Homogenization of Penrose tilings

Andrea Braides; Giuseppe Riey; Margherita Solci

doi:10.1016/j.crma.2009.03.019

Calculus of Variations

Homogenization of Penrose tilings

Presented by: Philippe G. Ciarlet

Andrea Braides ¹; Giuseppe Riey ²; Margherita Solci ³

¹ Dipartimento di Matematica, Università di Roma Tor Vergata, via della ricerca scientifica 1, 00133 Roma, Italy
² Dipartimento di Matematica, Università della Calabria, via P. Bucci, 87036 Arcavacata di Rende (CS), Italy
³ DAP, Università di Sassari, piazza Duomo 6, 07041 Alghero (SS), Italy

Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 697-700.

Abstract
Résumé

A homogenization theorem is proved for energies which follow the geometry of an a-periodic Penrose tiling. The result is obtained by proving that the corresponding energy densities are $W^{1}$ -almost periodic and hence also Besicovitch almost periodic, so that existing general homogenization theorems can be applied (Braides, 1986). The method applies to general quasicrystalline geometries.

On démontre un théorème d'homogénéisation pour des énergies qui suivent la géométrie d'un pavage apériodique de Penrose. Nos résultats, applicables à des géométries quasicristallines générales, sont obtenus en démontrant que les densités d'énergie correspondantes sont $W^{1}$ – et donc Besicovitch – quasi-périodiques, de sort que l'on peut appliquer les théorèmes d'homogénéisation de Braides, 1986.

Received: 2009-03-10
Accepted: 2009-03-20
Published online: 2009-04-21

DOI: 10.1016/j.crma.2009.03.019

Author's affiliations:

Andrea Braides ¹; Giuseppe Riey ²; Margherita Solci ³

¹ Dipartimento di Matematica, Università di Roma Tor Vergata, via della ricerca scientifica 1, 00133 Roma, Italy
² Dipartimento di Matematica, Università della Calabria, via P. Bucci, 87036 Arcavacata di Rende (CS), Italy
³ DAP, Università di Sassari, piazza Duomo 6, 07041 Alghero (SS), Italy

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     title = {Homogenization of {Penrose} tilings},
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     language = {en},
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Andrea Braides; Giuseppe Riey; Margherita Solci. Homogenization of Penrose tilings. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 697-700. doi : 10.1016/j.crma.2009.03.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.03.019/

References
Cited by

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