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Published on: September 26, 2014
Continuum limits of discrete isoperimetric problems and Wulff shapes in lattices and quasicrystal tilings
Giacomo Del Nin1, Mircea Petrache2
1Mathematics Institute, University of Warwick, Zeeman Building, Coventry, CV4 7HP UK.
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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