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
wjjing@tsinghua.edu.cn
Hung V. Tran
Dept. of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A.
hung@math.wisc.edu
Yifeng Yu
Dept. of Mathematics, University of California, Irvine, CA 92697, U.S.A.
yyu1@math.uci.edu
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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