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Fractional Laplacians on ellipsoids
- Publication Year :
- 2020
-
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.<br />Comment: 6 pictures, 27 pages
- Subjects :
- Mathematics - Analysis of PDEs
35R11, 35B50, 35G15, 35C05, 35S15
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2005.09514
- Document Type :
- Working Paper