Homogenization in 3D thin domains with oscillating boundaries of different orders

TL;DR

This paper extends the unfolding operator technique to analyze the asymptotic behavior of reaction-diffusion equations in three-dimensional thin domains with oscillating boundaries, providing a unified approach to complex geometries.

Contribution

It introduces a novel extension of the unfolding operator method to 3D thin domains with oscillating boundaries, enabling effective analysis of complex geometries.

Findings

Derived effective problems for various oscillation scenarios.

Enhanced understanding of reaction-diffusion behavior in complex 3D geometries.

Provided a unified analytical framework for oscillating boundary problems.

Abstract

This paper presents an extension of the unfolding operator technique, initially applied to two-dimensional domains, to the realm of three-dimensional thin domains. The advancement of this methodology is pivotal, as it enhances our understanding and analysis of three-dimensional geometries, which are crucial in various practical fields such as engineering and physics. Our work delves into the asymptotic behavior of solutions to a reaction-diffusion equation with Neumann boundary conditions set within such a oscillatory 3-dimensional thin domain. The method introduced enables the deduction of effective problems across all scenarios, tackling the intrinsic complexity of these domains. This complexity is especially pronounced due to the possibility of diverse types of oscillations occurring along their boundaries.