[Homogénéisation d'un pavage de Penrose]
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 – 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 -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.
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@article{CRMATH_2009__347_11-12_697_0,
author = {Braides, Andrea and Riey, Giuseppe and Solci, Margherita},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.03.019/}
}
TY - JOUR AU - Braides, Andrea AU - Riey, Giuseppe AU - Solci, Margherita TI - Homogenization of Penrose tilings JO - Comptes Rendus. Mathématique PY - 2009 SP - 697 EP - 700 VL - 347 IS - 11-12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.03.019/ DO - 10.1016/j.crma.2009.03.019 LA - en ID - CRMATH_2009__347_11-12_697_0 ER -
%0 Journal Article %A Braides, Andrea %A Riey, Giuseppe %A Solci, Margherita %T Homogenization of Penrose tilings %J Comptes Rendus. Mathématique %D 2009 %P 697-700 %V 347 %N 11-12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.03.019/ %R 10.1016/j.crma.2009.03.019 %G en %F CRMATH_2009__347_11-12_697_0
Braides, Andrea; Riey, Giuseppe; Solci, Margherita. Homogenization of Penrose tilings. Comptes Rendus. Mathématique, Tome 347 (2009) no. 11-12, pp. 697-700. doi : 10.1016/j.crma.2009.03.019. http://www.numdam.org/articles/10.1016/j.crma.2009.03.019/
[1] Almost Periodic Functions, Dover, Cambridge, 1954
[2] A homogenization theorem for weakly almost periodic functionals, Rend. Accad. Naz. Sci. XL, Volume 104 (1986), pp. 261-281
[3] Γ-convergence for Beginners, Oxford University Press, Oxford, 2002
[4] A handbook of Γ-convergence (Chipot, M.; Quittner, P., eds.), Handbook of Differential Equations, Stationary Partial Differential Equations, vol. 3, Elsevier, 2006
[5] Homogenization of Multiple Integrals, Oxford University Press, Oxford, 1998
[6] An Introduction to Γ-convergence, Birkhauser, Boston, 1993
[7] Algebraic theory of Penrose's nonperiodic tilings of the plane, Proc. K. Ned. Akad. Wet. Ser. A, Volume 43 (1981), pp. 39-66
[8] Almost periodic functions and partial differential operators, Russ. Math. Surv., Volume 33 (1978), pp. 1-52
[9] 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] Some generalized Penrose patterns from projections of n-dimensional lattices, Acta Cryst. A, Volume 44 (1988), pp. 105-112
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