Three representations of the fractional p-Laplacian: semigroup, extension and Balakrishnan formulas
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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 W^s,p(ℝ^n) seminorms.
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