The p-Laplacian operator in oscillating thin domains

TL;DR

This paper investigates the asymptotic behavior of nonlinear elliptic solutions in oscillating thin domains, deriving homogenized equations and convergence results using homogenization theory and monotone operator methods.

Contribution

It introduces a novel analysis of the p-Laplacian in highly oscillating thin domains, establishing strong convergence and corrector functions for the solutions.

Findings

Derivation of homogenized equations for oscillating thin domains

Proof of convergence of solutions as the domain degenerates

Construction of a corrector function ensuring strong convergence

Abstract

In this paper we study the asymptotic behavior of the solutions of a class of nonlinear elliptic problems posed in a 2-dimensional domain that degenerates into a line segment (a thin domain) when a positive parameter $\varepsilon$ goes to zero. We also allow high oscillating behavior on the upper boundary of the thin domain as $\varepsilon \to 0$. Combining methods from classic homogenization theory for reticulated structures and monotone operators we obtain the homogenized equation proving convergence of the solutions and establishing a corrector function which guarantees strong convergence in $W^{1,p}$ for $1<p<+\infty$.