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On isotropic cloaking and interior transmission eigenvalue problems

Part of: Elliptic equations and systems Miscellaneous topics - Partial differential equations Partial differential equations, boundary value problems Optics, electromagnetic theory - General Equations of mathematical physics and other areas of application Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  22 May 2017

XIA JI  and
HONGYU LIU
XIA JI
Affiliation:
LSEC, Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing, 100190, P. R. China email: jixia@lsec.cc.ac.cn
HONGYU LIU
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, P. R. China HKBU Institute of Research and Continuing Education, Virtual University Park, Shenzhen, P. R. China email: hongyu.liuip@gmail.com
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Abstract

This paper is concerned with the invisibility cloaking in acoustic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. It is shown that an interior transmission eigenvalue problem arises in our study, which is the one considered theoretically in Cakoni et al. (Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Problems and Imaging, 6 (2012), 373–398). Based on such an observation, we propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish that if a certain non-transparency condition is satisfied, then there exists an infinite set of incident waves such that the cloaking device is nearly invisible under the corresponding wave interrogation. The set of waves is generated from the Herglotz approximation of the associated interior transmission eigenfunctions. We provide both theoretical and numerical justifications.

Keywords

Acoustic wave scatteringinvisibility cloakingisotropic and regularinterior transmission eigenvalues

MSC classification

Primary: 35J57: Boundary value problems for second-order elliptic systems 35P25: Scattering theory 35Q60: PDEs in connection with optics and electromagnetic theory 78A46: Inverse scattering problems
Secondary: 35R30: Inverse problems 65N25: Eigenvalue problems 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 2 , April 2018 , pp. 253 - 280
Copyright
Copyright © Cambridge University Press 2017 

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