Multiscale Relaxation of Convex Functionals

The

-limit of a family of functionals

u\mapsto \int_{\Omega}f\left(\frac{x}{\varepsilon},\frac{x}{\varepsilon^2},D^su\right)\, dx u ↦ ∫ Ω f ( x ε , x ε 2 , D s u )   d x

is obtained for

s=1,2 s = 1 , 2

and when the integrand

f=f(x,y,v) f = f ( x , y , v )

is a continuous function, periodic in

x x

and

y y

, and convex with respect to

v v

. The

3 3

-scale limits of second order derivatives are characterized.