Creating and controlling band gaps in periodic media with small resonators

We investigate spectral properties of the Neumann Laplacian ${\mathcal{A}}_{\epsilon }$ on a periodic unbounded domain ${\mathrm{\Omega }}_{\epsilon }$ depending on a small parameter $\epsilon >0$. The domain ${\mathrm{\Omega }}_{\epsilon }$ is obtained by removing from ${\mathbb{R}}^n$ $m\in \mathbb{N}$ families of $\epsilon$-periodically distributed small resonators. We prove that the spectrum of ${\mathcal{A}}_{\epsilon }$ has at least $m$ gaps. The first $m$ gaps converge as $\epsilon \to 0$ to some intervals whose location and lengths can be controlled by a suitable choice the resonators; other gaps (if any) go to infinity. An application to the theory of photonic crystals is discussed.