Cloaking of Small Objects by Anomalous Localized Resonance

We investigate solutions of ∇ · (a∇u) = 0 with various boundary conditions. The coefficient a is assumed to have a real part with changing sign and a small, non-negative imaginary part of order η. We investigate a two-dimensional ring geometry with unit inner radius and outer radius R. We use Fourier expansions in polar coordinates to analyze the qualitative behaviour of solutions when boundary conditions are imposed on a small circular inclusion, centred at x 0 . Our result is that u depends qualitatively on the position of the inclusion. If lx 0 l is larger than the cloaking radius R * = R 3/2 , then u behaves as if no ring were present. If, instead, lx 0 l < R*, then the small inclusion is invisible in the limit η→ 0.