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

Abstract

Cloaking using complementary media was suggested by Lai et al. in 2009. This was proved by H.-M. Nguyen (2015) in the quasistatic regime. 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, H.-M. Nguyen introduced the technique of removing localized singularity and used a standard three spheres inequality. The method used 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; the cloak works for an arbitrary fixed frequency provided that the loss is sufficiently small. To successfully apply the above approach of Nguyen, we establish a new three spheres inequality. A modification of the cloaking setting to obtain illusion optics is also discussed.