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<p>In 1990, Pütter showed that the nodal line of any second eigenfunction of the Dirichlet Laplacian on a planar bounded simply connected domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> intersects the boundary <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\partial \Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> provided <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> has circular symmetry. By adapting the method of moving polarization, we establish similar information on the nodal set of second eigenfunctions of the Dirichlet <i>p</i>-Laplacian on circularly symmetric domains in all higher dimensions.</p>

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On the nodal set conjecture for the p-Laplacian in circularly symmetric domains

  • 期刊论文

  • 发表日期：

    2026年7月6日

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  • Vladimir Bobkov

摘要

In 1990, Pütter showed that the nodal line of any second eigenfunction of the Dirichlet Laplacian on a planar bounded simply connected domain \(\Omega \) Ω intersects the boundary \(\partial \Omega \) ∂ Ω provided \(\Omega \) Ω has circular symmetry. By adapting the method of moving polarization, we establish similar information on the nodal set of second eigenfunctions of the Dirichlet p-Laplacian on circularly symmetric domains in all higher dimensions.