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Annals of Global Analysis and Geometry

, Volume 55, Issue 4, pp 805–817 | Cite as

Generalization of Philippin’s results for the first Robin eigenvalue and estimates for eigenvalues of the bi-drifting Laplacian

  • Abdolhakim ShoumanEmail author
Article
  • 76 Downloads

Abstract

In the present paper, we first consider the weighted eigenvalue problem \(\Delta _f u=\lambda _{f}u\) in M with the Robin boundary condition \(\frac{\partial u}{\partial \nu }+\beta u=0\) on \(\partial M\), where \((M^n,g,e^{-f})\) is a compact n-dimensional weighted Riemannian manifold of nonnegative Bakry–Émery Ricci curvature. We derive under some convexity condition of the boundary \(\partial M\), an explicit lower bound of the first weighted Robin eigenvalue \(\lambda _{1,f}(\beta )\) depending only on the geometry of M and the constant \(\beta \) appearing in the boundary condition. Another new estimate for \(\lambda _{1,f}(\beta )\) with respect to the first nonzero Neumann eigenvalue \(\mu _{2,f}\) of the weighted Laplacian \(\Delta _f\) is also obtained. Furthermore, we provide some lower bounds for the first buckling and clamped plate eigenvalues of the bi-drifting Laplacian on weighted manifolds.

Keywords

Drifting Laplacian Bakry–Émery curvature Lower bounds Robin boundary condition Neumann problem Riemannian manifolds Convex boundary Reilly’s formula Bi-drifting Laplacian 

Mathematics Subject Classification

35P15 35J05 35J25 53C21 58C40 

Notes

Acknowledgements

The author would like to thank Mr Saïd Ilias for helpful discussions and useful remarks. The author would also like to thank the anonymous referee for the careful reading and the valuable comments on the first version of the submitted manuscript.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Physique Théorique, CNRS-UMR 7350, Fédération Denis Poisson, FR-CNRS 2964Université de ToursToursFrance

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