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<p>We study three types of fourth-order Steklov eigenvalue problems. For the first two of them, we derive the asymptotic expansion of the eigenvalues on Euclidean annular domains <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10277_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {B}^n_1\setminus \overline{\mathbb {B}^n_\epsilon }\)</EquationSource> </InlineEquation> as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10277_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(\epsilon \rightarrow 0\)</EquationSource> </InlineEquation>, in turn yielding some interesting results regarding the shape optimization of the eigenvalues. For these two problems, we also compute the respective spectra on cylinders over closed Riemannian manifolds. For the third problem, we obtain a sharp upper bound for its first non-zero eigenvalue on star-shaped and mean convex Euclidean domains.</p>

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On Shape Optimization for Fourth Order Steklov eigenvalue Problems

  • 期刊论文

  • 发表日期：

    2025年6月20日

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  • Changwei Xiong,
  • Jinglong Yang,
  • Jinchao Yu

摘要

We study three types of fourth-order Steklov eigenvalue problems. For the first two of them, we derive the asymptotic expansion of the eigenvalues on Euclidean annular domains \(\mathbb {B}^n_1\setminus \overline{\mathbb {B}^n_\epsilon }\) as \(\epsilon \rightarrow 0\) , in turn yielding some interesting results regarding the shape optimization of the eigenvalues. For these two problems, we also compute the respective spectra on cylinders over closed Riemannian manifolds. For the third problem, we obtain a sharp upper bound for its first non-zero eigenvalue on star-shaped and mean convex Euclidean domains.