On the honeycomb conjecture for Robin Laplacian eigenvalues

Bucur, Dorin; Fragalà, Ilaria

doi:10.1142/s0219199718500074

Cited by 8 publications

(13 citation statements)

References 32 publications

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“…One can notice immediately that the cells are not necessarily convex, for instance when D is a square and n = 5. The results in the periodic case are in accordance with results in [7,6]. • Computation of optimal packings.…”

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confidence: 84%

“…The numerical results presented in Figure 1 are performed for 8, 16 and 32 cells in the torus (the choice of the torus instead of a k-cell is based on the observation that, if Conjecture 4 would fail on some k-cell which tiles the plane, then it will also fail on the torus). Concerning the asymptotic behaviour of α-Cheeger clusters for problem (6) when α → +∞, the same arguments used to obtain Proposition 3 can be used to show that, up to subsequences, a solution to such problem converges as well in L 1 to a partition of D into k mutually disjoint subsets of equal measure. Anyway in this case we have no conjecture about the optimization problem solved by this limit configuration.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 98%

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Phase Field Approach to Optimal Packing Problems and Related Cheeger Clusters

2018

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In a fixed domain of R N we study the asymptotic behaviour of optimal clusters associated to α-Cheeger constants and natural energies like the sum or maximum: we prove that, as the parameter α converges to the "critical" value N −1 N + , optimal Cheeger clusters converge to solutions of different packing problems for balls, depending on the energy under consideration. As well, we propose an efficient phase field approach based on a multiphase Gamma convergence result of Modica-Mortola type, in order to compute α-Cheeger constants, optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions.

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confidence: 84%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 98%

Phase Field Approach to Optimal Packing Problems and Related Cheeger Clusters

2018

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“…where P(m, n) is a polygon with area m and n edges, R(m, n) is a regular polygon centred at the origin with area m and n edges, and z ∈ R 2 . Combining (20) and (21) gives…”

Section: Sketch Of the Proofs Of Theorems 11 And 12mentioning

confidence: 99%

“…Our result also falls into the field of optimal partitions (see Remark 4.10). The optimality of hexagonal tilings, or Honeycomb conjectures, have been proved for example by [21][22][23]47]. Kelvin's problem of finding the optimal foam in 3D (the 'three-dimensional Honeycomb conjecture') remains to this day unsolved; for over 100 years it was believed that truncated octahedra gave the optimal tessellation, until the remarkable discovery of a better tessellation by Weire and Phelan [72].…”

Section: Literature On Crystallization Optimal Partitions and Quantizationmentioning

confidence: 99%

Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

Bourne

Cristoferi

2021

Commun. Math. Phys.

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We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure $$f \mathrm {d}x$$ f d x by a discrete probability measure $$\sum _i m_i \delta _{z_i}$$ ∑ i m i δ z i , subject to a constraint on the particle sizes $$m_i$$ m i . The locations $$z_i$$ z i of the particles, their sizes $$m_i$$ m i , and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne et al. (Commun Math Phys, 329: 117–140, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté et al. (J Math Pures Appl, 95:382–419, 2011). Finally, we prove a crystallization result which states that optimal configurations with energy close to that of a triangular lattice are geometrically close to a triangular lattice.

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“…Here µ is a Borel measure on Ω δ,α of the form µ = Nµ i=1 m i δ z i with Nµ i=1 m i = V δ,α . By (21) we have…”

Section: Introductionmentioning

confidence: 99%

Asymptotic optimality of the triangular lattice for a class of optimal location problems

Bourne,

Cristoferi

2020

Preprint

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We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure f dx by a discrete probability measure i miδz i , subject to a constraint on the particle sizes mi. The locations zi of the particles, their sizes mi, and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous crystallization result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne, Peletier and Theil (Communications in Mathematical Physics, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté, Jimenez and Mahadevan (Journal de Mathématiques Pures et Appliquées, 2011).

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On the honeycomb conjecture for Robin Laplacian eigenvalues

Cited by 8 publications

References 32 publications

Phase Field Approach to Optimal Packing Problems and Related Cheeger Clusters

Phase Field Approach to Optimal Packing Problems and Related Cheeger Clusters

Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

Asymptotic optimality of the triangular lattice for a class of optimal location problems

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