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

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.