Existence of gaps in the spectrum of periodic dielectric structures on a lattice

We consider the finite-difference counterpart, i.e., the true lattice analog, of Maxwell's equations and equations that govern the propagation of acoustic waves in a medium with a periodic dielectric structure. In particular, the vector nature of electromagnetic waves is fully taken into account. The existence of true gaps for these lattice models is proved for a two-component medium for which the dielectric constant is everywhere real and positive, and the dielectric constant of the background is essentially larger than the one corresponding to the embedded component.