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

In their paper, Braides, Buttazzo and Fragala proved the density of Riemannian energies in the class of Finsler energy functionals with respect to $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }$-approximated by Riemannian energies.