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On the upper bound of second eigenvalues for uniformly elliptic operators of any orders. (English) Zbl 1035.35086

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)]

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations

Citations:

Zbl 0814.35082
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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
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