Fractional Laplacians on ellipsoids
Nicola Abatangelo, 1 Institut für Mathematik, Goethe-Universität Frankfurt am Main, Robert-Mayer-Straße 10, 60325 Frankfurt am Main, Germany, Sven Jarohs, Alberto Saldaña, 2 Instituto de Matemáticas, Universidad Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510 Coyoacán, Ciudad de México, México, <sup>†</sup><b>This contribution is part of the Special Issue:</b> Partial Differential Equations from theory to applications—Dedicated to Alberto Farina, on the occasion of his 50th birthday, Guest Editors: Serena Dipierro; Luca Lombardini, Link: <a href="www.aimspress.com/mine/article/5752/special-articles" target="_blank">www.aimspress.com/mine/article/5752/special-articles</a> · Mathematics in Engineering · 2020
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 $s\in(1, \sqrt{3}+3/2)$ 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.