Singular $p$-Homogenization for Highly Conductive Fractal Layers

We consider a quasi-linear homogenization problem in a two-dimensional prefractal domain Ωn, for n ∈ N, surrounded by thick fibers of amplitude ε. We introduce a sequence of “pre-homogenized” energy functionals and we prove that this sequence converges in a suitable sense to a quasi-linear fractal energy functional involving a p-energy on the fractal boundary. We prove existence and uniqueness results for (quasi-linear) pre-homogenized and homogenized fractal problems. The convergence of the solutions is also investigated.