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

Emanuele Caglioti (1) , François Golse (2) , Mikaela Iacobelli (3)

1 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]
2 CMLS - Centre de Mathématiques Laurent Schwartz
3 DPMMS - Department of Pure Mathematics and Mathematical Statistics

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.

Keywords: Quantization of measures; Monge-Kantorovich distance; Gradient flow; Parabolic systems of PDEs

Domains: Mathematics [math]; Analysis of PDEs [math.AP]

Emanuele Caglioti, François Golse, Mikaela Iacobelli. Quantization of Measures and Gradient Flows: a Perturbative Approach in the 2-Dimensional Case. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2018, 35, pp.1531-1555. ⟨10.1016/j.anihpc.2017.12.003⟩. ⟨hal-01341841⟩