Cloaking via change of variables for the Helmholtz equation

The transformation optics approach to cloaking uses a singular change of coordinates, which blows up a point to the region being cloaked. This paper examines a natural regularization, obtained by (1) blowing up a ball of radius ρ rather than a point, and (2) including a well-chosen lossy layer at the inner edge of the cloak. We assess the performance of the resulting near-cloak as the regularization parameter ρ tends to 0, in the context of (Dirichlet and Neumann) boundary measurements for the time-harmonic Helmholtz equation. Since the goal is to achieve cloaking regardless of the content of the cloaked region, we focus on estimates that are uniform with respect to the physical properties of this region. In three space dimensions our regularized construction performs relatively well: the deviation from perfect cloaking is of order ρ. In two space dimensions it does much worse: the deviation is of order 1/(log ρ(. In addition to proving these estimates, we give numerical examples demonstrating their sharpness. Some authors have argued that perfect cloaking can be achieved without losses by using the singular change-of-variable-based construction. In our regularized setting the analogous statement is false: without the lossy layer, there are certain resonant inclusions (depending in general on ρ) that have a huge effect on the boundary measurements.