Estimates on the eigenvalues of the clamped plate problem on domains in Euclidean spaces
Selma Yıldırım Yolcu, Türkay Yolcu · Journal of Mathematical Physics · 2013
The purpose of this article is two-fold. First, we obtain some certain bounds for the sums of (positive and negative) powers of the eigenvalues of the clamped plate problem of the Dirichlet bi-Laplacian operator \documentclass[12pt]{minimal}\begin{document}$\Delta ^{2}|_{{\mathcal {D}}}$\end{document}Δ2|D, restricted to a bounded domain \documentclass[12pt]{minimal}\begin{document}${\mathcal {D}}\subset {\mathbb {R}}^d$\end{document}D⊂Rd with d ⩾ 2. Second, we establish lower bounds for the sums of eigenvalues of \documentclass[12pt]{minimal}\begin{document}$\Delta ^2|_{{\mathcal {D}}}$\end{document}Δ2|D sharper than the bounds recently obtained by Cheng and Wei [“A lower bound for eigenvalues of a clamped plate problem,” Calculus Var. Partial Differ. Equ. 42(3–4), 579–590 (2011)]10.1007/s00526-011-0399-6. All these estimates are sharp in the sense of Weyl asymptotics.