Cloaking via change of variables in electric impedance tomography

A recent paper by Pendry et al ( 2006 Science 312 1780 - 2) 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 et al ( 2003 Math. Res. Lett. 10 685 - 93, 2003 Physiol. Meas. 24 413 - 9) 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 >= 2.