Numerical simulations for nodal domains and spectral minimal partitions

Bonnaillie‐Noël, Virginie; Helffer, Bernard; Vial, Grégory

doi:10.1051/cocv:2008074

Cited by 41 publications

(76 citation statements)

References 17 publications

(33 reference statements)

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“…As a matter of fact, optimal spectral partitions have received an increasing attention in the last decade, including also the case when the energy of the partition is the maximal eigenvalue among the chambers (in particular by Helffer and coauthors); without any attempt of completeness, let us quote the papers [1,2,3,10,11,15,16,17,25]. In particular, in [17], a similar honeycomb conjecture for the maximal eigenvalue problem is attributed to Van den Berg.…”

Section: Introductionmentioning

confidence: 99%

“…the well-known case solved by Hales in the celebrated paper [14], when the cost is the total perimeter of the partition, see also [23]). We are interested in studying for which kind of variational energies F optimal partition problems of the kind (1), (2), or (3) satisfies the "honeycomb conjecture". Roughly, it can be stated as the fact that, in the limit for k very large, an optimal packing will be made of translations of an identical shape, given precisely by a regular hexagon H. A simple mathematical formulation can be given as the asymptotic law (4) lim…”

Section: Introductionmentioning

confidence: 99%

“…Our main results are stated in Section 2 below: Theorems 2, 3, and 6 deal respectively with problems (1), (2) and (3), under suitable hypotheses on a generic shape functional F . As a consequence, the honeycomb conjecture is obtained for the Cheeger constant and for logarithmic capacity (see Corollaries 9-10 and 13); moreover, we provide some relatively simple sufficient conditions for its validity also to the case of the first Dirichlet eigenvalue (see Proposition 11).…”

Section: Introductionmentioning

confidence: 99%

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On the honeycomb conjecture for a class of minimal convex partitions

Bucur

Fragalà

Velichkov

et al. 2018

Trans. Amer. Math. Soc.

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Abstract. We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the energies of the cells, the cost being a shape functional which satisfies a few assumptions. They are: monotonicity under inclusions; homogeneity under dilations; a Faber-Krahn inequality for convex hexagons; a convexity-type inequality for the map which associates with every n ∈ N the minimizers of F among convex n-gons with given area. In particular, our result allows to obtain the honeycomb conjecture for the Cheeger constant and for the logarithmic capacity (still assuming the cells to be convex). Moreover we show that, in order to get the conjecture also for the first Dirichlet eigenvalue of the Laplacian, it is sufficient to establish some facts about the behaviour of λ1 among convex pentagons, hexagons, and heptagons with prescribed area.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the honeycomb conjecture for a class of minimal convex partitions

Bucur

Fragalà

Velichkov

et al. 2018

Trans. Amer. Math. Soc.

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“…Roughly speaking this estimate says that, far from ∂D, a tiling by regular hexagons of area |D| n is asymptotically close to the optimal partition. A close problem, still for k = 1, was considered by Bonnaillie-Noël, Helffer and Vial in [4], where the cost functional is replaced by L n (Ω 1 , . .…”

mentioning

confidence: 99%

Optimal Partitions for Eigenvalues

Bourdin¹,

Bucur

Oudet³

2010

SIAM J. Sci. Comput.

134

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Abstract. We introduce a new numerical method to approximate partitions of a domain minimizing the sum of DirichletLaplacian eigenvalues of any order. First we prove the equivalence of the original problem and a relaxed formulation based on measures. Using this result, we build a numerical algorithm to approximate optimal configurations. We describe numerical experiments aimed at studying the asymptotic behavior of optimal partitions with large numbers of cells.

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“…It is explored numerically in [14] why the second conjecture looks reasonable. Note that Caffarelli & Lin [16] mention conjecture 4.6 in relation with L k,1 (Ω).…”

Section: Conjecture 46mentioning

confidence: 98%

Remarks on the boundary set of spectral equipartitions

Bérard

Helffer

2014

Phil. Trans. R. Soc. A.

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Given a bounded open set in (or in a Riemannian manifold), and a partition of Ω by k open sets ω j , we consider the quantity , where λ ( ω j ) is the ground state energy of the Dirichlet realization of the Laplacian in ω j . We denote by ℒ k ( Ω ) the infimum of over all k -partitions. A minimal k -partition is a partition that realizes the infimum. Although the analysis of minimal k -partitions is rather standard when k =2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of k becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies λ ( ω j ) are all equal. The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. We consider estimates involving the cardinality of the partition.

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Numerical simulations for nodal domains and spectral minimal partitions

Abstract: Abstract.We recall here some theoretical results of Helffer et al.

Cited by 41 publications

References 17 publications

On the honeycomb conjecture for a class of minimal convex partitions

On the honeycomb conjecture for a class of minimal convex partitions

Optimal Partitions for Eigenvalues

Remarks on the boundary set of spectral equipartitions

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