Krahn’s proof of the Rayleigh conjecture revisited

Daners, Daniel

doi:10.1007/s00013-010-0218-x

Cited by 8 publications

(6 citation statements)

References 28 publications

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“…Now, concerning the proof of our results, the maximization of the eigenvalue λ among Lipschitz sets is performed introducing a weighted isoperimetric problem which involves modified Bessel functions. It could be of some interest to mention that inequality λ D (Ω) ≥ λ D (Ω ♯ ) was conjectured by Lord Rayleigh in 1877 and, as described in [18], he provided a proof in the case of nearly circular sets (in the plane) using perturbation series involving Bessel functions.…”

Section: Conjecturementioning

confidence: 99%

“…, where ν is the unit outer normal to ∂Ω. Then, to go from (20) to (18) we just use the explicit representation of u in terms of modified Bessel functions together with the explicit representation of the boundary of Ω in terms of v(ξ) according to Definition 1.3.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

“…Isoperimetric inequalities for eigenvalues of elliptic operators (Laplacian above all) is an active field of research [30,32]. Without presuming to give an exhaustive picture on the state-of-the-art, we consider just four remarkable examples (see for instance [5,7,10,12,16,17,18,22,30,31,32,33,41,45,46] for more details): (Faber-Krahn) λ D (Ω) = min…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue

Ferone¹,

Nitsch²,

Trombetti³

2014

CPAA

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For a given bounded Lipschitz set Ω, we consider a Steklov-type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding H 1 (Ω) ֒→ L 2 (∂Ω). We prove that a conjectured reverse Faber-Krahn inequality holds true at least in the class of Lipschitz sets which are "close" to a ball in a Hausdorff metric sense. The result implies that among sets of prescribed measure, balls are local minimizers of the embedding constant.2010 Mathematics Subject Classification. 46E35, 35P15, 28A75.

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Section: Conjecturementioning

confidence: 99%

Section: Proof Of Theorem 14mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue

Ferone¹,

Nitsch²,

Trombetti³

2014

CPAA

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“…Lord Rayleigh conjectured in 1877 that the first eigenvalue is minimal on the disc, among all other planar domains of the same area. The proof was given in 1923 by Faber in two dimensions and three years later extended by Krahn in any dimension of the Euclidean space (see [19] for a description of the history of the problem and [33,32] for a survey of the topic).…”

Section: Introductionmentioning

confidence: 99%

On the polygonal Faber-Krahn inequality

Bogosel,

Bucur

2023

Journal De L’École Polytechnique — Mathématiques

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It has been conjectured by Pólya and Szegö seventy years ago that the planar set which minimizes the first eigenvalue of the Dirichlet-Laplace operator among polygons with n sides and fixed area is the regular polygon. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this paper we prove that for each n ⩾ 5 the proof of the conjecture can be reduced to a finite number of certified numerical computations. Moreover, the local minimality of the regular polygon can be reduced to a single numerical computation. For n = 5, 6, 7, 8 we perform this computation and certify the numerical approximation by finite elements, up to machine errors.Résumé (Sur l'inégalité Faber-Krahn polygonale). -Il y a soixante-dix ans, Pólya et Szegö ont conjecturé que l'ensemble du plan qui minimise la première valeur propre du laplacien avec conditions de Dirichlet au bord parmi les polygones de n côtés et aire fixée est le polygone régulier. Malgré sa simplicité apparente, cette conjecture a été démontrée seulement pour les triangles et les quadrilatères. Dans cet article, nous démontrons que pour chaque n ⩾ 5 la preuve de la conjecture peut être réduite à un nombre fini de calculs numériques certifiés. En particulier, la minimalité locale du polygone régulier est réduite à un seul calcul certifié. Pour n = 5, 6, 7, 8 nous faisons ce calcul et nous certifions l'approximation par éléments finis, aux erreurs d'arrondi près.

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“…Lord Rayleigh conjectured in 1877 that the first eigenvalue is minimal on the disc, among all other planar domains of the same area. The proof was given in 1923 by Faber in two dimensions and three years later extended by Krahn in any dimension of the Euclidean space (see [14] for a description of the history of the problem and [25,24] for a survey of the topic).…”

Section: Introductionmentioning

confidence: 99%