Spectral properties of periodic media in the large coupling limit

We investigate the band-gap structure and the integrated density of states for a class of periodic divergence type operators which, in the simplest case, are given by T-lambda = -del . (1 + lambda(chi Omega)) del, lambda greater than or equal to 1, acting in L-2(R-m) with m greater than or equal to 2. We assume here that Omega is an open, connected, periodic subset of R-m and that the complement M of Omega does not intersect the boundary of the fundamental period cell Q(o). Operators of this type occur in simple models for heat conduction (or propagation of acoustic waves) in a metal with impurities like grains of sand or air bubbles, and in connection with photonic crystals. Among other results, we find that T-lambda will always have at least one open gap, for lambda large (except for trivial cases). We also establish a connection between the band-gap structure of T-lambda and the Dirichlet eigenvalue problem on M-o = M boolean AND Q(o).