Calculus of Variations and Geometric Measure Theory

E. Davoli - P. Piovano - U. Stefanelli

Sharp $N^{3/4}$ law for the minimizers of the Edge-Isoperimetric Problem on the triangular lattice

created by davoli on 14 Dec 2015
modified on 04 Sep 2020


Published Paper

Inserted: 14 dec 2015
Last Updated: 4 sep 2020

Journal: Journal of Nonlinear Science
Year: 2016


We investigate the Edge-Isoperimetric Problem (EIP) for sets of $n$ points in the triangular lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. By introducing a suitable notion of perimeter and area, EIP minimizers are characterized as extremizers of an isoperimetric inequality: they attain maximal area and minimal perimeter among connected configurations. The maximal area and minimal perimeter are explicitly quantified in terms of $n$. In view of this isoperimetric characterizations EIP minimizers $M_n$ are seen to be given by hexagonal configurations with some extra points at their boundary. By a careful computation of the cardinality of these extra points, minimizers $M_n$ are estimated to deviate from such hexagonal configurations by at most $K_t\, n^{3/4}+{\rm o}(n^{3/4})$ points. The constant $K_t$ is explicitly determined and shown to be sharp.

Keywords: isoperimetric inequality, wulff shape, triangular lattice, Edge-Isoperimetric Problem, Edge-perimeter, $N^{3/4}$ law