Some evaluations of the fractional $p$-Laplace operator on radial functions

TL;DR

This paper investigates the fractional p-Laplace operator's behavior on radial functions, revealing that certain functions are not eigenfunctions as in the linear case, through numerical evaluation.

Contribution

It provides the first numerical evidence that the fractional p-Laplace operator does not preserve certain radial functions as eigenfunctions, extending understanding beyond the linear case.

Findings

The fractional p-Laplace operator does not produce constant functions for specific radial functions.

Numerical methods confirm the non-constancy of the operator on these functions.

Results suggest differences between linear and nonlinear fractional Laplace operators.

Abstract

We face a rigidity problem for the fractional $p$-Laplace operator to extend to this new framework some tools useful for the linear case. It is known that $(-\Delta)^s(1-|x|^{2})^s_+$ and $-\Delta_p(1-|x|^{\frac{p}{p-1}})$ are constant functions in $(-1,1)$ for fixed $p$ and $s$. We evaluated $(-\Delta_p)^s(1-|x|^{\frac{p}{p-1}})^s_+$ proving that it is not constant in $(-1,1)$ for some $p\in (1,+\infty)$ and $s\in (0,1)$. This conclusion is obtained numerically thanks to the use of very accurate Gaussian numerical quadrature formulas.