Higher order Cheeger inequalities for Steklov eigenvalues

Hassannezhad, Asma; Miclo, Laurent

doi:10.24033/asens.2417

Cited by 22 publications

(36 citation statements)

References 22 publications

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“…One could also obtain similar lower bounds for σ k from the higher order Cheeger inequality for λ k [22,10], which should be compared with the higher order Cheeger type inequality proved in [14]. The lower bounds for σ k given in [17,14] depends on the global geometry of the manifold and not only the geometry of the manifold near the boundary.…”

Section: Introductionmentioning

confidence: 75%

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Colbois

Girouard

Hassannezhad

2020

Journal of Functional Analysis

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Given two compact Riemannian manifolds M 1 and M 2 such that their respective boundaries Σ 1 and Σ 2 admit neighbourhoods Ω 1 and Ω 2 which are isometric, we prove the existence of a constant C such that |σ k (M 1 ) − σ k (M 2 )| ≤ C for each k ∈ N. The constant C depends only on the geometry of Ω 1 ∼ = Ω 2 . This follows from a quantitative relationship between the Steklov eigenvalues σ k of a compact Riemannian manifold M and the eigenvalues λ k of the Laplacian on its boundary. Our main result states that the difference |σ k − √ λ k | is bounded above by a constant which depends on the geometry of M only in a neighbourhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant C is given explicitly in terms of bounds on the geometry of Ω 1 ∼ = Ω 2 .1991 Mathematics Subject Classification. 35P15 (primary), 58C40, 35P20 (secondary).1 The notation O(k −∞ ) designates a quantity which tends to zero faster than any power of k.

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Section: Introductionmentioning

confidence: 75%

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Colbois

Girouard

Hassannezhad

2020

Journal of Functional Analysis

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“…In [9], the author finds a lower bound for the first non trivial Steklov eigenvalue. Lower bounds for higher Steklov eigenvalues are given in [7].…”

Section: Definitionmentioning

confidence: 99%

Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs

Tschanz

2021

Ann Glob Anal Geom

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We study the Steklov problem on a subgraph with boundary $$(\Omega ,B)$$ ( Ω , B ) of a polynomial growth Cayley graph $$\Gamma$$ Γ . For $$(\Omega _l, B_l)_{l=1}^\infty$$ ( Ω l , B l ) l = 1 ∞ a sequence of subgraphs of $$\Gamma$$ Γ such that $$|\Omega _l| \longrightarrow \infty$$ | Ω l | ⟶ ∞ , we prove that for each $$k \in {\mathbb {N}}$$ k ∈ N , the kth eigenvalue tends to 0 proportionally to $$1/|B|^{\frac{1}{d-1}}$$ 1 / | B | 1 d - 1 , where d represents the growth rate of $$\Gamma$$ Γ . The method consists in associating a manifold M to $$\Gamma$$ Γ and a bounded domain $$N \subset M$$ N ⊂ M to a subgraph $$(\Omega , B)$$ ( Ω , B ) of $$\Gamma$$ Γ . We find upper bounds for the Steklov spectrum of N and transfer these bounds to $$(\Omega , B)$$ ( Ω , B ) by discretizing N and using comparison theorems.

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“…If we are to interpret the Neumann problem as finding the frequencies and modes of vibrations of a free boundary membrane, this means that the Steklov problem represents the frequencies and modes of a membrane whose mass is concentrated at the boundary. The reader should also refer to the work of Hassannezhad-Miclos [25,Section 4], where a similar construction is used to prove a Cheeger-type inequality for Steklov eigenvalues of a compact Riemannian manifold with boundary. Our primary goal in this paper is to establish a link in the reverse direction, by realizing the Neumann problem as a limit of appropriate Steklov problems.…”

Section: From Steklov To Neumann : Heuristicsmentioning

confidence: 99%

“…We thank both referees for a careful reading and many comments which greatly helped in improving the paper. In particular, we thank one of them for comments that led to Lemma 16 , and the other for pointing out reference [ 25 ]. A.G. acknowledges support from NSERC.…”

Section: Acknowledgementsmentioning

confidence: 99%

From Steklov to Neumann via homogenisation

Girouard

Henrot

Lagacé

2020

Arch Rational Mech Anal

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We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann eigenvalues from known isoperimetric bounds for Steklov eigenvalues. The interpolation also leads to the construction of planar domains with first perimeter-normalized Stekov eigenvalue that is larger than any previously known example. The proofs are based on a modification of the energy method. It requires quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.

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Higher order Cheeger inequalities for Steklov eigenvalues

Cited by 22 publications

References 22 publications

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs

From Steklov to Neumann via homogenisation

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