Eigenvalue bounds for the clamped plate problem of L^2_xi operator
Lingzhong Zeng, Ziyi Zhou · Electronic Journal of Differential Equations · 2025
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. For more information see https://ejde.math.txstate.edu/Volumes/2025/31/abstr.html