Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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THE LOWER BOUNDS FOR THE FIRST EIGENVALUES OF THE (p, q)-LAPLACIAN ON FINSLER MANIFOLDS

  • Sakineh Hajiaghasi (Department of Mathematics, Imam Khomeini International University) ;
  • Shahroud Azami (Department of Mathematics, Imam Khomeini International University)
  • Received : 2022.05.20
  • Accepted : 2022.12.16
  • Published : 2023.03.25
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Abstract

In this paper, we study the nonlinear eigenvalue problem for some of the (p, q)-Laplacian on compact Finsler manifolds with zero boundary condition, and estimate the lower bound of the first eigenvalues for (p, q)-Laplace operators on Finsler manifolds.

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References

  1. A. Abolarinwa, The first eigenvalue of p-Laplacian and geometric estimates, Nonl. Analysis and Differential Equations 2 (2014), no. 3, 105-115. https://doi.org/10.12988/nade.2014.455
  2. A. Abolarinwa and S. Azami, Comparison estimates on the first eigenvalue of a quasilinear elliptic system, Journal of Applied Analysis (2020), 273-285.
  3. G. A. Afrouzi, M. Mirzapour, and Q. Zhang, Simplicity and stability of the first eigenvalue of a (p, q) Laplacian system, Electr. J. Diff Eqn. 8 (2012), 1-6.
  4. H. Akbar-Zadeh, Sur les espaces de Finsler a courbures sectionnelles constantes, Acad. Roy. Belg. Bull. Cl. Sci. 74 (1988), no. 5, 281-322.
  5. S. Azami, The first eigenvalue of some (p, q)-Laplacian and geometric estimates, Commun. Korean Math. Soc. 33 (2018), no. 1, 317-323.
  6. M. Belloni, V. Ferone, and B. Kawohl,Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, J. Appl. Math. Phys. 54 (2003), 771-783.
  7. M. Belloni, B. Kawohl, and P. Juutinen, The p-Laplace eigenvalue problem as p → ∞ in a Finsler metric, J. Europ. Math. Soc. 8 (2006), 123-138. https://doi.org/10.4171/jems/40
  8. N. Benouhiba and Z. Belyacine, A class of eigenvalue problems for the (p, q)-Laplacian in RN , Int. J. Pure Appl. Math. 80 (2012), no. 5, 727-737.
  9. F. Bonder and J. P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems, Part II, J. Diff. Eq. 245 (2008), 875-891. https://doi.org/10.1016/j.jde.2008.04.019
  10. B. Chen, Some geometric and analysis problems in Finsler geometry, Report of Postdoctoral Research, Zhejiang Univ., 2010.
  11. S. Y. Cheng, Eigenfunctions and eigenvalues of Laplacian in differential geometry, Proc. Sympos. Pure Math., Vol. XXVII, Stanford, Calif., Part 2, 185-193, Amer. Math. Soc., Providence, RI, 1973.
  12. Q. M. Cheng and H. Yang, Estimates on eigenvalues of Laplacian, Math. Ann. 331 (2005), no. 2, 445-460. https://doi.org/10.1007/s00208-004-0589-z
  13. Y. X. Ge and Z. M. Shen, Eigenvalues and eigenfunctions of metric measure manifolds, Proc. London Math. Soc. 82 (2001), 725-746. https://doi.org/10.1112/plms/82.3.725
  14. G. Y. Huang and Z. Li, Upper bounds on the first eigenvalue for the P-Laplacian, preprint (2016); arXiv:1602.03610v2.
  15. A. E. Khalil, S. E. Manouni, and M. Ouanan, Simplicity and stability of the first eigenvalue of a nonlinear elliptic system, Int. J. Math. and Math. Sci. 10 (2005), 1555-1563. https://doi.org/10.1155/IJMMS.2005.1555
  16. A. M. Matei, First eigenvalue for the p-Laplace operator, Nonlinear Anal. 39 (2000), no. 8, 1051-1068. https://doi.org/10.1016/S0362-546X(98)00266-1
  17. P. L. De N'apoli and J. P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems, J. Diff. Eq. 227 (2006), 102-115. https://doi.org/10.1016/j.jde.2006.01.004
  18. Z. M. Shen, Lectures on Finsler geometry, World Sci., Singapore, 2001.
  19. Z. M. Shen, The nonlinear Laplacian for Finsler manifolds, The theory of Finslerian Laplacians and applications (edited by P. Antonelli), Proc. Conf. On Finsler Laplacian, Kluwer Acad. Press, Netherlads, 1998.
  20. H. Takeuchi, On the first eigenvalue of the p-Laplacian in a Riemannian manifold, Tokyo J. Math. 21 (1998), no. 1, 135-140. https://doi.org/10.3836/tjm/1270041991
  21. B. Y. Wu and Y. L. Xin, Comparison theorems in Finsler geometry and their applications, Math. Ann, 337 (2007), 177-196.
  22. S. T. Yin and Q. He, Some comparison theorems and their applications in Finsler geometry, J. Ineq. Appl. 2014 (2014), Article Number 107.
  23. S.T. Yin and Q. He, Eigenvalue comparison theorems on Finsler manifolds, Chin. Ann. Math. 36B (2015), no. 1, 31-44.
  24. S. T. Yin and Q. He, The first eigenvalue of Finsler p-Laplacian, Diff. Geom. Appl. 35 (2014), 30-49. https://doi.org/10.1016/j.difgeo.2014.04.009
  25. S. T. Yin and Q. He, The first eigenfunctions and eigenvalue of the p-Laplacian on Finsler manifolds, Sci. China Math. 59 (2016), 1769-1794. https://doi.org/10.1007/s11425-015-0411-9
  26. S. T. Yin and Q. He, Some eigenvalue comparison theorems of Finsler p-Laplacian, International Journal of Mathematics 29 (2018), no. 6, 1850044.
  27. S. T. Yin, Q. He, and Y. B. Shen, On lower bounds of the first eigenvalue of Finsler-Laplacian, Publ. Math. Debrecen 83 (2013), 385-405. https://doi.org/10.5486/PMD.2013.5532