Fractional Laplacian

In mathematical analysis, 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.

Definition

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.

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 C₀-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 .

Relation to other operators

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.

See also

• Fractional calculus
• Riemann-Liouville integral

References

External links

• "Fractional Laplacian". Nonlocal Equations Wiki, Department of Mathematics, The University of Texas at Austin.