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<p>We prove the existence of an open set <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_721_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mi mathvariant="double-struck">S</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$\Omega \subset \mathbb{S}^{2}$</EquationSource> </InlineEquation> for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving the maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary.</p>

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Spherical Caps do Not Always Maximize Neumann Eigenvalues on the Sphere

  • 期刊论文

  • 发表日期：

    2025年10月20日

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  • Dorin Bucur,
  • Richard S. Laugesen,
  • Eloi Martinet,
  • Mickaël Nahon

摘要

We prove the existence of an open set Ω ⊂ S 2 $\Omega \subset \mathbb{S}^{2}$ for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving the maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary.