Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 394–403
We establish some sharper inequalities for eigenvalues of a system of higher-order differential equations. Moreover, we present some sharper estimates for the upper bound of the (k +1)th eigenvalue and the gaps of its consecutive eigenvalues.
Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 999-1008
We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in $R^n$. We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the $(k + 1)$-th eigenvalue in terms of the first $k$ eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang -Xia inequality (J. Funct. Anal. - 2007. - 245) for the clamped plate problem to a fourth-order elliptic operator with variable coefficients.