College of Mathematics and Statistics
Jiangxi Normal University
Nanchang 330022, China
email: Lingzhongzeng@yeah.net
College of Mathematics and Statistics
Jiangxi Normal University
Nanchang 330022, China
email: zyzhouu@163.com
Lingzhong Zeng, Ziyi Zhou
Abstract:
The operator \(L_{II}\) is an important extrinsic
differential operator, which is elliptic
of divergence type and plays significant roles in the study
of translating solitons. In this article, we extend
\(L_{II}\) to a more general elliptic differential
operator \(L_{\xi}\), for studying the clamped plate problem of
the bi-\(L_{\xi}\) operator, denoted by \(L_{\xi}^2\),
on the complete Riemannian manifolds.
By establishing a general formula of eigenvalues for
\(L_{\xi}^2\), we give a new estimate for
the eigenvalues of bi-\(L_{\xi}\) operator.
Some further applications of this result includes
obtaining some universal inequalities for bi-\(L_{II}\)
operator on translators, and studying the eigenvalues on the
submanifolds of the Euclidean spaces, unit spheres, and projective spaces.
Submitted December 3, 2024. Published March 31, 2025.
Math Subject Classifications: 35P15, 53C40.
Key Words: L^2_xi operator; clamped plate problem; eigenvalues; submanifolds; translating solitons
DOI: 10.58997/ejde.2025.31
Show me the PDF file (443 KB), TEX file for this article.
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Lingzhong Zeng College of Mathematics and Statistics Jiangxi Normal University Nanchang 330022, China email: Lingzhongzeng@yeah.net |
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address: Ziyi Zhou College of Mathematics and Statistics Jiangxi Normal University Nanchang 330022, China email: zyzhouu@163.com |
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