Jia, Gao; Yang, Xiaoping; Qian, Chunlin On the upper bound of second eigenvalues for uniformly elliptic operators of any orders. (English) Zbl 1035.35086 Acta Math. Appl. Sin., Engl. Ser. 19, No. 1, 107-116 (2003). 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 Z. Chen and C. Qian [J. Math. Anal. Appl. 186, 821–834 (1994; Zbl 0814.35082)] Reviewer: Bodo Dittmar (Halle) Cited in 3 Documents MSC: 35P15 Estimates of eigenvalues in context of PDEs 35J25 Boundary value problems for second-order elliptic equations Keywords:eigenvalues; upper bounds Citations:Zbl 0814.35082 PDFBibTeX XMLCite \textit{G. Jia} et al., Acta Math. Appl. Sin., Engl. Ser. 19, No. 1, 107--116 (2003; Zbl 1035.35086) Full Text: DOI References: [1] Chen, Z.C., Qian, C.L. On the difference of consecutive eigenvalues of uniformly elliptic operators of higher orders. Chin. Ann. of Math., 14B(4): 435–442 (1993) · Zbl 0782.35046 [2] Chen, Z.C., Qian, C.L. On the upper bound of eigenvalues for elliptic equation with higher orders. J. of Math. Anal. & Appl., 186(3): 821–834 (1994) · Zbl 0814.35082 · doi:10.1006/jmaa.1994.1335 [3] Hile, G.N., Yeh, R.Z. Inequalities for eigenvalues of the biharmonic operator. Pacific J. Math., 112: 115–133 (1984) · Zbl 0541.35059 [4] Jia, G. Estimation of eigenvalues for croslwise vibration equations of the beam. J. of Anhui Univ., 2: 29–33 (1997) [5] Jia, G. On the upper bound of second eigenvalues for differential equations with higher orders. J. of Math. Study, 3: 30–35 (1999) [6] Protter, M.H. Can one hear the shape of a drum? SIAM Rev., 29: 185–197 (1987) · Zbl 0645.35074 · doi:10.1137/1029041 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.