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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
till.knoke@rwth-aachen.de



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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