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Mckean-type Estimates for the First Eigenvalue of the p-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds

Year 2023, , 358 - 366, 30.04.2023
https://doi.org/10.36890/iejg.1133383

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, , 358 - 366, 30.04.2023
https://doi.org/10.36890/iejg.1133383

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

Sakineh Hajiaghasi 0000-0002-0139-8642

Shahroud Azami 0000-0002-8976-2014

Early Pub Date April 26, 2023
Publication Date April 30, 2023
Acceptance Date March 2, 2023
Published in Issue Year 2023

Cite

APA Hajiaghasi, S., & Azami, S. (2023). Mckean-type Estimates for the First Eigenvalue of the p-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds. International Electronic Journal of Geometry, 16(1), 358-366. https://doi.org/10.36890/iejg.1133383
AMA Hajiaghasi S, Azami S. Mckean-type Estimates for the First Eigenvalue of the p-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds. Int. Electron. J. Geom. April 2023;16(1):358-366. doi:10.36890/iejg.1133383
Chicago Hajiaghasi, Sakineh, and Shahroud Azami. “Mckean-Type Estimates for the First Eigenvalue of the P-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds”. International Electronic Journal of Geometry 16, no. 1 (April 2023): 358-66. https://doi.org/10.36890/iejg.1133383.
EndNote Hajiaghasi S, Azami S (April 1, 2023) Mckean-type Estimates for the First Eigenvalue of the p-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds. International Electronic Journal of Geometry 16 1 358–366.
IEEE S. Hajiaghasi and S. Azami, “Mckean-type Estimates for the First Eigenvalue of the p-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds”, Int. Electron. J. Geom., vol. 16, no. 1, pp. 358–366, 2023, doi: 10.36890/iejg.1133383.
ISNAD Hajiaghasi, Sakineh - Azami, Shahroud. “Mckean-Type Estimates for the First Eigenvalue of the P-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds”. International Electronic Journal of Geometry 16/1 (April 2023), 358-366. https://doi.org/10.36890/iejg.1133383.
JAMA Hajiaghasi S, Azami S. Mckean-type Estimates for the First Eigenvalue of the p-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds. Int. Electron. J. Geom. 2023;16:358–366.
MLA Hajiaghasi, Sakineh and Shahroud Azami. “Mckean-Type Estimates for the First Eigenvalue of the P-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds”. International Electronic Journal of Geometry, vol. 16, no. 1, 2023, pp. 358-66, doi:10.36890/iejg.1133383.
Vancouver Hajiaghasi S, Azami S. Mckean-type Estimates for the First Eigenvalue of the p-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds. Int. Electron. J. Geom. 2023;16(1):358-66.