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<p>We generalize a classical inequality between the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, planar domains: in 1955, Payne proved that below the <i>k-</i>th eigenvalue of the Dirichlet Laplacian there exist at least <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3216_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(k + 2\)</EquationSource> </InlineEquation> eigenvalues of the Neumann Laplacian, provided the domain is convex. It has, however, been conjectured that this should hold for any domain. Here we show that the statement indeed remains true for all simply connected planar Lipschitz domains.</p>

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Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年8月1日

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  • Jonathan Rohleder

摘要

We generalize a classical inequality between the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, planar domains: in 1955, Payne proved that below the k-th eigenvalue of the Dirichlet Laplacian there exist at least \(k + 2\) eigenvalues of the Neumann Laplacian, provided the domain is convex. It has, however, been conjectured that this should hold for any domain. Here we show that the statement indeed remains true for all simply connected planar Lipschitz domains.