论文标题
椭圆形的分数laplacians
Fractional Laplacians on ellipsoids
论文作者
论文摘要
我们展示了用于评估椭圆形支持某些功能的（可能是高阶）分数laplacian的明确公式。特别是，我们得出了扭转功能的明确表达，并提供了$ s $ harmonic函数的示例。作为一个应用程序，我们推断出弱椭圆形的弱最大原理在（1，\ sqrt {3} +3/2）$中以任何维度$ n \ geq 2 $中的$ s \ in（1，\ sqrt {3} +3/2）$失败。我们根据扭转函数时间构建了一个反例2度的多项式。使用点反转转换，可以得出的是，各种有界和无界的域不能满足保持阳性的属性，我们给出了一些示例。
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.
