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

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Code:
83/2021
Title:
Some evaluations of the fractional p-Laplace operator on radial functions
Date:
Wednesday 15th December 2021
Author(s):
Colasuonno, F.; Ferrari F.; Gervasio, P.; Quarteroni, A.
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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-\bar x\bar ^2)^s_+ and -\Delta_p (1-\bar x \bar^(p/(p-1))) are constant functions in (-1,1) for fixed p and s. We evaluated (-\Delta_p)^s(1-\bar x \bar^(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.