# Periodic homogenization of integral energies under space-dependent differential constraints

### Elisa Davoli

Universität Wien, Austria### Irene Fonseca

Carnegie Mellon University, Pittsburgh, United States

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## Abstract

A homogenization result for a family of oscillating integral energies

is presented, where the fields $u_{\epsilon}$ are subjected to first order linear differential constraints depending on the space variable $x$. The work is based on the theory of $\mathscr A$-quasiconvexity with variable coefficients and on two-scale convergence techniques, and generalizes the previously obtained results in the case in which the differential constraints are imposed by means of a linear first order differential operator with constant coefficients. The identification of the relaxed energy in the framework of $\mathscr A$-quasiconvexity with variable coefficients is also recovered as a corollary of the homogenization result.

## Cite this article

Elisa Davoli, Irene Fonseca, Periodic homogenization of integral energies under space-dependent differential constraints. Port. Math. 73 (2016), no. 4 pp. 279–317

DOI 10.4171/PM/1988