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Portugaliae Mathematica


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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) Universit├Ąt 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