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Journal of Convex Analysis 14 (2007), No. 1, 205--226
Copyright Heldermann Verlag 2007

Multiscale Homogenization of Convex Functionals with Discontinuous Integrand

Marco Barchiesi
S.I.S.S.A., Via Beirut 2-4, 34014 Trieste, Italy


\newcommand{\e}{\varepsilon} This article is devoted to obtain the $\Gamma$-limit, as $\e$ tends to zero, of the family of functionals \begin{equation*} u\mapsto\int_{\Omega}f\Bigl(x,\frac{x}{\e}, \ldots, \frac{x}{\e^n}, \nabla u(x)\Bigr)dx , \end{equation*} where $f=f(x, y^1, \ldots, y^n, z)$ is periodic in $y^1, \ldots, y^n$, convex in $z$ and satisfies a very weak regularity assumption with respect to $x, y^1, \ldots, y^n$. We approach the problem using the multiscale Young measures.

Keywords: convexity, discontinuous integrands, iterated homogenization, periodicity, multiscale convergence, Young measures, Gamma-convergence

MSC: 28A20, 35B27, 35B40, 74Q05

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