FUNDAMENTAL TONE ESTIMATES ON FINSLER MANIFOLDS
Abstract
We study the fundamental tone of Laplacian operators on Finsler manifold $M$ evolved by a special function $u:\Omega\subset M \rightarrow \mathbb{R}$, and we give some geometric estimates of the first eigenvalue of p-laplace and (p,q)-Laplace operators depend on this function for simply connected manifolds, a class of warped product manifolds, and a class of Finsler submersions. Under a similar setting, we also study these results on a quasi-linear operator $Lu=-\Delta_{p}u+X\vert u\vert^{p-2}u$.
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S. Azami: The first eigenvalue of some (p, q)-Laplacian and geometric estimates. Commun. Korean Math. Soc. 33, No. 1, pp. 317-323 (2018).
S. Azami: Geometric estimates of the first eigenvalue of a quasilinear operator. Mathematical reports 23 (1-2), 107-121, 2021.
J. Cheeger: A lower bound for the smallest eigenvalue of the Laplacian. Princeton Univ. Press. Princeton, NJ, 195-199 (1970).
S. Y. Cheng: Eigenfunctions and eigenvalues of Laplacian in Differential geometry. Amer. Math. Soc, pp. 185-193 (1973).
S. Y. Cheng: Eigenvalue comparison theorems and its geometric applications. Math. Z. 143 (3): 289-297 (1975).
D. Chen, Q. Cheng, Q. Wang and C. Xia: On eigenvalues of a system of elliptic equations and biharmonic operator. J. Math. Anal. Appl. 387, 1146-1159 (2012).
Q.- M. Cheng and H. Yang: Estimates on eigenvalues of Laplacian. Math. Ann. no. 2, pp. 445-460 (2005).
D. Egloff: Uniform Finsler Hadamard manifolds. Annal. H. P., section A, 66, 323-357 (1997).
G. Francisco and P. Marcos: On the fundamental tone of the p-Laplacian on Riemannian manifolds and applications. J. Math. Anal. Appl. 506 (2022).
M. J. Habibi Vosta Kolaei and S. Azami: Lichnerowics-type estimates for the first eigenvalue of biharmonic operator. Complex Variable and Elliptic Equations, 67 (10), 2352-2359 (2022).
S. Hajiaghasi and S. Azami: The lower bounds for the first eigenvalues of the (p, q)-laplacian on Finsler manifolds. Honam Mathematical Journal, 45(1) (2023), 82-91.
Y. X. Huang: Existence of positive solutions for a class of the p-Laplace equations. J. Aust. Math. Soc. Ser. B 36 (2) 249-264 (1994).
A. Naber and D. Valtorta: Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound. Math Z. 227. pp. 867-891 (2014).
S. Ohta: Finsler interpolation inequalities. Calc. Var. Partial Differential Equation 36, 211-249 (2009).
H. Shah and E. H. Taha: Busemann functions in asymptotically harmonic Finsler manifolds. arXiv:2108.08466v1 [math.DG] 19 Aug 2021.
Z. Shen: Lectures on Finsler geometry. World Scientific publishing Co., Singapore, 2001.
Y-B. Shen and Z. Shen: Introduction to modern Finsler geometry. World Scientific Publishing Co., 2016.
K. Shiohama: Riemannian and Finsler geometry in the large. Recent Advances in Mathematics, RMS-Lecture Note Series 21, 163-179 (2015).
B-Y. Wu: Comparison theorems in Riemann-Finsler geometry with line radial integral curvature bounds and related results. J. Korean Math. Soc. 56, No. 2, pp: 421-437 (2019).
B. Y. Wu and Y. L. Xin: Comparison theorem in Finsler geometry and their applications. Math. Ann. 337, no.1, 177-196 (2007).
S. T. Yin and Q. He: The first eigenvalue of Finsler p-Laplacian. Differential Geometry and its Applications, 35 pp. 30-49 (2014).
S. T. Yin and Q. He: Some eigenvalue comparison theorems of Finsler p-Laplacian. International Journal of Mathematics, 29 (3) (2018).
L. Zhang and Y. Zhao: The lower bounds of the first eigenvalues for the biharmonic operator on manifolds. Journal of Inequalities and Applications, 5 (2016).
DOI: https://doi.org/10.22190/FUMI231008006H
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