The EMS Publishing House is now **EMS Press** and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

# Portugaliae Mathematica

Full-Text PDF (237 KB) | Metadata | Table of Contents | PM summary

**Volume 73, Issue 4, 2016, pp. 279–317**

**DOI: 10.4171/PM/1988**

Published online: 2016-11-30

Periodic homogenization of integral energies under space-dependent differential constraints

Elisa Davoli^{[1]}and Irene Fonseca

^{[2]}(1) Universität Wien, Austria

(2) Carnegie Mellon University, Pittsburgh, United States

A homogenization result for a family of oscillating integral energies $$u_{\epsilon} \mapsto \int_{\Omega} f(x,\frac{x}{\epsilon},u_{\epsilon}(x))\,dx,\quad \epsilon \to 0^+$$ 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.

*Keywords: *Homogenization, two-scale convergence, $\mathscr A$-quasiconvexity

Davoli Elisa, Fonseca Irene: Periodic homogenization of integral energies under space-dependent differential constraints. *Port. Math.* 73 (2016), 279-317. doi: 10.4171/PM/1988