Fractional Laplacians on ellipsoids

Nicola Abatangelo; Sven Jarohs; Alberto Saldaña; Nicola Abatangelo; Sven Jarohs; Alberto Saldaña

doi:10.3934/mine.2021038

Mathematics in Engineering

2021, Volume 3, Issue 5: 1-34. doi: 10.3934/mine.2021038

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Fractional Laplacians on ellipsoids

1.
Institut für Mathematik, Goethe-Universität Frankfurt am Main, Robert-Mayer-Straße 10, 60325 Frankfurt am Main, Germany
2.
Instituto de Matemáticas, Universidad Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510 Coyoacán, Ciudad de México, México

Received: 15 May 2020 Accepted: 08 September 2020 Published: 22 September 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.
- positivity preserving property,
- torsion function,
- point inversion
Citation: Nicola Abatangelo, Sven Jarohs, Alberto Saldaña. Fractional Laplacians on ellipsoids[J]. Mathematics in Engineering, 2021, 3(5): 1-34. doi: 10.3934/mine.2021038

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Abstract

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.

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