Multiscale Relaxation of Convex Functionals

The $\Gamma$-limit of a family of functionals $$u\mapsto \int_{\Omega}f\left(\frac{x}{\varepsilon},\frac{x}{\varepsilon^2},D^su\right)\, dx$$ 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.