Abstract
We show explicit formulas for the evaluation of (possibly higher-order) fractional Laplacians (-△)<sup><i>s</i></sup> 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.
Highlights
The fractional Laplacian (−∆)s, s > 0, is a pseudodifferential operator with Fourier symbol | · |2s which can be evaluated pointwisely via a hypersingular integral (see (2.1) below)
We show some explicit formulas for the evaluation of the fractional Laplacian of polynomial-like functions supported in ellipsoids
Our first result concerns the explicit expression of the torsion function of an ellipsoid
Summary
The fractional Laplacian (−∆)s, s > 0, is a pseudodifferential operator with Fourier symbol | · |2s which can be evaluated pointwisely via a hypersingular integral (see (2.1) below). In contrast to the results in [31] and [32] which rely on explicit computations of polynomials that can be verified quickly with a computer, the fractional case is much more complex, even with the explicit form of the fractional Laplacian (−∆)sUε , since these formulas are given in terms of hypergeometric functions which are in general difficult to manipulate To overcome this difficulty, we use an asymptotic analysis as the length of one of the axis in the ellipsoid goes to zero; it turns out that a suitable normalization of the hypergeometric functions simplifies in the limit and its asymptotic behavior can be determined with precision (see Lemma A.1).
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