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Cloaking via change of variables in electric impedance tomography. (English) Zbl 1153.35406

Summary: A recent paper by Pendry et al. [Science 312, 1780–1782 (2006)] used the coordinate invariance of Maxwell’s equations to show how a region of space can be ‘cloaked’-in other words, made inaccessible to electromagnetic sensing-by surrounding it with a suitable (anisotropic and heterogeneous) dielectric shield. Essentially the same observation was made several years earlier by A. Greenleaf, M. Lassas and G. Uhlmann [Math. Res. Lett. 10, No. 5–6, 685–693 (2003; Zbl 1054.35127)] in the closely related setting of electric impedance tomography. These papers, though brilliant, have two shortcomings: (a) the cloaks they consider are rather singular; and (b) the analysis by Greenleaf, Lassas and Uhlmann does not apply in space dimension \(n = 2\). The present paper provides a fresh treatment that remedies these shortcomings in the context of electric impedance tomography. In particular, we show how a regular near-cloak can be obtained using a nonsingular change of variables, and we prove that the change-of-variable-based scheme achieves perfect cloaking in any dimension \(n \geq 2\).

MSC:

35R30 Inverse problems for PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78A25 Electromagnetic theory (general)
92C55 Biomedical imaging and signal processing

Citations:

Zbl 1054.35127
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