
Journal of Convex Analysis 10 (2003), No. 2, 325350 Copyright Heldermann Verlag 2003 Multiscale Relaxation of Convex Functionals Irene Fonseca Dept. of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A., fonseca@andrew.cmu.edu Elvira Zappale Dip. di Ingegneria dell'Informazione e Matematica Applicata, Università degli Studi di Salerno, 84084 Fisciano, Italy, zappale@diima.unisa.it [Abstractpdf] 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, Γconvergence, Aquasiconvexity, measurability selection criterion. MSC 2000: 35G99, 49J40, 49J45, 74G65. FullTextpdf (398 K) 