Homogenization of high-contrast composites under differential constraints

TL;DR

This paper analyzes the asymptotic behavior of energy functionals for high-contrast composites with differential constraints, using variational and two-scale convergence techniques to derive a comprehensive limit description.

Contribution

It introduces a novel $ ext{Gamma}$-convergence framework for high-contrast composites under differential constraints, including new results on potentials and extension operators for linear differential operators.

Findings

Derived a $ ext{Gamma}$-limit for high-contrast composite energies.

Established existence of potentials and extension operators for differential constraints.

Handled lack of coercivity using two-scale convergence techniques.

Abstract

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.