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
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