Lower order eigenvalues for the bi-drifting Laplacian on the Gaussian shrinking soliton
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It may very well be difficult to prove an eigenvalue inequality of Payne-P\'{o}lya-Weinberger type for the bi-drifting Laplacian on the bounded domain of the general complete metric measure spaces. Even though we suppose that the differential operator is bi-harmonic on the standard Euclidean sphere, this problem still remains open. However, under certain condition, a general inequality for the eigenvalues of bi-drifting Laplacian is established in this paper, which enables us to prove an eigenvalue inequality of Ashbaugh-Cheng-Ichikawa-Mametsuka type (which is also called an eigenvalue inequality of Payne-P\'{o}lya-Weinberger type) for the eigenvalues with lower order of bi-drifting Laplacian on the Gaussian shrinking soliton.
Abstract
It may very well be difficult to prove an eigenvalue inequality of Payne-P\'{o}lya-Weinberger type for the bi-drifting Laplacian on the bounded domain of the general complete metric measure spaces. Even though we suppose that the differential operator is bi-harmonic on the standard Euclidean sphere, this problem still remains open. However, under certain condition, a general inequality for the eigenvalues of bi-drifting Laplacian is established in this paper, which enables us to prove an eigenvalue inequality of Ashbaugh-Cheng-Ichikawa-Mametsuka type (which is also called an eigenvalue inequality of Payne-P\'{o}lya-Weinberger type) for the eigenvalues with lower order of bi-drifting Laplacian on the Gaussian shrinking soliton.
참고문헌 (26)
- M. S. Ashbaugh, "Isoperimetric and universal inequalities for eigenvalues, in Spec-tral theory and geometry", London Math. Soc. Lecture Note Ser, 273, pp.95~139 (1999), Cambridge Univ. Press
- M. S. Ashbaugh, "More bounds on eigenvalue ratios for Dirichlet Laplacians in n dimensions", SIAM J. Math. Anal, 24(6), pp.1622~1651 (1993)[DOI]
- D. Chen, "Extrinsic estimates for eigenvalues of the Laplace operator", J. Math. Soc. Japan, 60(2), pp.325~339 (2008)
- Z. C. Chen, "Estimates for discrete spectrum of Laplacian operator with any order", J. China Univ. Sci. Tech, 20(3), pp.259~266 (1990)
- D. Chen, "Bounds for ratios of the membrane eigenvalues", J. Dierential Equations, 250(3), pp.1575~1590 (2011)[DOI]
- Q. -M. Cheng, "Estimates for lower order eigenvalues of a clamped plate problem", Cal. Var. & PDEs, 38(3-4), pp.409~416 (2010)
- Q. -M. Cheng, "Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere", Cal. Var. & PDEs, 36(4), pp.507~523 (2009)
- Q. -M. Cheng, "Inequalities for eigenvalues of Laplacian with any order", Commun. Contemp. Math, 11(4), pp.639~655 (2009)[DOI]
- Q. -M. Cheng, "Estimates for eigenvalues of a clamped plate problem on Riemannian manifolds", J. Math. Soc. Japan, 62(2), pp.673~686 (2010)
- Q. -M. Cheng, "Inequalities for eigenvalues of a clamped plate problem", Trans. Amer. Math. Soc, 358(6), pp.2625~2635 (2006)[DOI]
- T. H. Colding, "Generic mean curvature ow I; generic singular-ities", Ann. Math, 175(2), pp.755~833 (2012)
- F. Du, "Eigenvalue inequalities for the buckling problem of the drifting Laplacian on Ricci solitons", J. Dierential Equations, 260(7), pp.5533~5564 (2016)[DOI]
- F. Du, "Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons", Ann. Global Anal. Geom, 48(3), pp.255~268 (2015)[DOI]
- F. Du, "Estimates for eigenvalues of the bi-drifting Laplacian operator", Z. Angew. Math. Phys, 66(3), pp.703~726 (2015)[DOI]
- R. S. Hamilton, "The Ricci ow on surfaces, in Mathematics and general relativity", Contemp. Math, 71, pp.237~262 (1988)[DOI]
- S. M. Hook, "Domain-independent upper bounds for eigenvalues of elliptic operators", Trans. Amer. Math. Soc, 318(2), pp.615~642 (1990)[DOI]
- G. Huisken, "Asymptotic behavior for singularities of the mean curvature ow", J. Dif-ferential Geom, 31(1), pp.285~299 (1990)
- X. Li, "Estimates for the eigenvalues of the bi-drifting Laplacian on cigar soliton", J. Geom. Phys, 145, pp.9~ (2019)[DOI]
- L. E. Payne, "On the ratio of consecutive eigenvalues", J. Math. and Phys, 35, pp.289~298 (1956)[DOI]
- H. Sun, "Lower order eigenvalues of Dirichlet Laplacian", Manuscripta Math, 125(2), pp.139~156 (2008)[DOI]
- Q. Wang, "Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds", J. Funct. Anal, 245(1), pp.334~352 (2007)[DOI]
- Q. Wang, "Inequlaities for eigenvalues of a clamped problem", Cal. Var. & PDEs, 40(1-2), pp.273~289 (2011)
- C. Xia, "Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds", Ann. Global Anal. Geom, 45(3), pp.155~166 (2014)[DOI]
- H.-C. Yang, "An estimate of the dierence between consecutive eigenvalues", Preprint IC, 60 of ICTP, Trieste, pp.1991, ~
- L. Zeng, "Eigenvalues of the drifting Laplacian on complete noncompact Riemannian manifolds", Nonlinear Anal, 141, pp.1~15 (2016)[DOI]
- L. Zeng, "Estimates for the eigenvalues of the drifting Laplacian on some complete Ricci solitons", Kyushu J. Math, 72(1), pp.143~156 (2018)[DOI]