Asymptotic behaviour of the Steklov spectrum on dumbbell domains

Bucur, Dorin; Henrot, Antoine; Michetti, Marco

doi:10.1080/03605302.2020.1840587

Cited by 5 publications

(3 citation statements)

References 24 publications

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“…For the other case, we choose Ω as the unit disc, then µ 1 (Ω)|Ω| = πj 2 11 (j 11 is the first zero of the derivative of the Bessel function J 1 ). Let ω be a set for which σ 1 (ω)P (ω) ≤ δ (for example a dumbbell shape domain with the channel very thin see [9]), using arguments similar at the ones above we conclude that…”

Section: Existence or Non-existence Of Extremal Domainsmentioning

confidence: 75%

“…So we constructed a continuous path for the value µ 1 (Ω y )|Ω y | starting from and arriving to µ 1 (D)π, we conclude because was arbitrary small. Using the same argument (and [9]) we can prove that for every x ∈ [0, 2π] there exists a simply connected domain Ω x for which σ 1 (Ω x )P (Ω x ) = x (2π is the value of P (D)σ 1 (D). ).…”

Section: Convex Case: Upper and Lower Bounds For F (H) And F (ω)mentioning

confidence: 82%

“…The aim is to prove that v n → v in L 2 (0, 1), this, by classical results (see [17]), will imply the convergence of the spectrum and in particular the convergence of the first eigenvalue. We know that up to a subsequence h n → h a. e. in [0, 1], now using the lower bound (9) we obtain an upper bound g n (x, y) ≤ C, for every n ∈ N and for every (x, y)…”

Section: Convex Case: Thin Domainsmentioning

confidence: 99%

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A comparison between Neumann and Steklov eigenvalues

Henrot,

Michetti

2021

Preprint

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This paper is devoted to a comparison between the normalized first (nontrivial) Neumann eigenvalue |Ω|µ1(Ω) for a Lipschitz open set Ω in the plane, and the normalized first (non-trivial) Steklov eigenvalue P (Ω)σ1(Ω). More precisely, we study the ratio F (Ω) := |Ω|µ1(Ω)/P (Ω)σ1(Ω). We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets Ω. Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bounds. The paper finishes with the plot of the corresponding Blaschke-Santaló diagrams (x, y) = (|Ω|µ1(Ω), P (Ω)σ1(Ω)). Contents 23 5.1. The Blaschke-Santaló diagram E 23 5.2. The Blaschke-Santaló diagram E C 25 References 28

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Section: Existence or Non-existence Of Extremal Domainsmentioning

confidence: 75%

Section: Convex Case: Upper and Lower Bounds For F (H) And F (ω)mentioning

confidence: 82%

Section: Convex Case: Thin Domainsmentioning

confidence: 99%

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A comparison between Neumann and Steklov eigenvalues

Henrot,

Michetti

2021

Preprint

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Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

2021

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We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is $$8\pi k$$ 8 π k , which is the best upper bound for the $$k^{\text {th}}$$ k th area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For $$k=1$$ k = 1 , the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

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Some recent developments on the Steklov eigenvalue problem

Colbois,

Girouard,

Gordon

et al. 2023

Rev Mat Complut

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The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments linking the Steklov eigenvalues and eigenfunctions of compact Riemannian manifolds to the geometry of the manifolds. Topics include isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the case of surfaces and then in higher dimensions), stability and instability of eigenvalues under deformations of the Riemannian metric, optimisation of eigenvalues and connections to free boundary minimal surfaces in balls, inverse problems and isospectrality, discretisation, and the geometry of eigenfunctions. We begin with background material and motivating examples for readers that are new to the subject. Throughout the tour, we frequently compare and contrast the behavior of the Steklov spectrum with that of the Laplace spectrum. We include many open problems in this rapidly expanding area.

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Asymptotic behaviour of the Steklov spectrum on dumbbell domains

Cited by 5 publications

References 24 publications

A comparison between Neumann and Steklov eigenvalues

A comparison between Neumann and Steklov eigenvalues

Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

Some recent developments on the Steklov eigenvalue problem

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