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<p>We introduce the discrete poly-Laplace operator on a subgraph with Dirichlet boundary condition. We obtain upper and lower bounds for the sum of the first <i>k</i> Dirichlet eigenvalues of the poly-Laplace operators on a finite subgraph of lattice graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z}^{d}\)</EquationSource> </InlineEquation> extending classical results of Li-Yau and Kröger. Moreover, we prove that, for any positive integer <InlineEquation ID="IEq888"> <EquationSource Format="TEX"> \(l\)</EquationSource> </InlineEquation>, the Dirichlet eigenvalues of the poly-Laplace of order <InlineEquation ID="IEq889"> <EquationSource Format="TEX"> \(2l\)</EquationSource> </InlineEquation> are bounded below by the squares of the Dirichlet eigenvalues of the poly-Laplace of order <InlineEquation ID="IEq890"> <EquationSource Format="TEX">\(l\)</EquationSource> </InlineEquation>.</p>

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Eigenvalue estimates for the poly-Laplace operator on lattice subgraphs

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Bobo Hua,
  • Ruowei Li

摘要

We introduce the discrete poly-Laplace operator on a subgraph with Dirichlet boundary condition. We obtain upper and lower bounds for the sum of the first k Dirichlet eigenvalues of the poly-Laplace operators on a finite subgraph of lattice graph \(\mathbb {Z}^{d}\) extending classical results of Li-Yau and Kröger. Moreover, we prove that, for any positive integer \(l\) , the Dirichlet eigenvalues of the poly-Laplace of order \(2l\) are bounded below by the squares of the Dirichlet eigenvalues of the poly-Laplace of order \(l\) .