Fractional Laplacian

For ${\displaystyle 0<s<1}$, the fractional Laplacian of order s ${\displaystyle (-\Delta )^{s}}$ can be defined on functions ${\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} }$ as a Fourier multiplier given by the formula

${\displaystyle {\mathcal {F}}((-\Delta )^{s}f))(\xi )=|\xi |^{2s}{\mathcal {F}}(f)(\xi )}$

where the Fourier transform ${\displaystyle {\mathcal {F}}(f)}$ of a function ${\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} }$ is given by

${\displaystyle {\mathcal {F}}(f)(\xi )=\int \limits _{\mathbb {R} ^{d}}{f(x)e^{-ix\cdot \xi }\,dx}.}$

More concretely, the fractional Laplacian can be written as a singular integral operator defined by

${\displaystyle (-\Delta )^{s}f(x)=c_{d,s}\int \limits _{\mathbb {R} ^{d}}{{\frac {f(x)-f(y)}{|x-y|^{d+2s}}}\,dy}}$

where ${\displaystyle c_{d,s}={\frac {4^{s}\Gamma (d/2+s)}{\pi ^{d/2}|\Gamma (-s)|}}}$. These two definitions, along with several other definitions,[1] are equivalent.

Some authors prefer to adopt the convention of defining the fractional Laplacian of order s as ${\displaystyle (-\Delta )^{s/2}}$ (as defined above), where now ${\displaystyle 0<s<2}$, so that the notion of order matches that of a (pseudo-)differential operator.

• Fractional calculus
• Nonlocal operator
• Riemann-Liouville integral

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