The fractional Laplacian operator

Question

Let $U$ an open set in $\bf R^n$, I define the fractional operator $\Delta^s$ with $s$ depending on $x \in U$:

$$\Delta^s =\lim_n \oplus_i \Delta^{s_i(n)} =\oplus_{x\in U} \Delta^{s(x)}$$

where $\Delta^{s_i(n)}$ is the fractional Laplacian on a hypercube of diameter $1/n$, with fraction, an approximation $s_i (n)$ of $s(x)$ on the hypercube.

Can we use of the fractional Laplacian operator with $s$ depending on $x$?

Answer 1

The solution is given by the integral form of the fractional Laplacian. We can see that $s$ can depend on $x$ if we use the formula :

$$\Delta^{-s} \Gamma (s)= \int_0^{\infty} t^{-s-1} e^{-t \Delta} dt$$