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Publication 126

Periodic Homogenization Of Integral Energies Under Space-Dependent Differential Constraints


Elisa Davoli
Department of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 1
1090 Vienna, Austria

CMUIrene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

A homogenization result for a family of oscillating integral energies $$ u_{\varepsilon} \rightarrow \int_{\Omega} f(x, \frac{x}{\varepsilon}, u_{\varepsilon}(x))dx, \ \ \ \varepsilon \rightarrow 0^+$$ is presented, where the fields $u_{\varepsilon}$ are subjected to first order linear differential constraints depending on the space variable $x$. The work is based on the theory of ${\cal A}$-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 ${\cal A}$-quasiconvexity with variable coefficients is also recovered as a corollary of the homogenization result.
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