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<p>The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the corresponding eigenvalue of a disjoint union of two equal disks, for Robin parameters in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_254_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\([-4\pi ,4\pi ].\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mo>-</mo> <mn>4</mn> <mi>π</mi> <mo>,</mo> <mn>4</mn> <mi>π</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> This sharp inequality was known previously only for negative parameters in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_254_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">\([-4\pi ,0],\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mo>-</mo> <mn>4</mn> <mi>π</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> by Girouard and Laugesen. Their proof fails for positive Robin parameters because the second eigenfunction on a disk has non-monotonic radial part. This difficulty is overcome for parameters in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_254_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\((0,4\pi ]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>4</mn> <mi>π</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> by means of a degree-theoretic approach suggested by Karpukhin and Stern that yields suitably orthogonal trial functions.</p>

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Two disks maximize the third Robin eigenvalue: positive parameters

  • 期刊论文

  • 发表日期：

    2025年9月17日

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  • Hanna N. Kim,
  • Richard S. Laugesen

摘要

The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the corresponding eigenvalue of a disjoint union of two equal disks, for Robin parameters in \([-4\pi ,4\pi ].\) [ - 4 π , 4 π ] . This sharp inequality was known previously only for negative parameters in \([-4\pi ,0],\) [ - 4 π , 0 ] , by Girouard and Laugesen. Their proof fails for positive Robin parameters because the second eigenfunction on a disk has non-monotonic radial part. This difficulty is overcome for parameters in \((0,4\pi ]\) ( 0 , 4 π ] by means of a degree-theoretic approach suggested by Karpukhin and Stern that yields suitably orthogonal trial functions.