F. Colasuonno, F. Ferrari, P. Gervasio, A. Quarteroni
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引用次数: 1
摘要
我们面对分数阶$ p $ -拉普拉斯算子的刚性问题,将一些对线性情况有用的工具推广到这个新的框架中。我们知道,对于固定的$ p $和$ s $, $ (-\Delta)^s(1-|x|^{2})^s_+ $和$ -\Delta_p(1-|x|^{\frac{p}{p-1}}) $是$ (-1, 1) $中的常数函数。我们对$ (-\Delta_p)^s(1-|x|^{\frac{p}{p-1}})^s_+ $求值,证明对于某些$ p\in (1, +\infty) $和$ s\in (0, 1) $,它在$ (-1, 1) $中不是常数。由于使用了非常精确的高斯数值求积公式,这一结论在数值上得到了。
Some evaluations of the fractional $ p $-Laplace operator on radial functions
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.