Multiscale Homogenization of Convex Functionals with Discontinuous Integrand

This article is devoted to obtain the

-limit, as

\varepsilon ε

tends to zero, of the family of functionals

u\mapsto\int_{\Omega}f\Bigl(x,\frac{x}{\varepsilon}, \ldots, \frac{x}{\varepsilon^n}, \nabla u(x)\Bigr)dx, u ↦ ∫ Ω f ( x , x ε , … , x ε n , ∇ u ( x ) ) d x ,

where

f=f(x, y^1, \ldots, y^n, z) f = f ( x , y 1 , … , y n , z )

is periodic in

y^1, \ldots, y^n y 1 , … , y n

, convex in

z z

and satisfies a very weak regularity assumption with respect to

x, y^1, \ldots, y^n x , y 1 , … , y n

. We approach the problem using the multiscale Young measures.