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<p>We study the set of critical points of a solution to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2024_240_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Delta u = \lambda \cdot u\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>λ</mi> <mo>·</mo> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> and in particular components of the critical set that have codimension 1. We show, for example, that if a second Neumann eigenfunction of a simply connected polygon <i>P</i> has infinitely many critical points, then <i>P</i> is a rectangle.</p>

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Some remarks on critical sets of Laplace eigenfunctions

  • 期刊论文

  • 发表日期：

    2025年1月9日

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  • Chris Judge,
  • Sugata Mondal

摘要

We study the set of critical points of a solution to \(\Delta u = \lambda \cdot u\) Δ u = λ · u and in particular components of the critical set that have codimension 1. We show, for example, that if a second Neumann eigenfunction of a simply connected polygon P has infinitely many critical points, then P is a rectangle.