Abstract:We show explicit formulas for the evaluation of (possibly higher-order) fractional Laplacians (−∆) s of some functions supported on ellipsoids. In particular, we derive the explicit expression of the torsion function and give examples of s-harmonic functions. As an application, we infer that the weak maximum principle fails in eccentric ellipsoids for s ∈ (1, √ 3 + 3/2) in any dimension n ≥ 2. We build a counterexample in terms of the torsion function times a polynomial of degree 2. Using point inversion trans… Show more
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