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<p>We prove the existence of an open set minimizing the first eigenvalue of the Dirichlet polylaplacian of order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2936_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(m\ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> under volume constraint. Moreover, the corresponding eigenfunction is shown to enjoy <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2936_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{m-1,\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> Hölder regularity. This is performed for dimension <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2936_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">\(2\le d\le 4m\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>d</mi> <mo>≤</mo> <mn>4</mn> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>. In particular, our analysis answers the question of the existence of an optimal shape for the clamped plate up to dimension 8.</p>

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Existence of an optimal shape for the first eigenvalue of polyharmonic operators

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Roméo Leylekian

摘要

We prove the existence of an open set minimizing the first eigenvalue of the Dirichlet polylaplacian of order \(m\ge 1\) m ≥ 1 under volume constraint. Moreover, the corresponding eigenfunction is shown to enjoy \(C^{m-1,\alpha }\) C m - 1 , α Hölder regularity. This is performed for dimension \(2\le d\le 4m\) 2 ≤ d ≤ 4 m . In particular, our analysis answers the question of the existence of an optimal shape for the clamped plate up to dimension 8.