Half-Laplacian has been discussed in this entry using harmonic extension. Today, we derive a more general form, called fractional laplacian denoted by , for a function
where the parameter
is a real number between 0 and 1 and
.
It is worth recalling the following fact known as Newtonian potential in theory of PDEs. The Newtonian potential of a compactly supported integrable function
, i.e.
, is defined as the convolution
where the Newtonian kernel in dimension
is defined by
.
Here is the volume of the unit in
. Coefficients in
, denoted by
, are usually called normalization constants. The Newtonian potential
of
is a solution of the Poisson equation
.
In case ,
is actually the Riesz potential
with where the Riesz potential is defined by
where is some normalization constant given by
.
Observe that at least formally the Riezs potentials verify the following rule
which implies
.
Thus if we set to be
(in fact, this is impossible) we get
which helps us to write down
.
Thus we wish to define the fractional Laplacian as follows
.
The above fractional Laplacian is also often called the Riesz fractional derivative [here]. In a paper entitled “An Extension Problem Related to the Fractional Laplacian” due to Luis Caffarelli et al. [here] published in Comm. Partial Differential Equations in 2007, the fractional Laplacian can also be defined using
.
Let us now study the terminology of weak solution to the following semilinear elliptic equation in the whole space
.
By a weak solution we mean a function such that
for any positive test function in the distribution sense. I will back to this stuff once I finish introducing the fractional Laplacian via pseudo-differential operators.
See also: Half-Laplacian in