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

Vladimir Bobkov

TL;DR

The paper addresses Payne's nodal set conjecture for the Dirichlet $p$-Laplacian in circularly symmetric domains of arbitrary dimension. It employs the moving polarization (a two-point rearrangement) method, along with a precise circular-symmetry framework, to control the location of the nodal set of second eigenfunctions. Under assumptions $(O_1)$--$(O_4)$, it proves that $\mathrm{dist}(\mathcal{Z}(u),\partial\Omega)=0$, extending known planar results to nonlinear settings and higher dimensions. The work provides a robust nonlinear/nodal-set framework via polarization that could extend to other symmetries and domain regularities, highlighting the versatility of two-point rearrangements in eigenvalue problems.

Abstract

In 1990, Pütter shown that the nodal line of any second eigenfunction of the Dirichlet Laplacian on a planar bounded simply connected domain $Ω$ intersects the boundary $\partialΩ$ provided $Ω$ has the circular symmetry. By adopting 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 arbitrary higher dimension.

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

TL;DR

The paper addresses Payne's nodal set conjecture for the Dirichlet -Laplacian in circularly symmetric domains of arbitrary dimension. It employs the moving polarization (a two-point rearrangement) method, along with a precise circular-symmetry framework, to control the location of the nodal set of second eigenfunctions. Under assumptions --, it proves that , extending known planar results to nonlinear settings and higher dimensions. The work provides a robust nonlinear/nodal-set framework via polarization that could extend to other symmetries and domain regularities, highlighting the versatility of two-point rearrangements in eigenvalue problems.

Abstract

In 1990, Pütter shown that the nodal line of any second eigenfunction of the Dirichlet Laplacian on a planar bounded simply connected domain intersects the boundary provided has the circular symmetry. By adopting the method of moving polarization, we establish similar information on the nodal set of second eigenfunctions of the Dirichlet -Laplacian on circularly symmetric domains in arbitrary higher dimension.
Paper Structure (3 sections, 8 theorems, 32 equations, 4 figures)

This paper contains 3 sections, 8 theorems, 32 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Polarization
  3. Proof of Theorem \ref{['thm:main']}

Key Result

Lemma 1.1

Let Then $\lambda_2 = \mu_2$, and for any minimizer $v$ of $\mu_2$ there exist $\alpha,\beta>0$ such that $\alpha v^+ + \beta v^-$ is a second eigenfunction of eq:D.

Figures (4)

  • Figure 1: Extensions of payne1973 for domains with reflection symmetry.
  • Figure 2: An example of a domain with circular symmetry.
  • Figure 3: Dark gray - $\Omega \cap \sigma_{\theta}(\Omega)$ in $\Sigma_{\theta}^-$, light gray - $\Omega \cup \sigma_{\theta}(\Omega)$ in $\Sigma_{\theta}^+$, so that $\widetilde{P}_\theta \Omega = \Omega$.
  • Figure 4: Notations in the proof.

Theorems & Definitions (13)

  • Lemma 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 3 more