Homogenization on parallelizable Riemannian manifolds

Abstract

We consider the problem of finding the homogenization limit of oscillating linear elliptic equations on an arbitrary parallelizable manifold $(M,g,\Gamma)$. We replicate the concept of two-scale convergence by pulling back tensors $T$ defined on the torus bundle $\mathbb{T}M$ to $M$. The process consist of two steps: localization in the slow variable through Voronoi domains, and inducing local periodicity in the fast variable from the local exponential map in combination with the geometry of the torus bundle. The procedure yields explicit cell formulae for the homogenization limit and as a byproduct a theory of two-scale convergence of tensors of arbitrary order.