  We derive, by means of variational techniques, a limiting description for a
class of integral functionals under linear differential constraints. The
functionals are designed to encode the energy of a high-contrast composite,
that is, a heterogeneous material which, at a microscopic level, consists of a
periodically perforated matrix whose cavities are occupied by a filling with
different physical properties. Our main result provides a $\Gamma$-convergence
analysis as the periodicity tends to zero, and shows that the variational limit
of the functionals at stake is the sum of two contributions, one resulting from
the energy stored in the matrix and the other from the energy stored in the
inclusions. As a consequence of the underlying high-contrast structure, the
study is faced with a lack of coercivity with respect to the standard
topologies in $L^p$, which we tackle by means of two-scale convergence
techniques. In order to handle the differential constraints, instead, we
establish new results about the existence of potentials and of
constraint-preserving extension operators for linear, $k$-th order, homogeneous
differential operators with constant coefficients and constant rank.
