New eigenvalue estimates involving Bessel functions

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Fida El Chami
Lebanese University
Georges Habib
Lebanese University
Georges Habib
Lebanese University
Nicolas Ginoux
Universit´e de Lorraine

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.

Keywords
Bessel functions, eigenvalues, Dirac operator, Yamabe operator, Robin Laplacian

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How to Cite
El Chami, Fida et al. “New eigenvalue estimates involving Bessel functions”. Publicacions Matemàtiques, vol.VOL 65, no. 2, pp. 681-26, https://raco.cat/index.php/PublicacionsMatematiques/article/view/390245.
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