Kelvin transformation

A transformation of functions defined in domains of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 3 $, under which harmonic functions are transformed to harmonic functions. It was obtained by W. Thomson (Lord Kelvin, [1]).

If $ u $ is a harmonic function in a domain $ D \subset \mathbf R ^ {n} $, then its Kelvin transform is the function

$$ v ( y) = \ \left ( \frac{R}{| y | } \right ) ^ {n-} 2 u \left ( \frac{R ^ {2} }{| y | ^ {2} } y \right ) ,\ \ v ( \infty ) = 0 , $$

which is harmonic in the domain $ D ^ {*} $ obtained from $ D $ by inversion in the sphere $ S _ {R} = \{ {x } : {| x | = R } \} $, that is, by the mapping of $ \mathbf R ^ {n} $ defined by

$$ x \rightarrow y = \ \frac{R ^ {2} }{| x | ^ {2} } x ,\ \ 0 \rightarrow \infty , $$

where

$$ x = ( x _ {1} \dots x _ {n} ) ,\ \ | x | = ( x _ {1} ^ {2} + \dots + x _ {n} ^ {2} ) ^ {1/2} . $$

Under the inversion, the point at infinity $ \infty $ of the Aleksandrov compactification $ \overline{ {\mathbf R ^ {n} }}\; $ is taken to the origin $ 0 $ and vice versa. Under the Kelvin transformation, harmonic functions $ u $ in domains $ D $ containing $ \infty $ that are regular at $ \infty $, that is, are such that $ \lim\limits _ {| x | \rightarrow \infty } u ( x) = 0 $, are transformed to harmonic functions $ v $ in bounded domains $ D ^ {*} $ containing the origin $ 0 $, moreover, $ v ( 0) = 0 $. 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 $ \mathbf R ^ {n} $, $ n \geq 3 $, is preserved under analytic transformations of the form $ v ( y) = \phi ( y) u ( \psi ( y) ) $ only in the case when $ \phi ( y) \equiv 1 $ and $ \psi $ is a homothety, a translation or a symmetry with respect to a plane; for $ n = 2 $ the large class of conformal mappings $ \psi $ 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 $ n = 2 $ as well. In this case, harmonicity of $ u $ at infinity corresponds to boundedness of $ u $ at $ 0 $. 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