Cloaking using complementary media for the Helmholtz equation and a three spheres inequality for second order elliptic equations

Cloaking using complementary media was suggested by Lai et al. This was proved in the quasistatic regime by H. M. Nguyen. One of the difficulties in the study of this problem is the appearance of the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others as the loss goes to 0. To this end, the author introduced the technique of removing localized singularity and used a standard three spheres inequality. This method also works for the Helmholtz equation. However, it requires small size of the cloaked region for large frequency due to the use of the (standard) three spheres inequality. In this paper, we give a proof of cloaking using complementary media in the finite frequency regime without imposing any condition on the cloaked region; hence the cloak works for all frequency. To successfully apply the approach of H.M. Nguyen, we establish a new three spheres inequality. A modification of the cloaking setting to obtain illusion optics is also discussed.