Fractional Laplacian

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 proofen by Kwaśnicki, M in.

Let $$p \in [1,\infty)$$, $$\mathcal{X} := L^p( \R^n)$$ and $$s \in (0,1)$$.

Fourier Definition
If we further restrict to $$p \in [1,2]$$, we get


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

This definition uses the Fourier transform for $$f \in L^p( \R^n)$$. This definition can also be broaden 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 $$\mathcal{X}$$.


 * $$ (-\Delta)^sf(x) = \frac{4^s\Gamma(d/2+s)}{\pi^{d/2} |\Gamma(-s)|} \lim_{r \to 0^+} \int\limits_{\mathbb{R}^d \setminus B_r (x) }{\frac{f(x)-f(y)}{|x-y|^{d+2s}}\,dy} $$

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.

$$ -(-\Delta)^sf(x) = \lim_{t \to 0^+} \frac{P_t f - f}{ t} $$

It is to note, that the generator is not the fractional Laplacian $$(-\Delta)^s$$ but the negativ 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 $$p_t := \mathcal{F}^{-1}_{ \xi}(e^{ - t | \xi|^{2s}})$$.