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Kelvin transform

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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 f of a function f, it is necessary to first consider the concept of inversion in a sphere in R 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 S(0, R) with centre 0 and radius R, the inversion of a point x in R is defined to be x ∗ = R 2 | x | 2 x . {\displaystyle x^{*}={\frac {R^{2}}{|x|^{2}}}x.}

A useful effect of this inversion is that the origin 0 is the image of ∞ {\displaystyle \infty } , and ∞ {\displaystyle \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 R which does not contain 0, then for any function f defined on D, the Kelvin transform f of f with respect to the sphere S(0, R) is f ∗ ( x ∗ ) = | x | n − 2 R 2 n − 4 f ( x ) = 1 | x ∗ | n − 2 f ( x ) = 1 | x ∗ | n − 2 f ( R 2 | x ∗ | 2 x ∗ ) . {\displaystyle f^{*}(x^{*})={\frac {|x|^{n-2}}{R^{2n-4}}}f(x)={\frac {1}{|x^{*}|^{n-2}}}f(x)={\frac {1}{|x^{*}|^{n-2}}}f\left({\frac {R^{2}}{|x^{*}|^{2}}}x^{*}\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 R 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 S(0, R) is harmonic, subharmonic or superharmonic in D.

This follows from the formula Δ u ∗ ( x ∗ ) = R 4 | x ∗ | n + 2 ( Δ u ) ( R 2 | x ∗ | 2 x ∗ ) . {\displaystyle \Delta u^{*}(x^{*})={\frac {R^{4}}{|x^{*}|^{n+2}}}(\Delta u)\left({\frac {R^{2}}{|x^{*}|^{2}}}x^{*}\right).}

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