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
[Homogénéisation d'un pavage de Penrose]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2009-03-10
Accepté le : 2009-03-20
Publié le : 2009-04-21

DOI : 10.1016/j.crma.2009.03.019

Affiliations des auteurs :

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

@article{CRMATH_2009__347_11-12_697_0,
     author = {Andrea Braides and Giuseppe Riey and Margherita Solci},
     title = {Homogenization of {Penrose} tilings},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {697--700},
     publisher = {Elsevier},
     volume = {347},
     number = {11-12},
     year = {2009},
     doi = {10.1016/j.crma.2009.03.019},
     language = {en},
}

TY  - JOUR
AU  - Andrea Braides
AU  - Giuseppe Riey
AU  - Margherita Solci
TI  - Homogenization of Penrose tilings
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 697
EP  - 700
VL  - 347
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2009.03.019
LA  - en
ID  - CRMATH_2009__347_11-12_697_0
ER  -

%0 Journal Article
%A Andrea Braides
%A Giuseppe Riey
%A Margherita Solci
%T Homogenization of Penrose tilings
%J Comptes Rendus. Mathématique
%D 2009
%P 697-700
%V 347
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2009.03.019
%G en
%F CRMATH_2009__347_11-12_697_0

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/

Bibliographie
Cité par

[1] A.S. Besicovitch Almost Periodic Functions, Dover, Cambridge, 1954

[2] A. Braides A homogenization theorem for weakly almost periodic functionals, Rend. Accad. Naz. Sci. XL, Volume 104 (1986), pp. 261-281

[3] A. Braides Γ-convergence for Beginners, Oxford University Press, Oxford, 2002

[4] A. Braides A handbook of Γ-convergence (M. Chipot; P. Quittner, eds.), Handbook of Differential Equations, Stationary Partial Differential Equations, vol. 3, Elsevier, 2006

[5] A. Braides; A. Defranceschi Homogenization of Multiple Integrals, Oxford University Press, Oxford, 1998

[6] G. Dal Maso An Introduction to Γ-convergence, Birkhauser, Boston, 1993

[7] N.G. de Bruijn Algebraic theory of Penrose's nonperiodic tilings of the plane, Proc. K. Ned. Akad. Wet. Ser. A, Volume 43 (1981), pp. 39-66

[8] M.A. Shubin Almost periodic functions and partial differential operators, Russ. Math. Surv., Volume 33 (1978), pp. 1-52

[9] E.J.W. Whittaker; R.M. Whittaker Graphic representation and nomenclature of the four-dimensional crystal classes. IV. Irrational crypto-rotation planes of non-crystallographic orders, Acta Cryst. A, Volume 42 (1986), pp. 387-398

[10] E.J.W. Whittaker; R.M. Whittaker Some generalized Penrose patterns from projections of n-dimensional lattices, Acta Cryst. A, Volume 44 (1988), pp. 105-112

Daniel Baratta; Luigi Muglia; Domenico Vuono Second order regularity for solutions to anisotropic degenerate elliptic equations, Journal of Differential Equations, Volume 435 (2025), p. 113250 | DOI:10.1016/j.jde.2025.113250
Romain Rieger; Alexandre Danescu Macroscopic elasticity of the hat aperiodic tiling, Mechanics of Materials, Volume 193 (2024), p. 104988 | DOI:10.1016/j.mechmat.2024.104988
Niklas Wellander; Sébastien Guenneau; Elena Cherkaev Two-scale cut-and-projection convergence for quasiperiodic monotone operators, European Journal of Mechanics - A/Solids, Volume 100 (2023), p. 104796 | DOI:10.1016/j.euromechsol.2022.104796
Andrea Braides; Lorenza D’Elia Homogenization of discrete thin structures, Nonlinear Analysis, Volume 231 (2023), p. 112951 | DOI:10.1016/j.na.2022.112951
Carlo Sbordone; Margherita Guida Mathematical aspects of quasicrystals, Rendiconti Lincei. Scienze Fisiche e Naturali, Volume 34 (2023) no. 3, p. 721 | DOI:10.1007/s12210-023-01176-y
Marek Tyburec; Jan Zeman Bounded Wang tilings with integer programming and graph-based heuristics, Scientific Reports, Volume 13 (2023) no. 1 | DOI:10.1038/s41598-023-31786-3
Giacomo Del Nin; Mircea Petrache Continuum limits of discrete isoperimetric problems and Wulff shapes in lattices and quasicrystal tilings, Calculus of Variations and Partial Differential Equations, Volume 61 (2022) no. 6 | DOI:10.1007/s00526-022-02318-0
M. Amar; G. Riey Homogenization of singular elliptic systems with nonlinear conditions on the interfaces, Journal of Elliptic and Parabolic Equations, Volume 6 (2020) no. 2, p. 633 | DOI:10.1007/s41808-020-00075-9
Elena Cherkaev; Sebastien Guenneau; Niklas Wellander, 2019 URSI International Symposium on Electromagnetic Theory (EMTS) (2019), p. 1 | DOI:10.23919/ursi-emts.2019.8931468
Daniele Castorina; Giuseppe Riey; Berardino Sciunzi Hopf Lemma and regularity results for quasilinear anisotropic elliptic equations, Calculus of Variations and Partial Differential Equations, Volume 58 (2019) no. 3 | DOI:10.1007/s00526-019-1528-x
Enrico Cicalò; Andrea Causin; Margherita Solci Mathegraphics, ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics, Volume 809 (2019), p. 1511 | DOI:10.1007/978-3-319-95588-9_134
Emilio Turco; Emilio Barchiesi Equilibrium paths of Hencky pantographic beams in a three-point bending problem, Mathematics and Mechanics of Complex Systems, Volume 7 (2019) no. 4, p. 287 | DOI:10.2140/memocs.2019.7.287
ANDREA BRAIDES; MARGHERITA SOLCI INTERFACIAL ENERGIES ON PENROSE LATTICES, Mathematical Models and Methods in Applied Sciences, Volume 21 (2011) no. 05, p. 1193 | DOI:10.1142/s0218202511005295

Cité par 13 documents. Sources : Crossref

Commentaires - Politique