The p-Laplacian equation in thin domains: The unfolding approach

TL;DR

This paper uses the unfolding operator method to analyze the asymptotic behavior of solutions to the p-Laplacian equation in thin, oscillating domains, deriving homogenized limits and correctors for different oscillation regimes.

Contribution

It applies the unfolding approach to the p-Laplacian in thin domains with various oscillation scales, providing new homogenization results and correctors for each case.

Findings

Derived homogenized limits for all three oscillation regimes.

Obtained explicit correctors for the asymptotic solutions.

Unified approach for different boundary oscillation scales.

Abstract

In this work we apply the unfolding operator method to analyze the asymptotic behavior of the solutions of the $p$-Laplacian equation with Neumann boundary condition set in a bounded thin domain of the type $R^\varepsilon=\left\lbrace(x,y)\in\mathbb{R}^2:x\in(0,1)\mbox{ and }0<y<\varepsilon g\left({x}/{\varepsilon^\alpha}\right)\right\rbrace$ where $g$ is a positive periodic function. We study the three cases $0<\alpha<1$, $\alpha=1$ and $\alpha>1$ representing respectively weak, resonant and high osillations at the top boundary. In the three cases we deduce the homogenized limit and obtain correctors.