The p-Laplacian equation in a rough thin domain with terms concentrating on the boundary

TL;DR

This paper investigates the asymptotic behavior of solutions to the p-Laplacian equation in complex thin domains with boundary concentrations, using homogenization techniques to derive effective models.

Contribution

It introduces a homogenization framework for the p-Laplacian in rough thin domains with boundary concentration effects, covering weak, resonant, and high roughness scenarios.

Findings

Derived effective equations capturing geometric and boundary concentration effects.

Analyzed different roughness regimes and their impact on solution behavior.

Provided a unified approach using homogenization and unfolding operators.

Abstract

In this work we use reiterated homogenization and unfolding operator approach to study the asymptotic behavior of the solutions of the $p$-Laplacian equation with Neumann boundary conditions set in a rough thin domain with concentrated terms on the boundary. We study weak, resonant and high roughness, respectively. In the three cases, we deduce the effective equation capturing the dependence on the geometry of the thin channel and the neighborhood where the concentrations take place.