Minimal partitions for $p$-norms of eigenvalues

Abstract

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...

A theory of spectral partitions of metric graphs

“…In particular, the optimal partition is never the equipartition except for p = ∞ (which we recall corresponds to a ∞ = 0), and the cut point v 0 , as a function of p, moves smoothly and monotonically from its location at p = 1 towards the central vertex as p → ∞. This mirrors very much the numerically observed behaviour of the (conjectured) optimal partitions on domains in [BBN18].…”

“…But (7.1) is a direct consequence of the Hölder inequality, using the definition (4.1) of Λ N p (P). We continue discussing the dependence of optimal partitions and energies on p. To begin with, let us present a concrete example illustrating how the optimal partition, say in the simplest case for L D 2,p , can depend nontrivially on p. On domains, relatively little seems to be known, and most of the work to date seems to have been of (largely) numerical nature; see in particular [BBN18]. Our example, in addition to establishing that L D 2,p and the corresponding optimal partitions can, in fact, depend on p, should also demonstrate how in the case of metric graphs it seems possible to prove more properties (such as monotonicity of the deformation in p) analytically.…”

Phase Field Approach to Optimal Packing Problems and Related Cheeger Clusters

“…For α = 1, problem (3) has been firstly studied by Caroccia in the paper [10], where the existence of solutions and some regularity results for the free boundaries are obtained. In fact, for arbitrary α > N −1 N , the existence of solutions (E 1 , .…”

“…We choose to use a p-norm approach since this regularizes the non-smooth problem of minimizing the maximal radius of a family of disks. The minimization of a p-norm instead of the ∞-norm is a natural idea, already used in [3] for the study of partitions of a domain which minimize the largest fundamental eigenvalue of the Dirichlet-Laplace operator.…”

Optimal partitions for the sum and the maximum of eigenvalues

“…In cases where the previous criterion shows that candidates for the max are not optimal for the sum, we propose better candidates in Section 5. These candidates are either obtained with iterative methods already used in [6,3] or are constructed explicitly. §2.…”