We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in \( {\mathbb{R}^n} \). 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., 245, No. 1, 334–352 (2007)) for the clamped-plate problem to a fourth-order elliptic operator with variable coefficients.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 7, pp. 999–1008, July, 2011.
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Sun, HJ. Estimates for weighted eigenvalues of a fourth-order elliptic operator with variable coefficients. Ukr Math J 63, 1154–1164 (2011). https://doi.org/10.1007/s11253-011-0569-x
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DOI: https://doi.org/10.1007/s11253-011-0569-x