Calculus of Variations and Geometric Measure Theory
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G. Del Nin - M. Petrache

Continuum limits of discrete isoperimetric problems and Wulff shapes in lattices and quasicrystal tilings

created by delnin on 09 Feb 2021

[BibTeX]

preprint

Inserted: 9 feb 2021

Year: 2021

ArXiv: 2101.11977 PDF

Abstract:

We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the "multigrid construction" of quasiperiodic tilings (which is an extension of De Bruijn's "pentagrid" construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling.

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