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.
翻译:我们在整个参数范围内为分数 $p$-Laplace 操作员引入了三个代表公式:0 < 1美元和1美元 < p ⁇ infty$ 。请注意,对于 $p\ne 2美元,这是一个非线性操作员。第一个代表公式基于一个分离程序,该程序将非线性调整为线性热半组;第二个代表公式将非线性调整为Caffarelli-Silvestre线性扩展技术。第三个代表公式是Balakrishnan 公式的相应非线性版本。我们还讨论了分数 $p-Laplace 操作员的常数的正确选择,以便持续依赖$p\ to 2美元和$\ to 0 ⁇, 1 ⁇ - 美元。 由此可以得出一些后果和建议。 因此, 我们提议在域内建立一个自然光谱型操作员, 不同于小数位 $p$- Laplace 操作员在整个空间上的标准限制。 我们还提议了数字方案, 将分数价 $-Laplacean_r_rbs 的定出新的定义, 和 rmaps r_ rmax 和 r_ r_r_r_ 的替代描述。