Pólya-type inequalities on spheres and hemispheres
[Inégalités de type Pólya pour les sphères et hémisphères]
Freitas, Pedro1, 2 ; Mao, Jing3 ; Salavessa, Isabel2, 4
1 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa (Portugal)
2 Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, 1749-016 Lisboa (Portugal)
3 Faculty of Mathematics and Statistics, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, Wuhan, 430062 (China)
4 Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa (Portugal)
Annales de l'Institut Fourier, Online first, 73 p.

Soit λ une valeur propre de l’opérateur de Laplace–Beltrami sur des n-sphères ou -hemisphères, de multiplicité m telle que λ=λ k =⋯=λ k+m-1 . On caractérise les ordres les plus bas et les plus élevés dans l’ensemble k,⋯,k+m-1 pour lesquels la conjecture de Pólya est vraie ou échoue. En particulier, nous montrons que la conjecture de Pólya est vraie pour les hemisphères dans le cas de Neumann, mais pas dans le cas de Dirichlet lorsque n est supérieur à deux. Nous dérivons des inégalités de type Pólya avec un terme de correction fournissant des bornes inférieure et supérieure optimales pour toutes les valeurs propres. Cela nous permet de mesurer l’écart par rapport au terme principal de l’asymptotique de Weyl pour les valeurs propres des sphères et hemisphères. Nous obtenons des résultats similaires pour des domaines qui pavent des hémisphères, et des inégalités de Li–Yau directes et inverses pour 𝕊 2 et 𝕊 4 , respectivement.

Given an eigenvalue λ of the Laplace–Beltrami operator defined on n-spheres or -hemispheres, with multiplicity m such that λ=λ k =⋯=λ k+m-1 , we characterize the lowest and highest orders in the set k,⋯,k+m-1 for which Pólya’s conjecture holds and fails. In particular, we show that Pólya’s conjecture holds for hemispheres in the Neumann case, but not in the Dirichlet case when n is greater than two. We further derive Pólya-type inequalities by adding a correction term providing sharp lower and upper bounds for all eigenvalues. This allows us to measure the deviation from the leading term in the Weyl asymptotics for eigenvalues on spheres and hemispheres. As a direct consequence, we obtain similar results for domains which tile hemispheres. We also obtain direct and reversed Li–Yau inequalities for 𝕊 2 and 𝕊 4 , respectively.

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DOI : 10.5802/aif.3657
Classification : 35P15, 35J05, 35J25, 35P20, 58J50
Keywords: Eigenvalues, Laplace operator, Spheres and hemispheres, Pólya’s inequalities.
Mot clés : Valeurs propres, laplacien, sphères et hemisphères, inégalités de type Pólya.
Freitas, Pedro 1, 2 ; Mao, Jing 3 ; Salavessa, Isabel 2, 4

1 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa (Portugal)
2 Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, 1749-016 Lisboa (Portugal)
3 Faculty of Mathematics and Statistics, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, Wuhan, 430062 (China)
4 Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa (Portugal) 
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     title = {P\'olya-type inequalities on spheres and hemispheres},
     journal = {Annales de l'Institut Fourier},
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     note = {Online first},
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Freitas, Pedro; Mao, Jing; Salavessa, Isabel. Pólya-type inequalities on spheres and hemispheres. Annales de l'Institut Fourier, Online first, 73 p.

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