Payne nodal set conjecture for the fractional $p$-Laplacian in Steiner symmetric domains

TL;DR

This paper proves that for certain solutions of the fractional p-Laplacian in Steiner symmetric domains, the positive and negative regions of the solution touch the boundary, extending the Payne nodal set conjecture.

Contribution

It establishes the Payne nodal set conjecture for fractional p-Laplacian eigenfunctions and solutions in Steiner symmetric domains, using polarization inequalities.

Findings

Supports of positive and negative parts touch the boundary

Nodal set touches the boundary in connected domains

Analysis based on polarization inequalities

Abstract

Let $u$ be either a second eigenfunction of the fractional $p$-Laplacian or a least energy nodal solution of the equation $(-\Delta)^s_p \, u = f(u)$ with superhomogeneous and subcritical nonlinearity $f$, in a bounded open set $\Omega$ and under the nonlocal zero Dirichlet conditions. Assuming only that $\Omega$ is Steiner symmetric, we show that the supports of positive and negative parts of $u$ touch $\partial\Omega$. As a consequence, the nodal set of $u$ has the same property whenever $\Omega$ is connected. The proof is based on the analysis of equality cases in certain polarization inequalities involving positive and negative parts of $u$, and on alternative characterizations of second eigenfunctions and least energy nodal solutions.