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The aim of (passive) cloaking with respect to electromagnetic (or acoustic) sensing is to surround a region of space with a material layer -- the cloak -- that renders its contents and even the existence of the layer undetectable by such measurements. At least theoretically this can be achieved using the coordinate invariance of the underlying wave equation, through so-called cloaking by mapping. However, a practical realization of the cloaking by mapping schemes discussed in the literature frequently requires the design of highly anisotropic materials with extreme dielectric properties. In this work we consider, in the electrostatic case, a regularized, approximate cloaking by mapping scheme and discuss the problem of optimal choice of radial maps, that determine the conductivity distribution of the cloak. We consider two different optimality criteria: minimal maximal anisotropy and minimal mean anisotropy of this conductivity distribution. Using both criteria we show that it is possible to achieve significantly lower anisotropy (for a prescribed level of invisibility) or significantly lower visibility (for a prescribed level of anisotropy). For example, in two dimensions one may achieve exponentially small visibility with a cloak, that in terms of anisotropy (and lowest and highest conductivity) is no worse than the traditional affine map cloak, which only yields quadratically small visibility.

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Michael Vogelius