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

Homogenization of high-contrast composites under differential constraints