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A transformation of functions defined in domains of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k0551901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k0551902.png" />, under which harmonic functions are transformed to harmonic functions. It was obtained by W. Thomson (Lord Kelvin, [[#References|[1]]]).
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k0551903.png" /> is a [[Harmonic function|harmonic function]] in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k0551904.png" />, then its Kelvin transform is the function
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{{TEX|auto}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k0551905.png" /></td> </tr></table>
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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, [[#References|[1]]]).
  
which is harmonic in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k0551906.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k0551907.png" /> by inversion in the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k0551908.png" />, that is, by the mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k0551909.png" /> defined by
+
If  $  u $
 +
is a [[Harmonic function|harmonic function]] in a domain $  D \subset  \mathbf R  ^ {n} $,  
 +
then its Kelvin transform is the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519010.png" /></td> </tr></table>
+
$$
 +
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
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519011.png" /></td> </tr></table>
+
$$
 +
= ( x _ {1} \dots x _ {n} ) ,\ \
 +
| x |  = ( x _ {1}  ^ {2} + \dots + x _ {n}  ^ {2} )  ^ {1/2} .
 +
$$
  
Under the inversion, the point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519012.png" /> of the [[Aleksandrov compactification|Aleksandrov compactification]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519013.png" /> is taken to the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519014.png" /> and vice versa. Under the Kelvin transformation, harmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519015.png" /> in domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519016.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519017.png" /> that are regular at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519018.png" />, that is, are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519019.png" />, are transformed to harmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519020.png" /> in bounded domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519021.png" /> containing the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519022.png" />, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519023.png" />. Because of this property, the Kelvin transformation enables one to reduce exterior problems in potential theory to interior ones and vice versa (see [[#References|[2]]], [[#References|[3]]]).
+
Under the inversion, the point at infinity $  \infty $
 +
of the [[Aleksandrov compactification|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 [[#References|[2]]], [[#References|[3]]]).
  
Apart from under Kelvin transformation, harmonicity of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519025.png" />, is preserved under analytic transformations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519026.png" /> only in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519028.png" /> is a [[Homothety|homothety]], a translation or a symmetry with respect to a plane; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519029.png" /> the large class of conformal mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519030.png" /> has this property.
+
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|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Thomson,  "Extraits de deux letters adressées à M. Liouville"  ''J. Math. Pures Appl.'' , '''12'''  (1847)  pp. 256–264</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  pp. Chapt. 5  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Thomson,  "Extraits de deux letters adressées à M. Liouville"  ''J. Math. Pures Appl.'' , '''12'''  (1847)  pp. 256–264</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  pp. Chapt. 5  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
These results hold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519031.png" /> as well. In this case, harmonicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519032.png" /> at infinity corresponds to boundedness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519033.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055190/k05519034.png" />. See, e.g., [[#References|[a1]]] or [[#References|[a2]]].
+
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., [[#References|[a1]]] or [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.L. Helms,  "Introduction to potential theory" , Wiley  (1969)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Wermer,  "Potential theory" , ''Lect. notes in math.'' , '''408''' , Springer  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O.D. Kellogg,  "Foundations of potential theory" , Dover, reprint  (1954)  (Re-issue: Springer, 1967)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Doob,  "Classical potential theory and its probabilistic counterpart" , Springer  (1984)  pp. 390</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.L. Helms,  "Introduction to potential theory" , Wiley  (1969)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Wermer,  "Potential theory" , ''Lect. notes in math.'' , '''408''' , Springer  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O.D. Kellogg,  "Foundations of potential theory" , Dover, reprint  (1954)  (Re-issue: Springer, 1967)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Doob,  "Classical potential theory and its probabilistic counterpart" , Springer  (1984)  pp. 390</TD></TR></table>

Revision as of 22:14, 5 June 2020


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
How to Cite This Entry:
Kelvin transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kelvin_transformation&oldid=15293
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article