# On the Approximation of Anisotropic Energy Functionals by Riemannian Energies via Homogenization

## TL;DR

This paper extends homogenization results for Riemannian energies to higher dimensions but shows that not all anisotropic energies, like Finsler and Cartan functionals, can be approximated by Riemannian energies through $ ext{Gamma}$-convergence.

## Contribution

It generalizes the homogenization theorem to arbitrary dimensions and provides counterexamples demonstrating the limitations of Riemannian approximation for certain anisotropic energies.

## Key findings

- Homogenization theorem extended to higher dimensions.
- Counterexamples show limitations of Riemannian approximation.
- Not all anisotropic energies can be $ ext{Gamma}$-approximated by Riemannian energies.

## Abstract

In their paper, Braides, Buttazzo and Fragala proved the density of Riemannian energies in the class of Finsler energy functionals with respect to $\Gamma$-convergence in the one-dimensional case. In this thesis we prove that one of the main tools in that paper, a homogenization theorem, can be extended to arbitrary dimension, however, the density result cannot be generalized to higher dimensions. In fact, we construct counterexamples that show: there are anisotropic energy functionals, such as Finsler energies, Cartan functionals and their dominance functionals that cannot be $\Gamma$-approximated by Riemannian energies.