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Helmholtz Solutions for the Fractional Laplacian and Other Related Operators | | Vincent Guan
; Mathav Murugan
; Juncheng Wei
; | | Date: |
1 Jan 2022 | | Abstract: | We show that the bounded solutions to the fractional Helmholtz equation,
$(-Delta)^s u= u$ for $0<s<1$ in $mathbb{R}^n$, are given by the bounded
solutions to the classical Helmholtz equation $(-Delta)u= u$ 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 $Acos{x} + Bsin{x}$. We show that this
classification of fractional Helmholtz solutions extends for $1<s le 2$ and
$sin mathbb{N}$ 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 $psi(-Delta) u= psi(1)u$ in $mathbb{R}^n$, when $psi$ is complete
Bernstein and certain regularity conditions are imposed on the associated
weight $a(t)$. | | Source: | arXiv, 2201.00252 | | Services: | Forum | Review | PDF | Favorites | |
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