Homogenization of problems in ∞

We study the homogenization of functionals of the form $$\left\Vert f \left( \frac{x}{e}, \nabla u \right) \right\Vert_{L^\infty (\Omega)}$$. These functionals are related to a model of dielectric breackdown and can be used as a variational approximation of power-law energies. We compute optimal bounds in the case of the mixtures of two materials, which exhibit a different structure from the integral case. We prove a homogenization theorem in a general context and compare it with the corresponding result for the integral case.