JournalspmVol. 73 , No. 4pp. 279–317

Periodic homogenization of integral energies under space-dependent differential constraints

  • Elisa Davoli

    Universität Wien, Austria
  • Irene Fonseca

    Carnegie Mellon University, Pittsburgh, United States
Periodic homogenization of integral energies under space-dependent differential constraints cover
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Abstract

A homogenization result for a family of oscillating integral energies

uϵΩf(x,xϵ,uϵ(x))dx,ϵ0+u_{\epsilon} \mapsto \int_{\Omega} f(x,\frac{x}{\epsilon},u_{\epsilon}(x))\,dx,\quad \epsilon \to 0^+

is presented, where the fields uϵu_{\epsilon} are subjected to first order linear differential constraints depending on the space variable xx. The work is based on the theory of A\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 A\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