Multiscale Homogenization of Convex Functionals with Discontinuous Integrand
Marco Barchiesi · Journal of convex analysis · 2007
This article is devoted to obtain the \Gamma Γ -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}, abla 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.