Three representations of the fractional $p$-Laplacian: semigroup, extension and Balakrishnan formulas

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.