Kelvin transformation

A transformation of functions defined in domains of a Euclidean space , , under which harmonic functions are transformed to harmonic functions. It was obtained by W. Thomson (Lord Kelvin, [1]).

If is a harmonic function in a domain , then its Kelvin transform is the function

which is harmonic in the domain obtained from by inversion in the sphere , that is, by the mapping of defined by

where

Under the inversion, the point at infinity of the Aleksandrov compactification is taken to the origin and vice versa. Under the Kelvin transformation, harmonic functions in domains containing that are regular at , that is, are such that , are transformed to harmonic functions in bounded domains containing the origin , moreover, . Because of this property, the Kelvin transformation enables one to reduce exterior problems in potential theory to interior ones and vice versa (see [2], [3]).

Apart from under Kelvin transformation, harmonicity of functions in , , is preserved under analytic transformations of the form only in the case when and is a homothety, a translation or a symmetry with respect to a plane; for the large class of conformal mappings has this property.

References

[1]	W. Thomson, "Extraits de deux letters adressées à M. Liouville" J. Math. Pures Appl. , 12 (1847) pp. 256–264
[2]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) pp. Chapt. 5 (Translated from Russian)
[3]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)

Comments

These results hold for as well. In this case, harmonicity of at infinity corresponds to boundedness of at . See, e.g., [a1] or [a2].

References

[a1]	L.L. Helms, "Introduction to potential theory" , Wiley (1969) (Translated from German)
[a2]	J. Wermer, "Potential theory" , Lect. notes in math. , 408 , Springer (1974)
[a3]	O.D. Kellogg, "Foundations of potential theory" , Dover, reprint (1954) (Re-issue: Springer, 1967)
[a4]	J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390