Fractional Laplacians on ellipsoids

TL;DR

This paper derives explicit formulas for fractional Laplacians on ellipsoids, explores their properties, and demonstrates the failure of the maximum principle in certain eccentric ellipsoids, revealing limitations of positivity preservation.

Contribution

It provides explicit formulas for fractional Laplacians on ellipsoids and shows the failure of the maximum principle in specific eccentric domains, a novel insight in this area.

Findings

Explicit formulas for fractional Laplacians on ellipsoids

Counterexamples to the maximum principle in eccentric ellipsoids

Identification of domains lacking positivity preserving properties

Abstract

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.