TOPICS
Search

Fractional Laplacian

The fractional Laplacian (-Delta)^s, with 0<s<1, is the nonlocal operator defined for suitably regular functions on R^n by

(-Delta)^sf(x)=C_(n,s)PVint_(R^n)(f(x)-f(y))/(||x-y||^(n+2s))dy,

where ||·|| is the Euclidean norm, PV denotes the Cauchy principal value, and C_(n,s)>0 is the normalization constant. With the Fourier transform convention using e^(-ix·xi), its multiplier is ||xi||^(2s). Thus it is a fractional power of the negative Laplacian.

When 2s<n, the Riesz potential of order 2s gives its inverse on suitable functions. The fractional Lane-Emden conjecture concerns a nonlinear system formed from this operator (Mei and Wei 2026).

See also

Fourier Transform, Fractional Lane-Emden Conjecture, Laplacian, Riesz Potential

Explore with Wolfram|Alpha

References

Mei, L. and Wei, J. "A Proof of the Fractional Lane-Emden Conjecture." 7 Oct 2026. https://arxiv.org/abs/2610.10188.

Cite this as:

Weisstein, Eric W. "Fractional Laplacian." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FractionalLaplacian.html

Subject classifications