On some unexpected properties of radial and symmetric eigenvalues and eigenfunctions of the $p$-Laplacian on a disk

TL;DR

This paper investigates unexpected properties of eigenvalues and eigenfunctions of the p-Laplacian on a disk, revealing nonradial eigenfunctions, unusual nodal set shapes, and eigenvalue limits as p approaches 1 and infinity.

Contribution

It demonstrates the existence of nonradial eigenfunctions for radial eigenvalues and eigenfunctions with unique nodal set shapes, expanding understanding of p-Laplacian spectral properties.

Findings

Existence of nonradial eigenfunctions for radial eigenvalues.

Eigenfunctions with nodal sets not possible in the linear case.

As p approaches 1 and infinity, eigenvalues exhibit specific limit behaviors.

Abstract

We discuss several properties of eigenvalues and eigenfunctions of the $p$-Laplacian on a ball subject to zero Dirichlet boundary conditions. Among main results, in two dimensions, we show the existence of nonradial eigenfunctions which correspond to the radial eigenvalues. Also we prove the existence of eigenfunctions whose shape of the nodal set cannot occur in the linear case $p=2$. Moreover, the limit behavior of some eigenvalues as $p \to 1+$ and $p \to +\infty$ is studied.