Abstract

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 Γ\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 Γ\Gamma-approximated by Riemannian energies.

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Keywords

  • Gamma convergence
  • Riemannian metrics
  • Finsler metrics
  • Cartan functionals
  • dominance functions

Mathematics Subject Classification

49J45, 58B20, 53B40

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Suggested citation

T. Knoke. “On the Approximation of Anisotropic Energy Functionals by Riemannian Energies via Homogenization.” Journal of Convex Analysis 26 (2019), No. 2, 449–483. https://doi.org/10.68381/jca26023

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.