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

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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  ,   and  .

Fourier Definition

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If we further restrict to  , we get

 

This definition uses the Fourier transform for  . This definition can also be broaden through the Bessel potential to all  .

Singular Operator

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

 

Generator of C_0-semigroup

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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 negativ of it  . The operator   is defined by

 ,

where   is the convolution of two functions and  .

See also

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References

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  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. 44 (5): 397–415. arXiv:1806.06818. doi:10.1080/03605302.2018.1554675. ISSN 0360-5302.
  2. ^ Wettstein, Jerome D. (2023). "Half-harmonic gradient flow: aspects of a non-local geometric PDE". Mathematics in Engineering. 5 (3): 1–38. arXiv:2112.08846. doi:10.3934/mine.2023058. ISSN 2640-3501.
  3. ^ Kwaśnicki, Mateusz (2017). "Ten equivalent definitions of the fractional Laplace operator". Fractional Calculus and Applied Analysis. 20. arXiv:1507.07356. doi:10.1515/fca-2017-0002.
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  • "Fractional Laplacian". Nonlocal Equations Wiki, Department of Mathematics, The University of Texas at Austin.