%0 Journal Article
%A Del Teso Méndez, Félix
%A Gómez-Castro, D.
%A Vázquez, Juan Luis
%T Three representations of the fractional p-Laplacian:semigroup, extension and Balakrishnan formulas
%D 2021
%@ 1311-0454
%U https://hdl.handle.net/20.500.14352/4992
%X We introduce three representation formulas for the fractional p-Laplace operator in the whole range of parameters 0 < s < 1 and 1 < p < ∞. Note that for p ≠ 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 → 2 and s → 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 Ws, p(ℝn) seminorms.
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