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<p>In this paper, we investigate the eigenvalue problem of elliptic operators in weighted divergence form on smooth metric measure spaces. Firstly, we give a general inequality for eigenvalues of the elliptic operators in weighted divergence form on a compact smooth metric measure space with boundary (possibly empty). Then, applying this general inequality, we get some new universal inequalities for the eigenvalues of fourth-order elliptic operators in weighted divergence form on smooth metric measure spaces. Also, using these general inequalities and three generalized Cheeger–Gromoll splitting theorems, we give some new universal inequalities for the eigenvalues of vibration problem for a clamped plate on the smooth metric measure spaces that satisfy some curvature conditions. Moreover, our result can reveal the relationship between the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3328_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(k + 1)$</EquationSource> </InlineEquation>-th eigenvalue and the first <i>k</i> eigenvalues in a relatively quick way.</p>

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New universal inequalities for eigenvalues of elliptic operators in weighted divergence form on smooth metric measure spaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月11日

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  • Yanli Li,
  • Feng Du

摘要

In this paper, we investigate the eigenvalue problem of elliptic operators in weighted divergence form on smooth metric measure spaces. Firstly, we give a general inequality for eigenvalues of the elliptic operators in weighted divergence form on a compact smooth metric measure space with boundary (possibly empty). Then, applying this general inequality, we get some new universal inequalities for the eigenvalues of fourth-order elliptic operators in weighted divergence form on smooth metric measure spaces. Also, using these general inequalities and three generalized Cheeger–Gromoll splitting theorems, we give some new universal inequalities for the eigenvalues of vibration problem for a clamped plate on the smooth metric measure spaces that satisfy some curvature conditions. Moreover, our result can reveal the relationship between the ( k + 1 ) $(k + 1)$ -th eigenvalue and the first k eigenvalues in a relatively quick way.