Singular $p$-Homogenization for Highly Conductive Fractal Layers

Abstract

We consider a quasi-linear homogenization problem in a two-dimensional pre-fractal domain ${\Omega }_n$, for $n\in \mathbb{N}$, surrounded by thick fibers of amplitude $\epsilon$. 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.