I found the following interesting identity which is similar to what I have showed recent days [here]. In that entry, we showed that
where and are connected by
.
For we denote by the following
.
Then we show that
Lemma.
.
I found the following interesting identity which is similar to what I have showed recent days [here]. In that entry, we showed that
where and are connected by
.
For we denote by the following
.
Then we show that
Lemma.
.
This short note is to prove the following
where and are connected by
.
The proof is straightforward as follows.
We see that
Similarly, we get
In this topic, we consider the Kelvin transform for Laplaction operators. Precisely, what we get is the following
.
We now consider a different situation. The detail can be found in the following paper due to X.W. Xu published in Proc. Roy. Soc. Edinburgh Sect. A, 2000.
Theorem. If is a sufficiently good function then satisfies the equation
where is defined to be
.
For each point , denote and
is the inversion of with respect to the unit sphere. We have the following identities
and
.
Thus,
.
Next, think of as a system of orthogonal curvilinear coordinates for , we deduce that the metric tensor of the Euclidean space in curvilinear coordinates
.
This implies that the so-called Lame coefficients is
.
In this topic, we proved a very interesting property involving the Laplacian of the Kelvin transform of a function. Recall that, for a given function , its Kelvin transform is defined to be
.
We then have
where the inversion point of is defined to be
.
Therefore, the Kelvin transform can be defined to be
.
We also have another formula
.
The right hand side of the above identity involves , we can rewrite this one in terms of . Actually, we have the following
which gives
.
Today, we study the a more general form of the Kelvin transform. The above definition of the Kelvin transform is with respect to the origin and unit ball in . We are now interested in the case when the Kelvin transform is defined with respect to a fixed ball. The following result is adapted from a paper due to M.C. Leung published in Math. Ann. in 2003.
Define the reflection on the sphere with center at and radius by
.
It is direct to check that
.
We observe that
.
The Kelvin transform with center at and radius of a function is given by
.
We remark that if and are fixed, and if , then as well, where . In addition, sends a set of small diameter not too close to to a set of small diameter. We verify that double Kelvin transform is the identity map. We have