Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2020; 57(6): 1471-1484

Published online November 1, 2020 https://doi.org/10.4134/JKMS.j190737

Copyright © The Korean Mathematical Society.

Lower order eigenvalues for the bi-drifting Laplacian on the Gaussian shrinking soliton

Lingzhong Zeng

Jiangxi Normal University

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.

Keywords: Bi-drifting Laplacian, eigenvalues, Gaussian shrinking soliton

MSC numbers: Primary 53C23, 35P15

Supported by: The research work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11861036 and 11826213) and the Key Program of Natural Science Foundation of Jiangxi Province (Grant No. 20171ACB21023)