In mathematics, the fractional Laplacian is an operator, which generalizes the notion of Laplacian spatial derivatives to fractional powers.
This operator is often used to generalise certain types of Partial differential equation, two examples are [1] and [2] which both take known PDEs containing the Laplacian and replacing it with the fractional version.
In literature the definition of the fractional Laplacian often varies, but most of the time those definitions are equivalent.
The following is a short overview proven by Kwaśnicki, M in.
{\displaystyle {\mathcal {X}}:=C_{bu}(\mathbb {R} ^{n})}
, we get This definition uses the Fourier transform for
This definition can also be broadened through the Bessel potential to all
The Laplacian can also be viewed as a singular integral operator which is defined as the following limit taken in
Using the fractional heat-semigroup which is the family of operators
, we can define the fractional Laplacian through its generator.
{\displaystyle -(-\Delta )^{s}f(x)=\lim _{t\to 0^{+}}{\frac {P_{t}f-f}{t}}}
It is to note that the generator is not the fractional Laplacian
The operator
is defined by
is the convolution of two functions and
ξ
ξ
For all Schwartz functions
, the fractional Laplacian can be defined in a distributional sense by where
is defined as in the Fourier definition.
The fractional Laplacian can be expressed using Bochner's integral as where the integral is understood in the Bochner sense for
Alternatively, it can be defined via Balakrishnan's formula: with the integral interpreted as a Bochner integral for
Another approach by Dynkin defines the fractional Laplacian as with the limit taken in
, the fractional Laplacian can be characterized via a quadratic form: where When
, the fractional Laplacian satisfies The fractional Laplacian can also be defined through harmonic extensions.
Specifically, there exists a function
α
− α
α 2
that depends continuously on