On the number of Courant-sharp Dirichlet eigenvalues

Berg, M. van den; Gittins, Katie

doi:10.4171/jst/139

Cited by 6 publications

(19 citation statements)

References 12 publications

(24 reference statements)

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“…This result proves the existence of such k 0 but is not quantitative. Recently, Bérard-Helffer [3] and van den Berg-Gittins [31] exhibit an explicit bound for k 0 .…”

Section: Nodal Partitionmentioning

confidence: 99%

Minimal partitions for $p$-norms of eigenvalues

Bonnaillie‐Noël

Bogosel

2018

Interfaces Free Bound.

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In this article we are interested in studying partitions of the square, the disk and the equilateral triangle which minimize a p-norm of eigenvalues of the Dirichlet-Laplace operator. The extremal case of the infinity norm, where we minimize the largest fundamental eigenvalue of each cell, is one of our main interests. We propose three numerical algorithms which approximate the optimal configurations and we obtain tight upper bounds for the energy, which are better than the ones given by theoretical results. A thorough comparison of the results obtained by the three methods is given. We also investigate the behavior of the minimal partitions with respect to p. This allows us to see when partitions minimizing the 1-norm and the infinity-norm are different.With a little abuse of notation, we notice thatThe index ∞ is omitted when there is no confusion. The optimization problem we consider is to determine the infimum of the p-energy (1 ≤ p ≤ ∞) among the partitions of P k (Ω):This optimization problem has been a subject of great interest in the last twenty years. Two cases are especially studied: the sum which corresponds to p = 1 and the max, corresponding to p = ∞. General aspects concerning existence results for optimal partitions problems are presented in [15,14]. Existence and regularity results for optimal partitioning problems regarding non-linear eigenvalue problems, containing as a particular case the Dirichlet eigenvalues, are considered in [17]. In [16] the authors consider the minimization of the partitions minimizing the sum of the Dirichlet-Laplace eigenvalues, stating the spectral honeycomb conjecture and initiating many theoretical and numerical works on the subject. In [24] the authors consider the partitions minimizing the maximum of the fundamental eigenvalues and they provide results concerning connections between such optimal partitions and nodal partitions, for particular values of k. More recently, the link between these two optimization problems is taken into consideration in [23]. In particular, a criterion is established to assert that a ∞-minimal k-partition is not a 1-minimal k-partition. This criterion is given in Proposition 3.8 and applied in Section 4.4.There are few cases for which optimal partitions are known explicitly for the spectral quantities we consider here. This motivates the development of numerical algorithms which can find approximations of optimal partitions and suggest candidates as optimal partitions. The case p = 1, corresponding to the sum of the eigenvalues, was considered in [12], where an algorithm based on a relaxation procedure was presented. The algorithm allowed the study of partitions made of several hundreds of cells and shows that it is likely that partitions made of hexagons are a good candidate to being minimal as k → ∞. The numerical minimization of the largest eigenvalue has been considered in [8,5,9,10]. In [8], we exhibit some candidates for the 3-partition of the square and the disk by using a mixed Dirichlet-Neumann approach that will be used in Sec...

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“…This result proves the existence of such k 0 but is not quantitative. Recently, Bérard-Helffer [3] and van den Berg-Gittins [31] exhibit an explicit bound for k 0 .…”

Section: Nodal Partitionmentioning

confidence: 99%

Minimal partitions for $p$-norms of eigenvalues

Bonnaillie‐Noël

Bogosel

2018

Interfaces Free Bound.

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“…Following from Pleijel's result, natural questions are, for a given domain, how many such eigenvalues are there and how large are they? The recent articles [1,3] consider these questions and give upper bounds for the largest Courant-sharp Dirichlet eigenvalue and the number of such eigenvalues in terms of some of the geometric quantities of the underlying domain. In [3], such geometric upper bounds are obtained for an open set in R n with finite Lebesgue measure.…”

Section: Introductionmentioning

confidence: 99%

“…The recent articles [1,3] consider these questions and give upper bounds for the largest Courant-sharp Dirichlet eigenvalue and the number of such eigenvalues in terms of some of the geometric quantities of the underlying domain. In [3], such geometric upper bounds are obtained for an open set in R n with finite Lebesgue measure. In the case where the domain is convex, Example 1 of [3] shows that if it has a large number of Courant-sharp Dirichlet eigenvalues then its isoperimetric ratio is also large.…”

Section: Introductionmentioning

confidence: 99%

“…In [3], such geometric upper bounds are obtained for an open set in R n with finite Lebesgue measure. In the case where the domain is convex, Example 1 of [3] shows that if it has a large number of Courant-sharp Dirichlet eigenvalues then its isoperimetric ratio is also large.…”

Section: Introductionmentioning

confidence: 99%

“…We also observe that we extend results which were previously known in the Dirichlet case, with some additional hypotheses due to the difficulties in handling the Neumann boundary condition. In Section 3 of [3], the authors obtain an upper bound for the number of Courant-sharp Dirichlet eigenvalues of an open, bounded, convex set in any dimension, in terms of the isoperimetric ratio. In Theorem 1.3 of [1], the authors bound this number using the area, perimeter, maximal curvature and minimal cut-distance of the boundary, for a set in R 2 which is sufficiently regular but not necessarily convex.…”

Section: Introductionmentioning

confidence: 99%

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