The p-Laplacian in thin channels with locally periodic rough boundaries

TL;DR

This paper studies the asymptotic behavior of solutions to the p-Laplacian equation in thin, locally periodic rough boundary channels as the domain thickness approaches zero, revealing how boundary oscillations influence the limit problem.

Contribution

It provides a rigorous analysis of the p-Laplacian in thin domains with locally periodic boundaries, extending understanding of boundary effects in nonlinear PDEs in thin geometries.

Findings

Derived the limit behavior of solutions as the domain thickness tends to zero

Characterized the influence of locally periodic boundary oscillations on the limit problem

Established convergence results for the solutions in thin, rough domains

Abstract

In this work we analyze the asymptotic behavior of the solutions of the $p$-Laplacian equation with homogeneous Neumann boundary conditions set in bounded thin domains as $$R^\varepsilon=\left\lbrace(x,y)\in\mathbb{R}^2:x\in(0,1)\mbox{ and }0<y<\varepsilon G\left(x,{x}/{\varepsilon}\right)\right\rbrace.$$ We take a smooth function $G:(0,1)\times\mathbb{R} \mapsto \mathbb{R}$, $L$-periodic in the second variable, which allows us to consider locally periodic oscillations at the upper boundary. The thin domain situation is established passing to the limit in the solutions as the positive parameter $\varepsilon$ goes to zero.