Title
Sharp N^3/4 Law for the Minimizers of the Edge-Isoperimetric Problem on the Triangular Lattice
Author
Abstract
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 Mn 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 Mn are estimated to deviate from such hexagonal configurations by at most Ktn3/4+o(n3/4) points. The constant Kt is explicitly determined and shown to be sharp.
Keywords
Edge-isoperimetric problemEdge perimeterTriangular latticeIsoperimetric inequalityWulff shapeN3/4 law
Object type
Language
English [eng]
Persistent identifier
Appeared in
Title
Journal of Nonlinear Science
Volume
27
Issue
2
From page
627
To page
660
Publication
Springer Nature
Version type
Date issued
2016
Access rights
Rights
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