An Application of Periodic Unfolding on Manifolds

TL;DR

This paper applies the periodic unfolding method on Riemannian manifolds to homogenize elliptic PDEs, demonstrating invariance of the homogenization limit under coordinate changes and scalings.

Contribution

It introduces a novel application of periodic unfolding on manifolds for PDE homogenization, extending classical methods to curved spaces.

Findings

Homogenization limit is invariant under coordinate transformations.

Generalized limit- and cell-problems for elliptic PDEs on manifolds.

Method ensures independence from reference cell scalings.

Abstract

We show how the newly developed method of Periodic Unfolding on Riemannian manifolds can be applied to PDE problems: We consider the homogenization of an elliptic model problem. In the limit, we obtain a generalization of the well-known limit- and cell-problem. By constructing an equivalence relation of atlases, one can show the invariance of the limit problem with respect to this equivalence relation. This implies e.g. that the homogenization limit is independent of change of coordinates or scalings of the reference cell.