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Journal of Convex Analysis 26 (2019), No. 2, 449--483
Copyright Heldermann Verlag 2019

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

Till Knoke
Mathematisches Institut, RWTH Aachen, Templergraben 55, 52062 Aachen

A. Braides, G. Buttazzo and I. FragalÓ [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.

Keywords: Gamma convergence, Riemannian metrics, Finsler metrics, Cartan functionals, dominance functions.

MSC: 49J45, 58B20, 53B40

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