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