Quantization of Measures and Gradient Flows: a Perturbative Approach in the 2-Dimensional Case

Abstract : In this paper we study a perturbative approach to the problem of quantization of measures in the plane. Motivated by the fact that, as the number of points tends to infinity, hexagonal lattices are asymptotically optimal from an energetic point of view (see [Morgan, Bolton: Amer. Math. Monthly 109 (2002), 165-172]), we consider configurations that are small perturbations of the hexagonal lattice and we show that: (1) in the limit as the number of points tends to infinity, the hexagonal lattice is a strictly minimizer of the energy; (2) the gradient flow of the limiting functional allows us to evolve any perturbed configuration to the optimal one exponentially fast. In particular, our analysis provides a solid mathematical justification of the asymptotic optimality of the hexagonal lattice among its nearby configurations.
Type de document :
Pré-publication, Document de travail
24 pages, 6 figures. 2016
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https://hal-polytechnique.archives-ouvertes.fr/hal-01341841
Contributeur : François Golse <>
Soumis le : mardi 5 juillet 2016 - 01:19:45
Dernière modification le : jeudi 11 janvier 2018 - 06:12:13

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  • HAL Id : hal-01341841, version 1

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Emanuele Caglioti, François Golse, Mikaela Iacobelli. Quantization of Measures and Gradient Flows: a Perturbative Approach in the 2-Dimensional Case . 24 pages, 6 figures. 2016. 〈hal-01341841〉

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