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Journal of Convex Analysis 10 (2003), No. 2, 325--350
Copyright Heldermann Verlag 2003

Multiscale Relaxation of Convex Functionals

Irene Fonseca
Dept. of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.,

Elvira Zappale
Dip. di Ingegneria dell'Informazione e Matematica Applicata, UniversitÓ degli Studi di Salerno, 84084 Fisciano, Italy,


The $\Gamma$-limit of a family of functionals $$ u\mapsto \int_{\Omega}f\left(\frac{x}{\e},\frac{x}{\e^2},D^su\right)\, dx $$ is obtained for $s=1,2$ and when the integrand $f=f(x,y,v)$ is a continuous function, periodic in $x$ and $y$, and convex with respect to $v$. The $3$-scale limits of second order derivatives are characterized.

Keywords: Convexity, periodicity, multiscale limits, Γ-convergence, A-quasiconvexity, measurability selection criterion.

MSC 2000: 35G99, 49J40, 49J45, 74G65.

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