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

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 $R^\varepsilon=\left\lbrace(x,y)\in\mathbb{R}^2:x\in(0,1)\mbox{ and }0<y<\varepsilon g\left({x}/{\varepsilon^α}\right)\right\rbrace$ 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.