Effective Fronts of Polytope Shapes
Keywords:
Homogenization, front propagation, effective Hamiltonian, effective fronts, centrally symmetric polytopes, optimal rate of convergence.Abstract
We study the periodic homogenization of first order front propagations. Based on PDE methods, we provide a simple proof that, for n ≥ 3, the class of centrally symmetric polytopes with rational coordinates and nonempty interior is admissible as effective fronts, which was also established by I. Babenko and F. Balacheff [Sur la forme de la boule unité de la norme stable unidimensionnelle, Manuscripta Math. 119(3) (2006) 347–358] and M. Jotz [Hedlund metrics and the stable norm, Diff. Geometry Appl. 27(4) (2009) 543–550] in the form of stable norms as an extension of Hedlund’s classical result [G. A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. Math. 33 (1932) 719–739]. Besides, we obtain the optimal convergence rate of the homogenization problem for this class.
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