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^* = \frac{R^2}{|x|^2} x.

A useful effect of this inversion is that the origin 0 is the image of \infty, and \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^*) = \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 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

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

See also

References

  • William Thomson, Lord Kelvin (1845) "Extrait d'une lettre de M. William Thomson à M. Liouville", Journal de Mathématiques Pures et Appliquées 10: 364–7
  • William Thompson (1847) "Extraits deux lettres adressees à M. Liouville, par M. William Thomson", Journal de Mathématiques Pures et Appliquées 12: 556–64
  • {{cite book | author = J. L. Doob | author-link = Joseph Leo Doob | title = Classical Potential Theory and Its Probabilistic Counterpart |page=26| publisher = Springer-Verlag | year = 2001 | isbn=3-540-41206-9 }}
  • {{cite book | author = L. L. Helms | title = Introduction to potential theory | publisher = R. E. Krieger | year = 1975 | isbn=0-88275-224-3 }}
  • {{cite book | author = O. D. Kellogg | author-link = Oliver Dimon Kellogg | title = Foundations of potential theory | publisher = Dover | year = 1953 | isbn=0-486-60144-7 }}
  • John Wermer (1981) Potential Theory 2nd edition, page 84, Lecture Notes in Mathematics #408 {{ISBN|3-540-10276-0}}

Category:Harmonic functions

Category:Transforms

Transform