论文标题
Helmholtz解决方案，用于分数Laplacian和其他相关操作员
Helmholtz Solutions for the Fractional Laplacian and Other Related Operators
论文作者
论文摘要
我们表明，分数Helmholtz方程的有限解决方案，$（δ）^s U = U $ $ 0 <s <s <1 $ in $ \ MATHBB {r}^n $，由经典的Helmholtz方程$（-De）等式$（ - δ）U = $ n $ \ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n}以$ \ infty $消失。当$ n = 1 $时，我们表明经典解决方案$ a \ cos {x} + b \ sin {x} $再次给出了有界的分数Helmholtz解决方案。我们表明，当$ u \ in c^\ infty（\ mathbb {r}^n）$时，这种分数Helmholtz解决方案的分类延长了$ 1 <s \ le 2 $和$ s \ in \ mathbb {n} $。最后，我们证明了经典的解决方案是$ \ Mathbb {r}^n $中更通用方程式$ψ（-Δ）U =ψ（1）U $的独特界面解决方案，当$ \ n $是完整的伯恩斯坦，并且某些规律性条件在相关的重量$ a（t）$上施加了某些规律性条件。
We show that the bounded solutions to the fractional Helmholtz equation, $(-Δ)^s u= u$ for $0<s<1$ in $\mathbb{R}^n$, are given by the bounded solutions to the classical Helmholtz equation $(-Δ)u= u$ in $\mathbb{R}^n$ for $n \ge 2$ when $u$ is additionally assumed to be vanishing at $\infty$. When $n=1$, we show that the bounded fractional Helmholtz solutions are again given by the classical solutions $A\cos{x} + B\sin{x}$. We show that this classification of fractional Helmholtz solutions extends for $1<s \le 2$ and $s\in \mathbb{N}$ when $u \in C^\infty(\mathbb{R}^n)$. Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation $ψ(-Δ) u= ψ(1)u$ in $\mathbb{R}^n$, when $ψ$ is complete Bernstein and certain regularity conditions are imposed on the associated weight $a(t)$.
