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<p>In this paper, we establish a priori log-concavity estimates for the first Dirichlet eigenfunction of convex domains of a Riemannian manifold. Specifically, we focus on cases where the principal eigenfunction <i>u</i> is assumed to be log-concave and our primary goal is to obtain quantitative estimates for the Hessian of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_10004_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\(\log u\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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A priori log-concavity estimates for Dirichlet eigenfunctions

  • 期刊论文

  • 发表日期：

    2025年5月29日

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  • Gabriel Khan,
  • Soumyajit Saha,
  • Malik Tuerkoen

摘要

In this paper, we establish a priori log-concavity estimates for the first Dirichlet eigenfunction of convex domains of a Riemannian manifold. Specifically, we focus on cases where the principal eigenfunction u is assumed to be log-concave and our primary goal is to obtain quantitative estimates for the Hessian of \(\log u\) log u .