Cloaking via change of variables in electric impedance tomography

A recent paper by Pendry, Schurig, and Smith [Science 312, 2006, 1780-1782] 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 heterogenous) dielectric shield. Essentially the same observation was made several years earlier by Greenleaf, Lassas, and Uhlmann [Mathematical Research Letters 10, 2003, 685-693 and Physiological Measurement 24, 2003, 413-419] 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 might leave the impression that the phenomenon is limited to space dimension n ≥ 3. 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 ≥ 2.