Spectre du laplacien et écrasement d'anses

Anné, Colette

doi:10.24033/asens.1533

Cited by 43 publications

(41 citation statements)

References 1 publication

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“…For convenience we assume that f (ε) = 1. If we denote by ν(A ε ) the first eigenvalue of the mixed eigenvalue problem on the annulus A ε , with Dirichlet conditions on the outer boundary and Neumann conditions on the inner boundary, then it is well known that ν(A ε ) converges to λ * as ε → 0 (see [1]). Thus, using the min-max, we will have for sufficiently small ε,…”

Section: 2mentioning

confidence: 99%

Spectrum of the Laplacian with weights

Colbois

Soufi

2018

Ann Glob Anal Geom

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Given a compact Riemannian manifold (M, g) and two positive functions ρ and σ, we are interested in the eigenvalues of the Dirichlet energy functional weighted by σ, with respect to the L 2 inner product weighted by ρ. Under some regularity conditions on ρ and σ, these eigenvalues are those of the operator −ρ −1 div(σ∇u) with Neumann conditions on the boundary if ∂M = ∅. We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved.2010 Mathematics Subject Classification. 35P15, 58J50.

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Section: 2mentioning

confidence: 99%

Spectrum of the Laplacian with weights

Colbois

Soufi

2018

Ann Glob Anal Geom

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“…Let us denote by m(M, g, λ i ) the multiplicity of the eigenvalue λ i (M, g), i.e. how many times the value of λ i (M, g) appears in the sequence (1). Let us consider a functional…”

Section: Introductionmentioning

confidence: 99%

An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane

Nadirashvili

Penskoi

2018

Geom. Funct. Anal.

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We prove an isoperimetric inequality for the second non-zero eigenvalue of the Laplace-Beltrami operator on the real projective plane. For a metric of unit area this eigenvalue is not greater than 20π. This value is attained in the limit by a sequence of metrics of area one on the projective plane. The limiting metric is singular and could be realized as a union of the projective plane and the sphere touching at a point, with standard metrics and the ratio of the areas 3 : 2. It is also proven that the multiplicity of the second non-zero eigenvalue on the projective plane is at most 6.

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“…In general, a thin domain problem consists in a partial differential equation .E " / defined in a domain " of dimension n, which has k dimensions of negligible size with respect to the other n k dimensions. The aim is then to obtain an approximation of the problem by an equation .E/ defined in a domain of dimension n k. It seems that the first modern rigorous studies of such approximations mostly date back to the late 80's: [15], [1], [2], [13], [23], … There exists an enormous quantity of papers dealing with thin domain problems of many different types. We refer to [20] for a presentation of the subject and some references.…”

Section: The Thin Domains Techniquesmentioning

confidence: 99%

How opening a hole affects the sound of a flute

Joly¹

2011

J. Spectr. Theory

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Abstract. In this paper, we consider an open tube of diameter " > 0, on the side of which a small hole of size " 2 is pierced. The resonances of this tube correspond to the eigenvalues of the Laplacian operator with homogeneous Neumann condition on the inner surface of the tube and Dirichlet one on the open parts of the tube. We show that this spectrum converges when " goes to 0 to the spectrum of an explicit one-dimensional operator. At a first order of approximation, the limit spectrum describes the note produced by a flute, for which one of its holes is open. Mathematics Subject Classification (2010). 35P15, 35Q99.

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Spectre du laplacien et écrasement d'anses

Cited by 43 publications

References 1 publication

Spectrum of the Laplacian with weights

Spectrum of the Laplacian with weights

An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane

How opening a hole affects the sound of a flute

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