Kelvin transform

The Kelvin transform is a device used in classical potential theory to extend the concept of a harmonic function, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of subharmonic and superharmonic functions.

In order to define the Kelvin transform {{itco|f}}^* of a function f, it is necessary to first consider the concept of inversion in a sphere in Rⁿ as follows.

It is possible to use inversion in any sphere, but the ideas are clearest when considering a sphere with centre at the origin.

Given a fixed sphere {{nowrap|S(0, R)}} with centre 0 and radius R, the inversion of a point x in Rⁿ is defined to be $$x^*\; =\; \backslash frac\{R^2\}\{|x|^2\}\; x.$$

A useful effect of this inversion is that the origin 0 is the image of $\backslash infty$, and $\backslash infty$ is the image of 0. Under this inversion, spheres are transformed into spheres, and the exterior of a sphere is transformed to the interior, and vice versa.

The Kelvin transform of a function is then defined by:

If D is an open subset of Rⁿ which does not contain 0, then for any function f defined on D, the Kelvin transform {{itco|f}}^* of f with respect to the sphere {{nowrap|S(0, R)}} is

$$f^*(x^*)\; =\; \backslash frac\{|x|^\{n-2\}\}\{R^\{2n-4\}\}f(x)\; =\; \backslash frac\{1\}\{|x^*|^\{n-2\}\}f(x)\; =\; \backslash frac\{1\}\{|x^*|^\{n-2\}\}\; f\backslash left(\backslash frac\{R^2\}\{|x^*|^2\}\; x^*\backslash right).$$

One of the important properties of the Kelvin transform, and the main reason behind its creation, is the following result:

: Let D be an open subset in Rⁿ which does not contain the origin 0. Then a function u is harmonic, subharmonic or superharmonic in D if and only if the Kelvin transform u^* with respect to the sphere {{nowrap|S(0, R)}} is harmonic, subharmonic or superharmonic in D^*.

This follows from the formula

$$\backslash Delta\; u^*(x^*)\; =\; \backslash frac\{R^\{4\}\}\{|x^*|^\{n+2\}\}(\backslash Delta\; u)\backslash left(\backslash frac\{R^2\}\{|x^*|^2\}\; x^*\backslash right).$$