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

TL;DR

This paper extends the understanding of the nodal set of second eigenfunctions from the classical Laplacian to the nonlinear p-Laplacian in circularly symmetric domains across higher dimensions, using the method of moving polarization.

Contribution

It generalizes Pütter's 1990 result to the p-Laplacian and higher dimensions by employing the method of moving polarization.

Findings

Nodal line of second eigenfunction intersects the boundary in symmetric domains

Method of moving polarization effectively analyzes p-Laplacian eigenfunctions

Results hold in arbitrary higher dimensions

Abstract

In 1990, P\"utter shown that the nodal line of any second eigenfunction of the Dirichlet Laplacian on a planar bounded simply connected domain $\Omega$ intersects the boundary $\partial\Omega$ provided $\Omega$ 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.