Li, Hongjie; Li, Jingzhi; Liu, Hongyu On quasi-static cloaking due to anomalous localized resonance in \(\mathbb{R}^3\). (English) Zbl 1322.35176 SIAM J. Appl. Math. 75, No. 3, 1245-1260 (2015). The paper deals with a mathematical model for an invisibility cloaking (more precisely, for the cloaking due to anomalous localized resonance (CALR)) in a three-dimensional setting. In the model, time-harmonic waves propagate in the quasi-static regime and the wave pressure \(u(x)\) satisfies the equation \(\nabla (\epsilon_\eta(x)\nabla u(x)) = f(x)\), \(u(x)=O(|x|^{-1}\) as \(|x|\to\infty\), where \(\epsilon_\eta(x)=\epsilon(x)+i\eta\chi_D(x)\), \(D={\mathbb R}^3\) or \(D=\Omega\setminus\bar\Sigma\), and \(\epsilon(x)\) in the part of the domain \(\Sigma \subset\Omega\) associated with the shell is negatively valued (plasmonic material parameter). The CALR occurs when for the rate \(E_\eta=\int_D \frac{\eta}{2}|\nabla u(x)|^2dx\) at which the energy of the wave field is dissipated into the heat, one has \(E_\eta\to+\infty\) and \(|u(x)/\sqrt{E_\eta}|\to 0\) as \(\eta\to +0\) for all \(|x|\) big enough; then, both the source \(f\) and the cloaking device \((\Omega;\epsilon)\) are invisible to the observation made from outside. The paper aims at extending the related results known for dimension two, to the three-dimensional setting. By using variational arguments, it is shown that CALR does not occur if one takes a similar structure, but can still happen if the plasmonic parameters are chosen properly. Reviewer: Dmitry Shepelsky (Kharkov) Cited in 28 Documents MSC: 35R30 Inverse problems for PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs Keywords:anomalous localized resonance; plasmonic material; invisibility cloaking PDFBibTeX XMLCite \textit{H. Li} et al., SIAM J. Appl. Math. 75, No. 3, 1245--1260 (2015; Zbl 1322.35176) Full Text: DOI arXiv References: [1] H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, {\it Anomalous localized resonance using a folded geometry in three dimensions}, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 469 (2013), 20130048. · Zbl 1348.78006 [2] H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, {\it Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance}, Arch. Ration. Mech. Anal., 208 (2013), pp. 667-692. · Zbl 1282.78004 [3] H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, {\it Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance} II, Contemp. Math., 615 (2014), pp. 1-14. · Zbl 1330.35421 [4] H. Ammari, J. Garnier, W. Jing, H. Kang, M. Lim, K. Solna, and H. Wang, {\it Mathematical and Statistical Methods for Multistatic Imaging}, Lecture Notes in Math. 2098, Springer, Cham, Switzerland, 2013. · Zbl 1288.35001 [5] K. Ando and H. Kang, {\it Analysis of Plasmon Resonance on Smooth Domains Using Spectral Properties of the Neumann-Poincaré Operator}, preprint, arXiv:1412.6250, 2014. · Zbl 1327.81193 [6] G. Bouchitté and B. Schweizer, {\it Cloaking of small objects by anomalous localized resonance}, Quart. J. Mech. Appl. Math., 63 (2010), pp. 438-463. · Zbl 1241.78001 [7] O. P. Bruno and S. Lintner, {\it Superlens-cloaking of small dielectric bodies in the quasistatic regime}, J. Appl. Phys., 102 (2007), 124502. [8] D. Chung, H. Kang, K. Kim, and H. Lee, {\it Cloaking due to anomalous localized resonance in plasmonic structures of confocal ellipses}, SIAM J. Appl. Math., 74 (2014), pp. 1691-1707. · Zbl 1327.35433 [9] D. Colton and R. Kress, {\it Inverse Acoustic and Electromagnetic Scattering Theory}, 2nd ed., Springer-Verlag, Berlin, 1998. · Zbl 0893.35138 [10] H. Kettunen, M. Lassas, and P. Ola, {\it On Absence and Existence of the Anomalous Localized Resonace Without the Quasi-Static Approximation}, preprint, arXiv:1406.6224, 2014. · Zbl 1401.35033 [11] R. V. Kohn, J. Lu, B. Schweizer, and M. I. Weinstein, {\it A variational perspective on cloaking by anomalous localized resonance}, Comm. Math. Phys., 328 (2014), pp. 1-27. · Zbl 1366.78002 [12] G. W. Milton and N.-A. P. Nicorovici, {\it On the cloaking effects associated with anomalous localized resonance}, R. Soc. Lond. Proc. A Math. Phys. Eng. Sci., 462 (2006), pp. 3027-3059. · Zbl 1149.00310 [13] N.-A. P. Nicorovici, R. C. McPhedran, and G. W. Milton, {\it Optical and dielectric properties of partially resonant composites}, Phys. Rev. B (3), 49 (1994), pp. 8479-8482. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.