The clamped plate in Gauss space

Chasman, L. Mercredi; Langford, Jeffrey J.

doi:10.1007/s10231-016-0550-2

Cited by 20 publications

(2 citation statements)

References 22 publications

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“…The argument below parallels Talenti's argument for the Euclidean clamped plate. See [30] and also [14] for the analogous argument in Gauss space (where unbounded domains are considered). At this point the reader might find it useful to review the notation and definitions of Section 2.…”

Section: Existence Of the Spectrum And Regularity Of Solutionsmentioning

confidence: 99%

On Clamped Plates with Log-Convex Density

Chasman¹,

Langford²

2018

Preprint

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We consider the analogue of Rayleigh's conjecture for the clamped plate in Euclidean space weighted by a log-convex density. We show that the lowest eigenvalue of the bi-Laplace operator with drift in a given domain is bounded below by a constant C(V, n) times the lowest eigenvalue of a centered ball of the same volume; the constant depends on the volume V of the domain and the dimension n of the ambient space. Our result is driven by a comparison theorem in the spirit of Talenti, and the constant C(V, n) is defined in terms of a minimization problem following the work of Ashbaugh and Benguria. When the density is an "anti-Gaussian," we estimate C(V, n) using a delicate analysis that involves confluent hypergeometric functions, and we illustrate numerically that C(V, n) is close to 1 for low dimensions.

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Section: Existence Of the Spectrum And Regularity Of Solutionsmentioning

confidence: 99%

On Clamped Plates with Log-Convex Density

Chasman¹,

Langford²

2018

Preprint

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“…We note that the conjecture is still open in higher dimensions; however, almost simultaneously with [AB95], Ashbaugh and Laugesen [AL96] provided an asymptotically sharp estimate, i.e., Λ 0 (Ω) ≥ w n Λ 0 (Ω ) with w n ∈ [0.89, 1) for every n ≥ 4 and lim n→∞ w n = 1. Recently, Chasman and Langford [CL16] proved a non-sharp isoperimetric inequality for clamped plates on Gaussian spaces, stating that Γ w (Ω) ≥ cΓ w (Ω ) for some c = c(Ω, n) ∈ (0, 1), where Γ w (Ω) and Γ w (Ω ) are the fundamental tones of clamped plates with respect to the Gaussian density w.…”

Section: Introductionmentioning

confidence: 99%

Lord Rayleigh’s Conjecture for Vibrating Clamped Plates in Positively Curved Spaces

Kristály

2022

Geom. Funct. Anal.

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We affirmatively solve the analogue of Lord Rayleigh’s conjecture on Riemannian manifolds with positive Ricci curvature for any clamped plates in 2 and 3 dimensions, and for sufficiently large clamped plates in dimensions beyond 3. These results complement those from the flat (Ashbaugh and Benguria in Duke Math J 78(1):1–17, 1995; Nadirashvili in Arch Ration Mech Anal 129(1):1–10, 1995) and negatively curved (Kristály in Adv Math 367:107113, 2020) cases that are valid only in 2 and 3 dimensions, and at the same time also provide the first positive answer to Lord Rayleigh’s conjecture in higher dimensions. The proofs rely on an Ashbaugh–Benguria–Nadirashvili–Talenti nodal-decomposition argument, on the Lévy–Gromov isoperimetric inequality, on fine properties of Gaussian hypergeometric functions and on sharp spectral gap estimates of fundamental tones for both small and large clamped spherical caps. Our results show that positive curvature enhances genuine differences between low- and high-dimensional settings, a tacitly accepted paradigm in the theory of vibrating clamped plates. In the limit case—when the Ricci curvature is non-negative we establish a Lord Rayleigh-type isoperimetric inequality that involves the asymptotic volume ratio of the non-compact complete Riemannian manifold; moreover, the inequality is strongly rigid in 2 and 3 dimensions, i.e., if equality holds for a given clamped plate then the manifold is isometric to the Euclidean space.

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Fundamental tones of clamped plates in nonpositively curved spaces

Kristály

2020

Advances in Mathematics

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We study Lord Rayleigh's problem for clamped plates on an arbitrary n-dimensional (n ≥ 2) Cartan-Hadamard manifold (M, g) with sectional curvature K ≤ −κ 2 for some κ ≥ 0. We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in (M, g) is universally bounded from below by (n−1) 4 16 κ 4 whenever the κ-Cartan-Hadamard conjecture holds on (M, g), e.g. in 2-and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2-and 3-dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped plates, i.e. the fundamental tone of any domain in (M, g) of volume v > 0 is not less than the corresponding fundamental tone of a geodesic ball of the same volume v in the space of constant curvature −κ 2 provided that v ≤ cn/κ n with c2 ≈ 21.031 and c3 ≈ 1.721, respectively. In particular, Rayleigh's problem in Euclidean spaces resolved by Nadirashvili (1992) and Ashbaugh and Benguria (1995) appears as a limiting case in our setting (i.e. K ≡ κ = 0). Sharp asymptotic estimates of the fundamental tone of small and large geodesic balls of low-dimensional hyperbolic spaces are also given. The sharpness of our results requires the validity of the κ-Cartan-Hadamard conjecture (i.e. sharp isoperimetric inequality on (M, g)) and peculiar properties of the Gaussian hypergeometric function, both valid only in dimensions 2 and 3; nevertheless, some nonoptimal estimates of the fundamental tone of arbitrary clamped plates are also provided in high-dimensions. As an application, by using the sharp isoperimetric inequality for small clamped hyperbolic discs, we give necessarily and sufficient conditions for the existence of a nontrivial solution to an elliptic PDE involving the biharmonic Laplace-Beltrami operator.2000 Mathematics Subject Classification. Primary: 35P15, 53C21, 35J35, 35J40.

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The clamped plate in Gauss space

Cited by 20 publications

References 22 publications

On Clamped Plates with Log-Convex Density

On Clamped Plates with Log-Convex Density

Lord Rayleigh’s Conjecture for Vibrating Clamped Plates in Positively Curved Spaces

Fundamental tones of clamped plates in nonpositively curved spaces

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