Singular p-homogenization for highly conductive fractal layers

TL;DR

This paper develops a homogenization framework for quasi-linear problems in fractal domains, showing convergence of energy functionals and solutions to a fractal limit involving p-energy, with proofs of existence and uniqueness.

Contribution

It introduces a novel pre-homogenization approach for fractal layers and establishes convergence to a fractal p-energy functional, advancing homogenization theory in fractal geometries.

Findings

Convergence of pre-homogenized functionals to fractal p-energy functional.

Existence and uniqueness of solutions for the homogenized fractal problem.

Analysis of solution convergence in fractal homogenization context.

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 $\varepsilon$. 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.