On the upper bound of second eigenvalues for uniformly elliptic operators of any orders. (Q1873587)

The paper deals with inequalities between the first and second eigenvalue of the following eigenvalue problem \[ (-1)^t \sum_{i_1,i_2,\dots, i_t=1}^m D_{i_1i_2\dots i_t}(a_{i_1i_2\dots i_t}(x)D_{i_1i_2\dots i_t}u)=\lambda(-\Delta)^ru, \;\;x \in \Omega, \] \[ u=\frac{\partial u}{\partial \nu}= \dots = \frac{\partial^{t-1}u}{\partial \nu^{t-1}} =0, \;\;x \in \partial \Omega, \] where \(\Omega \subset R^m\) is a bounded domain. The results are generalizations of the inequalities proven by \textit{Z. Chen} and \textit{C. Qian} [J. Math. Anal. Appl. 186, 821--834 (1994; Zbl 0814.35082)]