Neumann Domains on Graphs and Manifolds

Alon, Lior; Band, Ram; Bersudsky, Michael; Egger, Sebastian

doi:10.1017/9781108615259.011

Cited by 6 publications

(4 citation statements)

References 51 publications

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“…The Neumann domains arise as the clusters of a partition cut at the points where u k (x) = 0; the number of Neumann domains behaves similarly as a function of k, at least in the "generic" case where (among other things) all cuts are made away from the vertices [AlBa19]. Perhaps most notably for us, it has been shown in the generic case that the difference between the number of nodal domains ν(k) and the number of Neumann domains ξ(k) of u k satisfies exactly the same bounds as the indices appearing in Theorems 1.1 and 1.2 [AlBa19, Proposition 3.1(1)] (see also [ABBE20,Proposition 11.2]):…”

Section: Introductionmentioning

confidence: 86%

“…Suppose u is an eigenfunction, with eigenvalue λ, of the (standard) Laplacian on G, and suppose that considering the total cut of G at all points where u reaches a local nonzero maximum or minimum generates a partition with k = ξ(u) clusters. (In the language of Section 5 and [ABBE20] this means u has ξ(u) Neumann domains.) Then λ equals the first nontrivial standard Laplacian eigenvalue on each cluster, with eigenfunction u (see [ABBE20, Lemma 8.1]).…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Just as the basic idea behind Theorem 1.1 is gluing together nodal domains of Neumann eigenfunctions of the partition clusters to construct a test partition, here we will be interested in gluing together the so-called Neumann domains of the cluster Dirichlet eigenfunctions (see, e.g., [AlBa19,ABBE20]).…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

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Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

Hofmann,

Kennedy

2021

Preprint

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We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in [Kennedy et al, arXiv:2005.01126]. These inequalities, which involve the first Betti number and the number of degree one vertices of the graph, recall both interlacing and other inequalities for the Laplacian eigenvalues of the whole graph, as well as estimates on the difference between the number of nodal and Neumann domains of the whole graph eigenfunctions. To this end we study carefully the principle of cutting a graph, in particular quantifying the size of a cut as a perturbation of the original graph via the notion of its rank. As a corollary we obtain an inequality between these energies and the actual Dirichlet and standard Laplacian eigenvalues, valid for all compact graphs, which complements a version for tree graphs of Friedlander's inequalities between Dirichlet and Neumann eigenvalues of a domain. In some cases this results in better Laplacian eigenvalue estimates than those obtained previously via more direct methods.

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Section: Introductionmentioning

confidence: 86%

Section: Proof Of Theorem 11mentioning

confidence: 99%

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Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

Hofmann,

Kennedy

2021

Preprint

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“…These same motivations remain valid in the case of quantum graphs, where comparatively little seems to be known, at least in terms of the profile of the eigenfunctions: some work has been done constructing so-called landscape functions to control their size [34,35], and relatively recently the concept of Neumann domains of the eigenfunctions, the regions separated by critical points of the eigenfunctions, was introduced and is now being studied [5,6,8,13]. But to date the "hot and cold spots" of a quantum graph do not seem to have received direct attention, a preliminary note of the current authors excluded [44].…”

mentioning

confidence: 99%

On the hot spots of quantum graphs

Kennedy¹,

Rohleder²

2021

CPAA

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We undertake a systematic investigation of the maxima and minima of the eigenfunctions associated with the first nontrivial eigenvalue of the Laplacian on a metric graph equipped with standard (continuity-Kirchhoff) vertex conditions. This is inspired by the famous hot spots conjecture for the Laplacian on a Euclidean domain, and the points on the graph where maxima and minima are achieved represent the generically "hottest" and "coldest" spots of the graph. We prove results on both the number and location of the hot spots of a metric graph, and also present a large number of examples, many of which run contrary to what one might naïvely expect. Amongst other results we prove the following: (i) generically, up to arbitrarily small perturbations of the graph, the points where minimum and maximum, respectively, are attained are unique; (ii) the minima and maxima can only be located at the vertices of degree one or inside the doubly connected part of the metric graph; and (iii) for any fixed graph topology, for some choices of edge lengths all minima and maxima will occur only at degree-one vertices, while for others they will only occur in the doubly connected part of the graph.

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Asymptotics and Estimates for Spectral Minimal Partitions of Metric Graphs

Hofmann

Kennedy

Mugnolo

et al. 2021

Integr. Equ. Oper. Theory

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We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in Kennedy et al. (Calc Var 60:6, 2021). We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metric graphs, the upper bounds are more involved and mirror the combinatorial structure of the metric graph as well. Combining them, we deduce that these spectral minimal energies also satisfy a Weyl-type asymptotic law similar to the well-known one for eigenvalues of quantum graph Laplacians with various vertex conditions. Drawing on two examples we show that in general no second term in the asymptotic expansion for minimal partition energies can exist, but show that various kinds of behaviour are possible. We also study certain aspects of the asymptotic behaviour of the minimal partitions themselves.

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Neumann Domains on Graphs and Manifolds

Cited by 6 publications

References 51 publications

Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

On the hot spots of quantum graphs

Asymptotics and Estimates for Spectral Minimal Partitions of Metric Graphs

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