Top Qs
Timeline
Chat
Perspective
Fractional Laplacian
From Wikipedia, the free encyclopedia
Remove ads
In mathematics, the fractional Laplacian is an operator that generalizes the notion of the Laplace operator to fractional powers of spatial derivatives. It is frequently used in the analysis of nonlocal partial differential equations, especially in geometry and diffusion theory. Applications include:
- Global dissipative half-harmonic flows into spheres: small data in critical Sobolev spaces [1]
- Half-harmonic gradient flow: aspects of a non-local geometric PDE [2]
- Well-posedness of half-harmonic map heat flows for rough initial data [3]
Each of these replaces the classical Laplacian in a geometric PDE with the half-Laplacian to account for nonlocal effects.
Remove ads
Definition
Summarize
Perspective
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.[4]
Let and or let or , where:
- denotes the space of continuous functions that vanish at infinity, i.e., compact such that for all .
- denotes the space of bounded uniformly continuous functions , i.e., functions that are uniformly continuous, meaning such that for all with , and bounded, meaning such that for all .
Additionally, let .
Fourier Definition
If we further restrict to , we get
This definition uses the Fourier transform for . This definition can also be broadened through the Bessel potential to all .
Singular Operator
The Laplacian can also be viewed as a singular integral operator which is defined as the following limit taken in .
Generator of C_0-semigroup
Using the fractional heat-semigroup which is the family of operators , we can define the fractional Laplacian through its generator.
It is to note that the generator is not the fractional Laplacian but the negative of it . The operator is defined by
,
where is the convolution of two functions and .
Distributional Definition
For all Schwartz functions , the fractional Laplacian can be defined in a distributional sense by
where is defined as in the Fourier definition.
Bochner's Definition
The fractional Laplacian can be expressed using Bochner's integral as
where the integral is understood in the Bochner sense for -valued functions.
Balakrishnan's Definition
Alternatively, it can be defined via Balakrishnan's formula:
with the integral interpreted as a Bochner integral for -valued functions.
Dynkin's Definition
Another approach by Dynkin defines the fractional Laplacian as
with the limit taken in .
Quadratic Form Definition
In , the fractional Laplacian can be characterized via a quadratic form:
where
Inverse of the Riesz Potential Definition
When and for , the fractional Laplacian satisfies
Harmonic Extension Definition
The fractional Laplacian can also be defined through harmonic extensions. Specifically, there exists a function such that
where and is a function in that depends continuously on with bounded for all .
Remove ads
Relation to other Operators
Summarize
Perspective
Riesz transforms and the half-Laplacian
In dimension one, the Hilbert transform satisfies the identity
This expresses the half-Laplacian as the composition of the Hilbert transform with the spatial derivative.
In higher dimensions , this generalizes naturally to the vector-valued Riesz transform. For a function , the -th Riesz transform is defined as the singular integral operator
Equivalently, it is a Fourier multiplier with symbol
Letting and , we obtain the key identity:
This follows directly from the Fourier symbols:
Summing over recovers , hence the identity holds in the sense of tempered distributions.
Remove ads
See also
References
External links
Wikiwand - on
Seamless Wikipedia browsing. On steroids.
Remove ads