We investigate the band-gap structure and the integrated density of states
for a class of periodic divergence type operators which, in the simplest ca
se, 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 O
mega 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 sa
nd or air bubbles, and in connection with photonic crystals.
Among other results, we find that T-lambda will always have at least one op
en gap, for lambda large (except for trivial cases). We also establish a co
nnection between the band-gap structure of T-lambda and the Dirichlet eigen
value problem on M-o = M boolean AND Q(o).