New eigenvalue estimates involving Bessel functions

• El Chami, Fida ^[1] ; Habib, Georges ^[1] ; Ginoux, Nicolas ^[2]
  1. [1] Lebanese University

     Lebanese University

     Líbano

     
  2. [2] Universit´e de Lorraine
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 65, Nº 2, 2021, págs. 681-726
• Idioma: inglés
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• Resumen

  • Given a compact Riemannian manifold (Mn, g) with boundary ∂M, we give an estimate for the quotient R ∂M f dµg R M f dµg , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [39], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a new inequality is given for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Indepen[1]dently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.