Fractional Laplacian in \(\mathbb R^d\)
Fractional Laplacian
\[
(-\Delta)^s u (x) = \frac{C(d,s)}2 \int_{\mathbb R^d } \frac{2u(x) - u(x+y) - u(x-y)}{|y|^{d+2s}} d y.
\]
The popular choice is \[
{C(d,s)} = \left( \int_{\mathbb R^d} \frac{1 - \cos \omega_1}{|\omega|^{d+2s}} d \omega\right)^{-1}
\]
Equivalent definitions (up-to ten (Kwasnicki, 2017))
The unique operator such that
\[
\mathcal F[ (-\Delta)^s u ] = |\xi|^{2s} \mathcal F[u]
\]
i.e. the spectral fractional power of \(-\Delta\).
The infinitesimal generator of a Lévy process \(X_h\), characterised by “long jumps”
\[
(-\Delta)^s u (x) = \lim_{h \to 0} \frac{ \mathbb E[ f(x) - f(x - X_h) ] }{ h }
\]
The Riesz potential
\[
\mathcal I_{\alpha} [u] (x) = C(d, \alpha) \int_{\mathbb R^d} \frac{u(y)}{|x-y|^{d-\alpha}} dy.
\]
It is defined for \(0 < \alpha < d\).
It is known that with the right constant \[
\mathcal F[ \mathcal I_{\alpha} u ] = |\xi|^{-\alpha} \mathcal F[u]
\]
Thus, \[
\mathcal I_{2s} = \Big( (-\Delta)^s \Big)^{-1}
\]
All of the above works on tempered functions, in particular \[
u(x) \to 0, \qquad \text{ as } |x| \to \infty.
\]
Laplace equation
\[
\begin{cases}
(-\Delta)^s u (x) = f(x) & \textrm{for all } x \in \mathbb R^d \\
u(x) \to 0 & \textrm{as } |x| \to \infty
\end{cases}
\]
Then when \(d > 2s\) we can recover through the Riesz potential \[
u(x) = C(d,s) \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-2s}} dy.
\]
For any \(d, 2s\) we can also find this solution by the minisation of the energy functional
\[
J(u) = \frac 1 2 \int_{\mathbb R^d} \int_{\mathbb R^d } \frac{|u(x) - u(y)|^2}{|x-y|^{d+2s}} dx \, dy - \int_\Omega f u
\]
The principal value
Notice that \[\begin{align*}
\int_{ \mathbb R^d \setminus B_{\varepsilon} (0) } &\frac{ 2u(x) - u(x+y) - u(x-y) }{|y|^{d+2s}} dy \\
&=
2\int_{ \mathbb R^d \setminus B_{\varepsilon} (0) } \frac{ u(x) - u(y)}{|x-y|^{d+2s}} dy \\
\end{align*}\]
Thus, it makes sense to define \[\begin{equation*}
\mathrm{P.V.}\int_{ \mathbb R^d } = \lim_{\varepsilon \to 0} \int_{ \mathbb R^d \setminus B_{\varepsilon} (0) }
\end{equation*}\]
Numerics
There are both Finite Difference schemes \[
(-\Delta)^s_h u(x) = \sum_{i \in \mathbb Z^d} \omega_{i-j} (u(x) - u(x+ih))
\]
Finite Element schemes are also possible. The un-bounded domain is tricky.
Fractional LaplacianS
in bounded domains
Restricted fractional Laplacian
Singular integral operator: \[
(-\Delta)^s_{\mathrm {RFL}} u (x)= C(d,s) \, \mathrm{P.V.}\int_{ \mathbb R^d } \frac{u(x) - u(y)}{|x-y|^{d+2s}} \; dy.
\]
If we work only in \(\Omega\), we must prescribe \(u\) in \(\Omega^c = \mathbb R^d \setminus \Omega\).
Spectral fractional Laplacian
Operational power. \[
-\Delta \varphi_m = \lambda_m \varphi_m \textrm{ in } \Omega, \qquad
\varphi_m = 0 \textrm{ on } \partial \Omega.
\]
one defines \[
u(x) = \sum_{m=1}^{+\infty} {u_m} \varphi_m(x) \qquad \longmapsto \qquad
(-\Delta)_{\mathrm{SFL}}^s u (x) = \sum_{m=1}^{+\infty} \lambda_m^s u_m \varphi_m (x).
\]
The “boundary condition” is \(u = 0\) on \(\partial \Omega\).
Censored fractional Laplacian (CFL)
For \(s > \frac 1 2\) \[\begin{equation}
\tag{CFL}
(-\Delta)^s_{\mathrm{CFL}} u (x) = C(d,s) \, \mathrm{P.V.} \int_{ \Omega } \frac{u(x) - u(y)}{|x-y|^{n+2s}} \; dy,
\end{equation}\]
We do not integrate over \(\Omega^c\), so it makes sense to pick simply \(u=0\) on \(\partial\Omega.\)
Laplace equation
\[
\begin{dcases}
\mathcal L u = f & \Omega \\
u = 0 & \partial \Omega \text{ or } \mathbb R^d \setminus \overline \Omega \\
\end{dcases}
\]
We observe
\(\mathcal L = (-\Delta)^s_{\mathrm{RFL}}, (-\Delta)^s_{\mathrm{CFL}}\) are sub-differentials of energies \[
J(u) = \int_A \int_A \frac{|u(x) - u(y)|^2 }{|x-y|^{d+2s}} dx \, dy.
\]
\(\mathcal L = (-\Delta)^s_{\mathrm{SFL}}\) is just a power.
The inverse is naturally \((-\Delta_\Omega)^{-s}\), and in works between “powers” of \(H_0^1(\Omega)\).
Self-adjoint compact operators. Furthermore \(\lambda_1 > 0\).
Nice theory of energy solutions.
Green kernels
The solution of the Laplace equation \[
\begin{dcases}
\mathcal L u = f & \Omega \\
u = 0 & \partial \Omega \text{ or } \mathbb R^d \setminus \overline \Omega \\
\end{dcases}
\]
Allows us to define \(\mathrm G : L^2 (\Omega) \to L^2(\Omega)\) we can represent it by a kernel \[
u(x) = \mathrm G[f] (x) = \int_\Omega \mathbb G(x,y) f(y) dy.
\]
The probabilistic approach provides in each of our cases a similar shape \[
\mathbb G(x,y) \asymp \frac{1}{|x-y|^{d-2s}} \left(1 \wedge \frac{\delta(x)}{|x-y|} \right)^\gamma\left(1 \wedge \frac{\delta(y)}{|x-y|} \right)^\gamma
\]
Since we deal only with self-adjoint \(\mathcal L\), then \(\mathbb G(x,y) = \mathbb G(y,x)\).
The probabilistic approach provides in each of our cases a similar shape \[
\mathbb G(x,y) \asymp \frac{1}{|x-y|^{d-2s}} \left(1 \wedge \frac{\delta(x)}{|x-y|} \right)^\gamma\left(1 \wedge \frac{\delta(y)}{|x-y|} \right)^\gamma
\]
And \(\gamma \in (0,1]\) depends on the setting
Numerics
Finite Differences
For the RFL and CFL we can re-use the weights of the whole space.
For the SFL: Make \(A\) numerical matrix for \(-\Delta\) problem and take \(A^s\): \[
\textrm{e.g. }
A =
\begin{pmatrix}
2 & -1 & \\
-1 & 2 & - 1 \\
& \ddots & \\
&&-1&2
\end{pmatrix}
\]
