A gap for eigenvalues of a clamped plate problem

Abstract

This paper studies eigenvalues of the clamped plate problem on a bounded domain in an $n$-dimensional Euclidean space. We give an estimate for the gap between $\sqrt{{\Gamma }_{k+1}-{\Gamma }_1}$ and $\sqrt{{\Gamma }_k-{\Gamma }_1}$, for any positive integer $k$. According to the asymptotic formula of Agmon and Pleijel, we know, the gap between $\sqrt{{\Gamma }_{k+1}-{\Gamma }_1}$ and $\sqrt{{\Gamma }_k-{\Gamma }_1}$ is bounded by a term with a lower order $k^{\frac{1}{n}}$ in the sense of the asymptotic formula of Agmon and Peijel, where ${\Gamma }_j$ denotes the $j^{{}^{\text{th}}}$ eigenvalue of the clamped plate problem.