New eigenvalue estimates involving Bessel functions

Abstract

Given a compact Riemannian manifold (M n , g) with boundary ∂M , we give an estimate for the quotient ∂M f dµ g M f dµ g, 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 direct proof is given of the Faber-Krahn inequalities 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. Independently, 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.

Eigenvalue Estimates on Weighted Manifolds

“…Now, take any p -eigenform of and consider the -form for each . By the Rayleigh min-max principle, we have that In the following, we will adapt some computations done in [ 16 ] to the context of the drifting Hodge Laplacian (see also [ 1 ], [ 8 , Thm. 5.8] for a similar computation).…”

Eigenvalue Curves for Generalized MIT Bag Models

“…More generally, the quest of geometrical upper and lower bounds for the spectral gap is a trending topic of research [5,24,30,56]. This quest is also addressed in the differential geometry literature for Dirac operators on spin manifolds, where sharp inequalities for spectral gaps in terms of geometric quantities are shown [2,4,12,13,35,45,46,47,53]. Despite the amount of works available on this topic, for the case of bounded domains in euclidean spaces the problem of minimizing the spectral gap under a volume constraint (and with no further restrictions on the geometry of the boundary) remains open.…”

Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones

“…In [39], related bounds on ˇ.M/ in the spirit of the Hersh inequality are proved for compact Riemannian manifolds in any dimension. For positive Robin parameters, ˇ> 0, a spectral isoperimetric inequality for ˇ.M/ in any dimension has been proved very recently in [15] under certain constraints on the curvatures of both the manifold M and its boundary @M. Due to the close connection between the Robin Laplacian and the Dirichlet-to-Neumann map, we are able to obtain as a consequence an analogue of (1.2) for the lowest eigenvalue of the Dirichlet-to-Neumann map, which is stated in Proposition 3.12. Second, we return back to the Euclidean setting and treat spectral optimization for the Robin Laplacian on a special class of unbounded three-dimensional domains.…”