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Minimax Theory and its Applications 05 (2020), No. 2, 347--360
Copyright Heldermann Verlag 2020

Effective Fronts of Polytope Shapes

Wenjia Jing
Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, P. R. China

Hung V. Tran
Dept. of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A.

Yifeng Yu
Dept. of Mathematics, University of California, Irvine, CA 92697, U.S.A.

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 unit\'e de la norme stable unidimensionnelle, Manuscripta Math. 119 (2006) 347--358] and M. Jotz [Hedlund metrics and the stable norm, Diff. Geometry Appl. 27 (2009) 543--550] in the form of stable norms as an extension of G. A. Hedlund's classical result [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.

Keywords: Homogenization, front propagation, effective Hamiltonian, effective fronts, centrally symmetric polytopes, optimal rate of convergence.

MSC: 35B40, 37J50, 49L25.

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