  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
very 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.
