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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.
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참고문헌 (26)
참고문헌 신청
M. S. Ashbaugh / 1999 / Isoperimetric and universal inequalities for eigenvalues, in Spec-tral theory and geometry / London Math. Soc. Lecture Note Ser 273:95~139
M. S. Ashbaugh / 1993 / More bounds on eigenvalue ratios for Dirichlet Laplacians in n dimensions / SIAM J. Math. Anal 24(6):1622~1651
D. Chen / 2008 / Extrinsic estimates for eigenvalues of the Laplace operator / J. Math. Soc. Japan 60(2):325~339
Z. C. Chen / 1990 / Estimates for discrete spectrum of Laplacian operator with any order / J. China Univ. Sci. Tech 20(3):259~266
D. Chen / 2011 / Bounds for ratios of the membrane eigenvalues / J. Dierential Equations 250(3):1575~1590
M. S. Ashbaugh / 1993 / More bounds on eigenvalue ratios for Dirichlet Laplacians in n dimensions / SIAM J. Math. Anal 24(6):1622~1651
D. Chen / 2008 / Extrinsic estimates for eigenvalues of the Laplace operator / J. Math. Soc. Japan 60(2):325~339
Z. C. Chen / 1990 / Estimates for discrete spectrum of Laplacian operator with any order / J. China Univ. Sci. Tech 20(3):259~266
D. Chen / 2011 / Bounds for ratios of the membrane eigenvalues / J. Dierential Equations 250(3):1575~1590
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