Unfolding operator method for thin domains with a locally periodic highly oscillatory boundary

TL;DR

This paper extends the unfolding operator method to analyze the homogenization of the Poisson equation in thin domains with complex, locally periodic oscillatory boundaries, providing new tools for variable-period media.

Contribution

It introduces an adapted unfolding operator technique for locally periodic media with variable oscillation amplitude and period, applicable to thin domains with oscillatory boundaries.

Findings

Derived the homogenized limit problem for the given domain.

Established a corrector result for the solutions.

Showed the method's adaptability to other locally periodic structures.

Abstract

We analyze the behavior of solutions of the Poisson equation with homogeneous Neumann boundary conditions in a two-dimensional thin domain which presents locally periodic oscillations at the boundary. The oscillations are such that both the amplitude and period of the oscillations may vary in space. We obtain the homogenized limit problem and a corrector result by extending the unfolding operator method to the case of locally periodic media. We emphasize the fact that the techniques developed in this paper can be adapted to other locally periodic cases like reticulated or perforated domains where the period may be space-dependent.