  We show explicit formulas for the evaluation of (possibly higher-order)
fractional Laplacians 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 $s\in(1,\sqrt{3}+3/2)$ in any
dimension $n\geq 2$. We build a counterexample in terms of the torsion function
times a polynomial of degree 2. Using point inversion transformations, it
follows that a variety of bounded and unbounded domains do not satisfy
positivity preserving properties and we give some examples.

Analysis of PDEs