Calculus of Variations and Geometric Measure Theory
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E. Mainini - P. Piovano - B. Schmidt - U. Stefanelli

$N^{3/4}$ law in the cubic lattice

created by mainini on 05 Jul 2018

[BibTeX]

Preprint

Inserted: 5 jul 2018

Year: 2018

ArXiv: 1807.00811 PDF

Abstract:

We investigate the Edge-Isoperimetric Problem (EIP) for sets with $n$ elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers $M_n$ of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most $O(n^{3/4})$ elements. The exponent $3/4$ is optimal. This extends to the cubic lattice analogous results that have already been established for the triangular, the hexagonal, and the square lattice in two space dimensions.

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