Cloaking property of a plasmonic structure in doubly complementary media and three-sphere inequalities with partial data

We investigate cloaking property of negative-index metamaterials in the time-harmonic electromagnetic setting for the so-called doubly complementary media. These are media consisting of negative-index metamaterials in a shell (plasmonic structure) and positive-index materials in its complement for which the shell is complementary to a part of the core and a part of the exterior of the core-shell structure. We show that an arbitrary object is invisible when it is placed close to a plasmonic structure of a doubly complementary medium as long as its cross section is smaller than a threshold given by the property of the plasmonic structure. To handle the loss of the compactness and of the ellipticity of the modeling Maxwell equations with sign-changing coefficients, we first obtain Cauchy's problems associated with two Maxwell systems using reflections. We then derive information from them, and combine it with the removing localized singularity technique to deal with the localized resonance. A central part of the analysis on the Cauchy's problems is to establish three-sphere inequalities with partial data for general elliptic systems, which are interesting in themselves. The proof of these inequalities first relies on an appropriate change of variables, inspired by conformal maps, and is then based on Carleman's estimates for a class of degenerate elliptic systems.