
zbMATH Keywords: interior transmission eigenfunction
zbMATH Keywords: corner
zbMATH Keywords: vanishing and localizing
zbMATH Keywords: non-scattering
reviewed by: D. Kapanadze
MaRDI profile type: Publication
OpenAlex ID: W2585160151
arXiv ID: 1701.07957
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DOI: 10.1016/J.JFA.2017.08.023
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description: scientific article