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

A. Braides, G. Buttazzo and I. Fragalà [Riemannian approximation of Finsler metrics, Asymptot. Anal. 31(2) (2002) 177–187] 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 article we prove that one of the main tools in the above-mentioned 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.