Research Article
Mckean-type Estimates for the First Eigenvalue of the p-Laplacian and (p,q)-Laplacian Operators on Finsler Manifolds
Abstract
In this paper, we use Hessian comparison and volume comparison theorems to investigate the Mckean-type estimate theorem for the first eigenvalue of p-Laplacian and (p,q)-Laplacian operators on Finsler manifolds.
References
- [1] S. Azami, The first eigenvalue of some $(p,q)$-Laplacian and geometric estimates, Commun. Korean Math. Soc. 33, no. 1, 317-323, 2018.
- [2] N. Benouhiba, Z. Belyacine, A class of eigenvalue problems for the $(p,q)$-Laplacian in $\mathbb{R}^{N}$, Int. J. Pure Appl. Math. 80, no. 5, 727-737, 2012.
- [3] Q. Ding, A new Laplacian comparison theorem and the estimate of eigenvalues, Chin. Ann. Math. 15B (1) 35-42, 1994.
- [4] A. Lichnerowicz, Geometric des groups de transformations, Truvaux et Recherches Mathemtiques, vol. III, Dunod, Paris, 1985. [5] H. P. Mckean, An upper bound for spectrum of $\Delta$ on a manifold of negative curvature, J. Differ. Geom. 4, 359-366, 1970.
- [6] S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differ. Equ. 36, 211-249, 2009.
- [7] B. Y. Wu, Volume form and its applications in Finsler geometry, Publ. Math. Debrecen 78, no. 3-4, 723-741, 2011. [8] B. Y. Wu, Comparison theorems and submanifolds in Finsler geometry, Sience Press Beijing, 2015. [9] B. Y. Wu, Comparison theorems in Riemann-Finsler geometry with line radial integral curvature bounds and related results, J. Korean Math. Soc. 56, 2, 421-437, 2019.
- [10] B. Y. Wu, Global Finsler geometry (Chinese), Tongji University Press, Shanghai, 2008. [11] B. Wu, Y. L. Xin, Comparison theorems in Finsler geometry and their applications, Math. Ann., 337, 177-196, 2007.
- [12] S. T. Yin, Q. He,The first eigenvalue of Finsler $p$-Laplacian,Diff. Geom. Appl., 35, 30-49, 2014.
- [13] S. T. Yin, Q. He, The first eigenfunctions and eigenvalue of the $p$-Laplacian on Finsler manifolds, Sci. China Math, 59, 1769-1794, 2016.
- [14] S. T. Yin, Q. He, Some eigenvalue comparison theorems of Finsler $p$-Laplacian, Int. J. Math. 2018.
- [15] S. T. Yin, Q. He, Y. B. Shen, On lower bounds of the first eigenvalue of Finsler-Laplacian, Publ. Math. Debrecen, 83, 385-405, 2013. [16] S. T. Yin, Q. He, Y. B. Shen, On the first eigenvalue of Finsler-Laplacian in a Finsler manifold with nonnegative weighted Ricci curvature, Science in China Math., 56, 2013.
Year 2023,
Volume: 16 Issue: 1, 358 - 366, 30.04.2023
Abstract
References
- [1] S. Azami, The first eigenvalue of some $(p,q)$-Laplacian and geometric estimates, Commun. Korean Math. Soc. 33, no. 1, 317-323, 2018.
- [2] N. Benouhiba, Z. Belyacine, A class of eigenvalue problems for the $(p,q)$-Laplacian in $\mathbb{R}^{N}$, Int. J. Pure Appl. Math. 80, no. 5, 727-737, 2012.
- [3] Q. Ding, A new Laplacian comparison theorem and the estimate of eigenvalues, Chin. Ann. Math. 15B (1) 35-42, 1994.
- [4] A. Lichnerowicz, Geometric des groups de transformations, Truvaux et Recherches Mathemtiques, vol. III, Dunod, Paris, 1985. [5] H. P. Mckean, An upper bound for spectrum of $\Delta$ on a manifold of negative curvature, J. Differ. Geom. 4, 359-366, 1970.
- [6] S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differ. Equ. 36, 211-249, 2009.
- [7] B. Y. Wu, Volume form and its applications in Finsler geometry, Publ. Math. Debrecen 78, no. 3-4, 723-741, 2011. [8] B. Y. Wu, Comparison theorems and submanifolds in Finsler geometry, Sience Press Beijing, 2015. [9] B. Y. Wu, Comparison theorems in Riemann-Finsler geometry with line radial integral curvature bounds and related results, J. Korean Math. Soc. 56, 2, 421-437, 2019.
- [10] B. Y. Wu, Global Finsler geometry (Chinese), Tongji University Press, Shanghai, 2008. [11] B. Wu, Y. L. Xin, Comparison theorems in Finsler geometry and their applications, Math. Ann., 337, 177-196, 2007.
- [12] S. T. Yin, Q. He,The first eigenvalue of Finsler $p$-Laplacian,Diff. Geom. Appl., 35, 30-49, 2014.
- [13] S. T. Yin, Q. He, The first eigenfunctions and eigenvalue of the $p$-Laplacian on Finsler manifolds, Sci. China Math, 59, 1769-1794, 2016.
- [14] S. T. Yin, Q. He, Some eigenvalue comparison theorems of Finsler $p$-Laplacian, Int. J. Math. 2018.
- [15] S. T. Yin, Q. He, Y. B. Shen, On lower bounds of the first eigenvalue of Finsler-Laplacian, Publ. Math. Debrecen, 83, 385-405, 2013. [16] S. T. Yin, Q. He, Y. B. Shen, On the first eigenvalue of Finsler-Laplacian in a Finsler manifold with nonnegative weighted Ricci curvature, Science in China Math., 56, 2013.
There are 11 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Early Pub Date | April 26, 2023 |
| Publication Date | April 30, 2023 |
| Acceptance Date | March 2, 2023 |
| Published in Issue | Year 2023 Volume: 16 Issue: 1 |
