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Publication 02-CNA-02

Multiscale Relaxation of Convex Functionals

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

Elvira Zappale
Dipartimento di Ingegneria dell'Informazione e Matematica Applicata,
Università degli Studi di Salerno,
Fisciano (Sa) 84084, Italy.

Abstract: The $ \Gamma$-limit of a family of functionals

$\displaystyle u\mapsto \int_{\Omega}f\left(\frac{x}{{\varepsilon}},\frac{x}{{\varepsilon}^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.


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