Fractional Laplacian - Misplaced Pages

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 and 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.

Let $p\in [1,\infty )$ and ${\mathcal {X}}:=L^{p}(\mathbb {R} ^{n})$ or let ${\mathcal {X}}:=C_{0}(\mathbb {R} ^{n})$ or ${\mathcal {X}}:=C_{bu}(\mathbb {R} ^{n})$ , where:

$C_{0}(\mathbb {R} ^{n})$ denotes the space of continuous functions $f:\mathbb {R} ^{n}\to \mathbb {R}$ that vanish at infinity, i.e., $\forall \varepsilon >0,\exists K\subset \mathbb {R} ^{n}$ compact such that $|f(x)|<\epsilon$ for all $x\notin K$ .
$C_{bu}(\mathbb {R} ^{n})$ denotes the space of bounded uniformly continuous functions $f:\mathbb {R} ^{n}\to \mathbb {R}$ , i.e., functions that are uniformly continuous, meaning $\forall \epsilon >0,\exists \delta >0$ such that $|f(x)-f(y)|<\epsilon$ for all $x,y\in \mathbb {R} ^{n}$ with $|x-y|<\delta$ , and bounded, meaning $\exists M>0$ such that $|f(x)|\leq M$ for all $x\in \mathbb {R} ^{n}$ .

Additionally, let $s\in (0,1)$ .

Fourier Definition

If we further restrict to $p\in$ , we get

(-\Delta )^{s}f:={\mathcal {F}}_{\xi }^{-1}(|\xi |^{2s}{\mathcal {F}}(f))

This definition uses the Fourier transform for $f\in L^{p}(\mathbb {R} ^{n})$ . This definition can also be broadened through the Bessel potential to all $p\in [1,\infty )$ .

Singular Operator

The Laplacian can also be viewed as a singular integral operator which is defined as the following limit taken in ${\mathcal {X}}$ .

(-\Delta )^{s}f(x)={\frac {4^{s}\Gamma ({\frac {d}{2}}+s)}{\pi ^{d/2}|\Gamma (-s)|}}\lim _{r\to 0^{+}}\int \limits _{\mathbb {R} ^{d}\setminus B_{r}(x)}{{\frac {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 $\{P_{t}\}_{t\in [0,\infty )}$ , we can define the fractional Laplacian through its generator.

$-(-\Delta )^{s}f(x)=\lim _{t\to 0^{+}}{\frac {P_{t}f-f}{t}}$

It is to note that the generator is not the fractional Laplacian $(-\Delta )^{s}$ but the negative of it $-(-\Delta )^{s}$ . The operator $P_{t}:{\mathcal {X}}\to {\mathcal {X}}$ is defined by

$P_{t}f:=p_{t}*f$ ,

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

Distributional Definition

For all Schwartz functions $\varphi$ , the fractional Laplacian can be defined in a distributional sense by

\int _{\mathbb {R} ^{d}}(-\Delta )^{s}f(y)\varphi (y)\,dy=\int _{\mathbb {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={\frac {1}{\Gamma (-{\frac {s}{2}})}}\int _{0}^{\infty }\left(e^{t\Delta }f-f\right)t^{-1-s/2}\,dt

where the integral is understood in the Bochner sense for ${\mathcal {X}}$ -valued functions.

Balakrishnan's Definition

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

(-\Delta )^{s}f={\frac {\sin \left({\frac {s\pi }{2}}\right)}{\pi }}\int _{0}^{\infty }(-\Delta )\left(sI-\Delta \right)^{-1}f\,s^{s/2-1}\,ds

with the integral interpreted as a Bochner integral for ${\mathcal {X}}$ -valued functions.

Dynkin's Definition

Another approach by Dynkin defines the fractional Laplacian as

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

with the limit taken in ${\mathcal {X}}$ .

Quadratic Form Definition

In ${\mathcal {X}}=L^{2}$ , the fractional Laplacian can be characterized via a quadratic form:

\langle (-\Delta )^{s}f,\varphi \rangle ={\mathcal {E}}(f,\varphi )

where

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

Inverse of the Riesz Potential Definition

When $s<d$ and ${\mathcal {X}}=L^{p}$ for $p\in [1,{\frac {d}{s}})$ , the fractional Laplacian satisfies

{\frac {\Gamma \left({\frac {d-s}{2}}\right)}{2^{s}\pi ^{d/2}\Gamma \left({\frac {s}{2}}\right)}}\int _{\mathbb {R} ^{d}}{\frac {(-\Delta )^{s}f(x+z)}{|z|^{d-s}}}\,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_{\alpha }^{2/\alpha }y^{2-2/\alpha }\partial _{y}^{2}u(x,y)=0&{\text{for }}y>0,\\u(x,0)=f(x),\\\partial _{y}u(x,0)=-(-\Delta )^{s}f(x),\end{cases}}

where $c_{\alpha }=2^{-\alpha }{\frac {|\Gamma \left(-{\frac {\alpha }{2}}\right)|}{\Gamma \left({\frac {\alpha }{2}}\right)}}$ and $u(\cdot ,y)$ is a function in ${\mathcal {X}}$ that depends continuously on $y\in [0,\infty )$ with $\|u(\cdot ,y)\|_{\mathcal {X}}$ bounded for all $y\geq 0$ .