论文标题
分数$ p $ -laplacian的三个表示：Semigroup，扩展和Balakrishnan公式
Three representations of the fractional $p$-Laplacian: semigroup, extension and Balakrishnan formulas
论文作者
论文摘要
我们在整个参数范围内为分数$ p $ -laplace操作员介绍了三个表示公式$ 0 <s <1 $和$ 1 <p <\ p <\ infty $。请注意，对于$ p \ ne 2 $，这是一个非线性操作员。第一个表示是基于分裂程序，该过程将重新归一化的非线性与线性热半群相结合。第二个使非线性适应了caffarelli-silvestre线性扩展技术。第三个是Balakrishnan公式的相应非线性版本。我们还讨论了分数$ p $ -laplace运算符的常数的正确选择，以便将持续依赖性为$ p \至2 $和$ s \ to 0^+，1^ - $。 得出了许多后果和建议。因此，我们在域中提出了一个自然的光谱型操作员，与在整个空间上作用的分数$ p $ laplace运算符的标准限制不同。我们还提出了数值方案，这是对流形的分数$ p $ -laplacian的新定义，以及$ w^{s，p}的替代特征（\ mathbb {r}^n）$ eminorms。
We introduce three representation formulas for the fractional $p$-Laplace operator in the whole range of parameters $0<s<1$ and $1<p<\infty$. Note that for $p\ne 2$ this a nonlinear operator. The first representation is based on a splitting procedure that combines a renormalized nonlinearity with the linear heat semigroup. The second adapts the nonlinearity to the Caffarelli-Silvestre linear extension technique. The third one is the corresponding nonlinear version of the Balakrishnan formula. We also discuss the correct choice of the constant of the fractional $p$-Laplace operator in order to have continuous dependence as $p\to 2$ and $s \to 0^+, 1^-$. A number of consequences and proposals are derived. Thus, we propose a natural spectral-type operator in domains, different from the standard restriction of the fractional $p$-Laplace operator acting on the whole space. We also propose numerical schemes, a new definition of the fractional $p$-Laplacian on manifolds, as well as alternative characterizations of the $W^{s,p}(\mathbb{R}^n)$ seminorms.
