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.

18 pages, 8 figures