Singular partial differential operators and pseudo-differential boundary conditions in invisibility cloaking

Transformation optics constructions have allowed the design of cloaking devices that steer electromagnetic, acoustic and quantum parameters waves around a region without penetrating it, so that this region is hidden from external observations. The material parameters used to describe these devices are anisotropic, and singular at the interface between the cloaked and uncloaked regions, making physical realization a challenge. These singular material parameters correspond to singular coefficient functions in the partial differential equations modeling these constructions and the presence of these singularities causes various mathematical problems and physical effects on the interface surface. In this paper, we give a review on mathematical theory of cloaking. Moreover, we consider the two-dimensional cloaking, that is, cylindrical cloaking, for Maxwell's equations. For this case, we present results that generalizes earlier analogous results for the two-dimensional cloaking for the scalar equations. In particular, we consider nonsingular approximate invisibility cloaks based on the truncation of the singular transformations. Using such truncation we analyze the limit when the approximate cloaking approaches the ideal cloaking. We show that a non-local pseudo-differential boundary condition appears on the inner cloak interface. This effect in the two-dimensional (or cylindrical) invisibility cloaks, which seems to be caused by the infinite phase velocity near the interface between the cloaked and uncloaked regions, is very different to the behavior of the solutions in the three-dimensional cloaks.