Inequalities for Lower Order Eigenvalues of Fourth-Order Elliptic System of Differential Equations

WANG Lin-lin, SUN He-jun, XIAO Meng-ge

doi:10.13371/j.cnki.chin.q.j.m.2026.02.002

Chinese Quarterly Journal of Mathematics >

2026 , Vol. 41 >Issue 2: 128 - 138

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2026.02.002

Inequalities for Lower Order Eigenvalues of Fourth-Order Elliptic System of Differential Equations

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School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210014, China)

WANG Lin-lin (2000-), female, native of Xuzhou, Jiangsu, postgraduate of Nanjing University of Science and Technology, engages in differential geometry; SUN He-jun (1976-), male, native of Lianyungang, Jiangsu, professor of Nanjing University of Science and Technology, Master supervisor, Ph.D, engages in differential geometry; XIAO Meng-ge (2002-), female, native of Macheng, Hubei, postgraduate of Nanjing University ofScience and Technology, engages in differential geometry.

SUN He-jun (1976-), male, native of Lianyungang, Jiangsu, professor of Nanjing University of Science and Technology, Master supervisor, Ph.D, engages in differential geometry;

Received date: 2025-08-30

Online published: 2026-06-30

Supported by

Supported by National Natural Science Foundation of China (Grant No. 11001130); Fundamental Research Funds for the Central Universities (Grant No. 30917011335).

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Abstract

In this paper, we investigate the Dirichlet eigenvalue problem of fourth-order elliptic system of differential equations on an n-dimensional Euclidean space as follows
\begin{equation*}
\left\{\begin{aligned}
&A\Delta^2\boldsymbol{u} = -\Gamma\Delta \boldsymbol{u}, && \text{in} \quad \Omega, \\
&\boldsymbol{u} = \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{\nu}} = \boldsymbol{0}, && \text{on} \quad \partial \Omega,
\end{aligned}\right.
\end{equation*}
where A is a symmetric coefficient matrix and ν is the outward unit normal vector field of ∂Ω. We derive some inequalities for lower order eigenvalues of this problem. Our results cover some previous results for the buckling problem.

Key words： Eigenvalue; Inequality; Elliptic system of differential equations

Cite this article

WANG Lin-lin, SUN He-jun, XIAO Meng-ge . Inequalities for Lower Order Eigenvalues of Fourth-Order Elliptic System of Differential Equations[J]. Chinese Quarterly Journal of Mathematics, 2026 , 41(2) : 128 -138 . DOI: 10.13371/j.cnki.chin.q.j.m.2026.02.002

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