Fractional Laplacians on ellipsoids

Abstract

We show explicit formulas for the evaluation of (possibly higher-order) fractional Laplacians~$\Ds$ 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 either and we give some examples.

Keywords

• positivity preserving property
• torsion function
• point inversion