Fractional Laplacians on ellipsoids

We show explicit formulas for the evaluation of (possibly higher-order) fractional Laplacians (-△)s some functions supported on ellipsoids. In particular, we derive expression torsion function and give examples $s$-harmonic functions. As an application, infer that weak maximum principle fails in eccentric ellipsoids $s\in(1, \sqrt{3}+3/2)$ any dimension $n\geq 2$. build a counterexample terms times polynomial degree 2. Using point inversion transformations, it follows variety bounded unbounded domains do not satisfy positivity preserving properties either examples.