Some examples of Fractional Laplacian on $\mathbb{R}^{2}$ : $(-\Delta)^{\alpha/2}u(x,y)$

Question

It is known that the fractional Laplacian of $u$ on $\mathbb{R}^2$ is defined as \begin{align}\label{fraclap2dim} (-\Delta)^{\alpha/2}u(x,y)=-c_{\alpha}\int \int_{\mathbb{R}^2}\dfrac{(p,q)\cdot \nabla u((x,y)+(p,q))}{||(p,q)||^{2+\alpha}}dp dq, \end{align} where $\alpha\in(0,2)$ and $c_{\alpha}=\dfrac{2^{\alpha-1}\Gamma(\alpha/2+1)}{\pi\Gamma(1-\alpha/2)}.$

I am trying to find the solution for some functions in the literature. I know that for the one-dimensional case, for different values of $\alpha$, they do exist, but for the two-dimensional case I couldn't find them.

Thank you very much in advance.

Answer 1

First of all (and maybe this is just me being silly), I've never seen this definition of the fractional Laplacian. The usual one, as can be found on the wiki page, is given by $$\tag{$\ast$}(-\Delta)^{\alpha/2}u(x) = c_{n,\alpha} \operatorname{PV} \int_{\mathbb R^n} \frac{u(x)-u(y)}{\vert x - y \vert^{n+\alpha}} \, dy $$ with $c_{n,\alpha} = \frac{\alpha 2^\alpha \Gamma \big( \frac{n+\alpha}2\big) }{\pi^{n/2}\Gamma (1-\alpha/2)}$. Even this paper which is devoted to several definitions of the fractional Laplacian doesn't mention this definition.

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Anyways, assuming that either your definition is equivalent to ($\ast$), or you've made some typo and actually meant $(\ast)$, let me answer your question. In general, doing computations with the fractional Laplacian is challenging, but there are definitely many known formulas. As you are only interested in $\mathbb R^2$, I will focus on this case, but many of these formulas hold in $\mathbb R^n$,

• Constant and linear functions: If $u(x,y) = \text{const.}$ then $(-\Delta)^{\alpha /2}u(x,y) = 0 $ and if $u(x,y) = ax+by$ then $(-\Delta)^{\alpha /2}u(x,y) = 0 $ provided that $\alpha >1$ (you need this assumption due to integrability issues).

• The fundament solution: if $u(x,y) = (x^2+y^2)^{\frac{\alpha-2}{2}}$ then $$(-\Delta)^{\alpha /2}u(x,y) = 0 $$ for all $(x,y) \neq (0,0)$. See Theorem 8.4 here

• Torsion function: If $u(x,y) = \gamma_{2,\alpha} (1-x^2-y^2)^{\alpha/2}_+$ with $\tau_+= \max \{ \tau , 0\}$ and $\gamma_{2,\alpha}$ a constant then $$ (-\Delta)^{\alpha/2}u(x,y) = 1 \qquad \text{if } x^2+y^2<1.$$ See Proposition 13.1 here. In fact, this holds more generally with ellipsoids, see this paper.

• 1-D functions: If $u(x,y)$ is a 1-D function, i.e. $u(x,y) = v(x)$ say then $$(-\Delta)^{\alpha/2}u(x,y) = C (-\Delta)^{\alpha/2}_{\mathbb R}v(x) $$ for some constant $C$, see Lemma 2.1. Hence, any formulas you know in 1D can be carried across to 2D. In particular, since $(-\Delta)^{\alpha/2}_{\mathbb R} \tau^{\alpha/2}_+=0$ for $\tau >0$, this implies that if $u(x,y) = x^{\alpha/2}_+$ then $(-\Delta)^{\alpha/2}u(x,y) =0$ for $x>0$. We also have by homogeneity that $(-\Delta)^{\alpha}\tau^\beta_+ = C\tau^{\beta-\alpha}$ for $\tau>0$ which gives you another solution.

• Radial functions: There is a formula for the fractional Laplacian of radial functions, see Section 7.

• Greens/Poisson kernel representations: There are well known Greens/Poisson kernel representations in a ball, so this gives you explicit solutions in terms of integrals, see Section 15.

• Special functions: the fractional Laplacian of Generalised hypergeometric functions and the more general Meijer G function is known, see this article. This gives you thousands of explicit examples, provided that you are comfortable working with special functions (which is pretty much unavoidable when dealing with the fractional Laplacian). For example, using that $e^{-x^2-y^2}={}_1F_1(1;1;-x^2-y^2)$ and Corollary 2, gives that $$(-\Delta)^{\alpha/2} e^{-x^2-y^2} = 2^\alpha \Gamma(1+\alpha/2){}_1F_1(1+\alpha/2;1;-x^2-y^2) $$ for all $(x,y)\in \mathbb R^2$.

These are all of the formula that I can think up off the top of my head, but I'll try update this if I think of more.