Fractional Laplacian Explained

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.

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.^[3]

Let

p \in [ 1 , inf ty) and l{X} : = L ^{p(R} ^{n)} or let l{X} : = C ^{n)} _{0 ( R} or l{X} : = C _{bu} ( R ^{n)} , where:C ^{n)} _{0 ( R} denotes the space of continuous functions f : R ^{n} \to R that vanish at infinity, i.e., \forall \varepsilon > 0 , \exists K \subset R ^{n} compact such that |f(x)| < \epsilon for all x \notin K .C _{bu} ( R ^{n)} denotes the space of bounded uniformly continuous functions f : R ^{n} \to R , i.e., functions that are uniformly continuous, meaning \forall \epsilon > 0 , \exists \delta > 0 such that |f(x) - f(y)| < \epsilon for all x, y \in R ^{n} with |x - y| < \delta , and bounded, meaning \exists M > 0 such that |f(x)| \leq M for all x \in R ^{n} .

Additionally, let

s \in ( 0 , 1 ) .

Fourier Definition

If we further restrict to

p \in [ 1 , 2 ] , we get( - \Delta) ^{s} f : = l{F} ^{-1} _{\xi} (| \xi| ^{2s} l{F}(f))

This definition uses the Fourier transform for

f \in L ^{p(} R ^{n)} . This definition can also be broadened through the Bessel potential to all p \in [ 1 , inf ty) .

Singular Operator

The Laplacian can also be viewed as a singular integral operator which is defined as the following limit taken in

l{X} .( - \Delta) ^{s} f(x) = (4 ^{s} \Gamma( (d)/(2) + s))/(\pi ^{d/2} |\Gamma( - s)|) \lim _{r \to 0 ^{+}} \int\limits _{R ^{d} \setminus B _{r} (x)} { (f(x) - f(y))/(|x - y| ^{d+2s}) dy}

Generator of C_0-semigroup

Using the fractional heat-semigroup which is the family of operators

\{P _{t} \} _{t} , we can define the fractional Laplacian through its generator.- ( - \Delta) ^{s} f(x) = \lim _{t \to 0 ^{+}} (P _{t} f - f)/(t)

It is to note that the generator is not the fractional Laplacian

( - \Delta) ^{s} but the negative of it - ( - \Delta) ^{s} . The operator P _{t} : l{X} \to l{X} is defined byP _{t} f : = p _{t} * f ,

where

* is the convolution of two functions and p _{t} : = l{F} ^{-1} _{\xi} (e ^{- t | \xi| ^{2s}} ) .

Distributional Definition

For all Schwartz functions

\varphi , the fractional Laplacian can be defined in a distributional sense by\int _{R ^{d}} ( - \Delta) ^{s} f(y) \varphi(y) dy = \int _{R ^{d}} f(x) ( - \Delta) ^{s} \varphi(x) dx

where

( - \Delta) ^{s} \varphi is defined as in the Fourier definition.

Bochner's Definition

The fractional Laplacian can be expressed using Bochner's integral as

( - \Delta) ^{s} f = (1)/(\Gamma( - (s)/() 2)) \int ^{inf ty} _{0} \left( e ^{t} f - f \right) t ^{-1} dt

where the integral is understood in the Bochner sense for

l{X} -valued functions.

Balakrishnan's Definition

Alternatively, it can be defined via Balakrishnan's formula:

( - \Delta) ^{s} f = (\sin \left( (s \pi)/(2) \right))/(\pi) \int ^{inf ty} _{0} ( - \Delta) \left( s I - \Delta \right) ^{-1} f s ^{s/2} ds

with the integral interpreted as a Bochner integral for

l{X} -valued functions.

Dynkin's Definition

Another approach by Dynkin defines the fractional Laplacian as

( - \Delta) ^{s} f = \lim _{r \to 0 ^{+}} (2 ^{s} \Gamma\left( (d + s)/(2) \right))/(\pi ^{d/ 2} \Gamma\left( - (s)/(2) \right)) \int _{R ^{d} \setminus \overline{B} (x, r)} (f(x + z) - f(x))/(|z| ^{d} \left( |z| ^{2} - r ^{2} \right) ^{s/2}) dz

with the limit taken in

l{X} .

Quadratic Form Definition

In

l{X} = L ^{2} , the fractional Laplacian can be characterized via a quadratic form:\langle ( - \Delta) ^{s} f, \varphi \rangle = l{E}(f, \varphi)

where

l{E}(f, g) = (2 ^{s} \Gamma\left( (d + s)/(2) \right))/(2 \pi ^{d/2} \Gamma\left( - (s)/(2) \right)) \int _{R ^{d}} \int _{R ^{d}} ((f(y) - f(x)) (\overline{g(y))/(-) \overline{g(x)})}{|x - y| ^{d} } \, dx \, dy

Inverse of the Riesz Potential Definition

When

s < d and l{X} = L ^{p} for p \in [ 1 , (d)/(s) ) , the fractional Laplacian satisfies(\Gamma\left( (d - s)/(2) \right))/(2 ^{s} \pi ^{d/2} \Gamma\left( (s)/(2) \right)) \int _{R ^{d}} (( - \Delta) ^{s} f(x + z))/(|z| ^{d}) dz = f(x)

Harmonic Extension Definition

The fractional Laplacian can also be defined through harmonic extensions. Specifically, there exists a function

u(x, y) such that\begin{cases} \Delta _{x} u(x, y) + \alpha ^{2} c ^{2 / \alpha} _{\alpha} y ^{2} \partial ^{2} _{y} u(x, y) = 0 & for y > 0 , \\ u(x, 0 ) = f(x), \\ \partial _{y} u(x, 0 ) = - ( - \Delta) ^{s} f(x), \end{cases}

where

c _{\alpha} = 2 ^{-\alpha} (|\Gamma\left( - (\alpha)/(2) \right)|)/(\Gamma\left( (\alpha)/(2) \right)) and u( ⋅ , y) is a function in l{X} that depends continuously on y \in [ 0 , inf ty) with \|u( ⋅ , y)\| _{l{X} } bounded for all y \geq 0 .

References

1. Melcher . Christof . Sakellaris . Zisis N. . 2019-05-04 . Global dissipative half-harmonic flows into spheres: small data in critical Sobolev spaces . Communications in Partial Differential Equations . en . 44 . 5 . 397–415 . 1806.06818 . 10.1080/03605302.2018.1554675 . 0360-5302.
2. Wettstein . Jerome D. . 2023 . Half-harmonic gradient flow: aspects of a non-local geometric PDE . Mathematics in Engineering . en . 5 . 3 . 1–38 . 2112.08846 . 10.3934/mine.2023058 . 2640-3501.
3. Kwaśnicki . Mateusz . 2017 . Ten equivalent definitions of the fractional Laplace operator . Fractional Calculus and Applied Analysis . 20 . 1507.07356 . 10.1515/fca-2017-0002.

See also

• Fractional calculus
• Nonlocal operator
• Riemann-Liouville integral

External links

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