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A concept in [[Potential theory|potential theory]] that is an analogue of the physical concept of the potential energy of a system of electric charges. For points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356601.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356603.png" />, let
+
 +
{{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/e/e035/e035660/e0356604.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A concept in [[Potential theory|potential theory]] that is an analogue of the physical concept of the potential energy of a system of electric charges. For points  $  x = ( x _ {1} \dots x _ {n} ) $
 +
of a Euclidean space  $  \mathbf R  ^ {n} $,
 +
$  n \geq  2 $,
 +
let
  
be the fundamental solution of the Laplace equation and let
+
\[
 +
    H(|x|) = \left\{ \begin{array}{rl}
 +
      \ln\frac{1}{|x|} & \text{for } n = 2 \\
 +
      \frac{1}{|x|^{n-2}} & \text{for } n \geq 3,
 +
    \end{array} \right.
 +
\]
 +
be (up to dimensional constants) the fundamental solution of the Laplace equation and let
  
<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/e/e035/e035660/e0356605.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
U _  \mu  (x)  = \int\limits H ( | x - y | )  d \mu (y)
 +
$$
  
be the Newton (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356606.png" />) or logarithmic (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356607.png" />) potential of a Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356608.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356609.png" />.
+
be the Newton (for $  n \geq  3 $)  
 +
or logarithmic (for $  n = 2 $)  
 +
potential of a Borel measure $  \mu $
 +
on $  \mathbf R  ^ {n} $.
  
Restricting from now on to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566010.png" />, one defines the mutual energy of two non-negative measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566012.png" /> by
+
Restricting from now on to the case $  n \geq  3 $,  
 +
one defines the mutual energy of two non-negative measures $  \mu $
 +
and $  \nu $
 +
by
  
<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/e/e035/e035660/e03566013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
( \mu , \nu )  = \int\limits H ( | x - y | )  d \mu (x)  d \nu (y) =
 +
$$
  
<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/e/e035/e035660/e03566014.png" /></td> </tr></table>
+
$$
 +
= \
 +
\int\limits U _  \mu  (y)  d \nu (y)  = \int\limits U _  \nu  (x)  d \mu (x) .
 +
$$
  
Now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566015.png" />, but it can happen that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566016.png" />. The energy of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566017.png" /> is the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566019.png" />. For two measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566021.png" /> of arbitrary sign one can use the canonical decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566023.png" /> (or any decomposition of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566025.png" />) and, provided these four measures have finite energy, define the mutual energy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566027.png" /> by
+
Now $  ( \mu , \nu ) \geq  0 $,  
 +
but it can happen that $  ( \mu , \nu ) = + \infty $.  
 +
The energy of the measure $  \mu $
 +
is the number $  ( \mu , \mu ) $,
 +
0 \leq  ( \mu , \mu ) \leq  + \infty $.  
 +
For two measures $  \mu $,  
 +
$  \nu $
 +
of arbitrary sign one can use the canonical decomposition $  \mu = \mu  ^ {+} - \mu  ^ {-} $,  
 +
$  \nu = \nu  ^ {+} - \nu  ^ {-} $(
 +
or any decomposition of the form $  \mu = \mu _ {1} - \mu _ {2} $,
 +
$  \mu _ {1} , \mu _ {2} \geq  0 $)  
 +
and, provided these four measures have finite energy, define the mutual energy of $  \mu $
 +
and $  \nu $
 +
by
  
<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/e/e035/e035660/e03566028.png" /></td> </tr></table>
+
$$
 +
( \mu , \nu )  = ( \mu  ^ {+} , \nu  ^ {+} ) +
 +
( \mu  ^ {-} , \nu  ^ {-} ) - ( \mu  ^ {+} , \nu  ^ {-} ) -
 +
( \mu  ^ {-} , \nu  ^ {+} ) ,
 +
$$
  
 
which may turn out to be negative, but
 
which may turn out to be negative, but
  
<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/e/e035/e035660/e03566029.png" /></td> </tr></table>
+
$$
 +
( \mu , \mu )  \geq  ( \sqrt {( \mu  ^ {+} , \mu  ^ {+} ) } -
 +
\sqrt {( \mu  ^ {-} , \mu  ^ {-} ) } )  ^ {2}  \geq  0 .
 +
$$
  
The totality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566030.png" /> of all measures with finite energy can be made into a pre-Hilbert vector space with the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566031.png" /> and the energy norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566032.png" />. Here the Bunyakovskii–Cauchy–Schwarz inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566033.png" /> holds as well as the energy principle: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566035.png" />. H. Cartan has shown that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566036.png" /> is not complete, but the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566037.png" /> of non-negative measures is complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566038.png" />.
+
The totality $  {\mathcal E} $
 +
of all measures with finite energy can be made into a pre-Hilbert vector space with the scalar product $  ( \mu , \nu ) $
 +
and the energy norm $  \| \mu \| _ {e} = \sqrt {( \mu , \mu ) } $.  
 +
Here the Bunyakovskii–Cauchy–Schwarz inequality $  | ( \mu , \nu ) | \leq  \| \mu \| _ {e} \cdot \| \nu \| _ {e} $
 +
holds as well as the energy principle: If $  \| \mu \| _ {e} = 0 $,  
 +
then $  \mu = 0 $.  
 +
H. Cartan has shown that the space $  {\mathcal E} $
 +
is not complete, but the set $  {\mathcal E}  ^ {+} \subset  {\mathcal E} $
 +
of non-negative measures is complete in $  {\mathcal E} $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566039.png" /> be a compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566041.png" />. Among all probability measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566042.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566043.png" /> (that is, those for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566045.png" />) there is an extremal capacitary measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566046.png" /> with minimal energy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566047.png" />, which is connected with the capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566049.png" /> by the relation
+
Let $  K $
 +
be a compact set in $  \mathbf R  ^ {n} $,  
 +
$  n \geq  3 $.  
 +
Among all probability measures $  \lambda $
 +
on $  K $(
 +
that is, those for which $  \lambda \geq  0 $,  
 +
$  \lambda (K) = 1 $)  
 +
there is an extremal capacitary measure $  \lambda _ {0} $
 +
with minimal energy $  ( \lambda _ {0} , \lambda _ {0} ) $,  
 +
which is connected with the capacity $  C (K) $
 +
of $  K $
 +
by the relation
  
<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/e/e035/e035660/e03566050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
( \lambda _ {0} , \lambda _ {0} )  = \int\limits
 +
U _ {\lambda _ {0}  } (x)  d \lambda _ {0} (x) = \
  
If the potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566051.png" /> of a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566052.png" /> has a square-summable gradient, then
+
\frac{1}{C (K) }
 +
.
 +
$$
  
<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/e/e035/e035660/e03566053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
If the potential  $  U _  \mu  $
 +
of a measure  $  \mu \in {\mathcal E} $
 +
has a square-summable gradient, then
 +
 
 +
$$ \tag{5 }
 +
c (n) \| \mu \| _ {e}  =  \| U _  \mu  \| ,
 +
$$
  
 
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/e/e035/e035660/e03566054.png" /></td> </tr></table>
+
$$
 +
\| U _  \mu  \|  = \left ( \int\limits _ {\mathbf R  ^ {n} }
 +
{\rm grad}  ^ {2}  U _  \mu  (x)  d x \right )  ^ {1/2}
 +
$$
  
is the Dirichlet norm and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566056.png" />. In fact, (5) remains valid for any measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566057.png" />, and the Dirichlet norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566058.png" /> can be defined by an appropriate limit transition.
+
is the Dirichlet norm and $  c (n) = ( n - 2 ) 2 \pi  ^ {n/2} / \Gamma ( n / 2 ) $,  
 +
$  n \geq  3 $.  
 +
In fact, (5) remains valid for any measure $  \mu \in {\mathcal E} $,  
 +
and the Dirichlet norm $  \| U _  \mu  \| $
 +
can be defined by an appropriate limit transition.
  
In the case of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566059.png" />, a direct application of (3) with the logarithmic potential (2) for the definition of the energy of measures is not possible because of the singular behaviour of the logarithmic kernel (1) at infinity. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566060.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566062.png" />, admitting a Green function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566063.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566064.png" /> be a Borel measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566065.png" />. When one applies Green potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566067.png" /> of the form
+
In the case of the plane $  \mathbf R  ^ {2} $,  
 +
a direct application of (3) with the logarithmic potential (2) for the definition of the energy of measures is not possible because of the singular behaviour of the logarithmic kernel (1) at infinity. Let $  \Omega $
 +
be a bounded domain in $  \mathbf R  ^ {n} $,  
 +
$  n \geq  2 $,  
 +
admitting a Green function $  g ( x , y ) $,  
 +
and let $  \mu $
 +
be a Borel measure on $  \Omega $.  
 +
When one applies Green potentials $  G _  \mu  $
 +
and $  G _  \nu  $
 +
of the form
  
<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/e/e035/e035660/e03566068.png" /></td> </tr></table>
+
$$
 +
G _  \mu  (x)  = \int\limits g ( x , y )  d \mu (y)
 +
$$
  
instead of Newton potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566070.png" /> in (3), one obtains for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566071.png" /> a definition of the energy of measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566072.png" /> that is equivalent to the one given above, but which turns out to be suitable also for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566073.png" />, with preservation of all properties described above (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e03566074.png" />).
+
instead of Newton potentials $  U _  \mu  $
 +
and $  U _  \nu  $
 +
in (3), one obtains for $  n \geq  3 $
 +
a definition of the energy of measures on $  \Omega $
 +
that is equivalent to the one given above, but which turns out to be suitable also for $  n = 2 $,  
 +
with preservation of all properties described above (and $  c (2) = 2 \pi $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Wermer,  "Potential theory" , ''Lect. notes in math.'' , '''408''' , Springer  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Wermer,  "Potential theory" , ''Lect. notes in math.'' , '''408''' , Springer  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR></table>

Latest revision as of 12:43, 17 March 2020


A concept in potential theory that is an analogue of the physical concept of the potential energy of a system of electric charges. For points $ x = ( x _ {1} \dots x _ {n} ) $ of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, let

\[ H(|x|) = \left\{ \begin{array}{rl} \ln\frac{1}{|x|} & \text{for } n = 2 \\ \frac{1}{|x|^{n-2}} & \text{for } n \geq 3, \end{array} \right. \] be (up to dimensional constants) the fundamental solution of the Laplace equation and let

$$ \tag{2 } U _ \mu (x) = \int\limits H ( | x - y | ) d \mu (y) $$

be the Newton (for $ n \geq 3 $) or logarithmic (for $ n = 2 $) potential of a Borel measure $ \mu $ on $ \mathbf R ^ {n} $.

Restricting from now on to the case $ n \geq 3 $, one defines the mutual energy of two non-negative measures $ \mu $ and $ \nu $ by

$$ \tag{3 } ( \mu , \nu ) = \int\limits H ( | x - y | ) d \mu (x) d \nu (y) = $$

$$ = \ \int\limits U _ \mu (y) d \nu (y) = \int\limits U _ \nu (x) d \mu (x) . $$

Now $ ( \mu , \nu ) \geq 0 $, but it can happen that $ ( \mu , \nu ) = + \infty $. The energy of the measure $ \mu $ is the number $ ( \mu , \mu ) $, $ 0 \leq ( \mu , \mu ) \leq + \infty $. For two measures $ \mu $, $ \nu $ of arbitrary sign one can use the canonical decomposition $ \mu = \mu ^ {+} - \mu ^ {-} $, $ \nu = \nu ^ {+} - \nu ^ {-} $( or any decomposition of the form $ \mu = \mu _ {1} - \mu _ {2} $, $ \mu _ {1} , \mu _ {2} \geq 0 $) and, provided these four measures have finite energy, define the mutual energy of $ \mu $ and $ \nu $ by

$$ ( \mu , \nu ) = ( \mu ^ {+} , \nu ^ {+} ) + ( \mu ^ {-} , \nu ^ {-} ) - ( \mu ^ {+} , \nu ^ {-} ) - ( \mu ^ {-} , \nu ^ {+} ) , $$

which may turn out to be negative, but

$$ ( \mu , \mu ) \geq ( \sqrt {( \mu ^ {+} , \mu ^ {+} ) } - \sqrt {( \mu ^ {-} , \mu ^ {-} ) } ) ^ {2} \geq 0 . $$

The totality $ {\mathcal E} $ of all measures with finite energy can be made into a pre-Hilbert vector space with the scalar product $ ( \mu , \nu ) $ and the energy norm $ \| \mu \| _ {e} = \sqrt {( \mu , \mu ) } $. Here the Bunyakovskii–Cauchy–Schwarz inequality $ | ( \mu , \nu ) | \leq \| \mu \| _ {e} \cdot \| \nu \| _ {e} $ holds as well as the energy principle: If $ \| \mu \| _ {e} = 0 $, then $ \mu = 0 $. H. Cartan has shown that the space $ {\mathcal E} $ is not complete, but the set $ {\mathcal E} ^ {+} \subset {\mathcal E} $ of non-negative measures is complete in $ {\mathcal E} $.

Let $ K $ be a compact set in $ \mathbf R ^ {n} $, $ n \geq 3 $. Among all probability measures $ \lambda $ on $ K $( that is, those for which $ \lambda \geq 0 $, $ \lambda (K) = 1 $) there is an extremal capacitary measure $ \lambda _ {0} $ with minimal energy $ ( \lambda _ {0} , \lambda _ {0} ) $, which is connected with the capacity $ C (K) $ of $ K $ by the relation

$$ \tag{4 } ( \lambda _ {0} , \lambda _ {0} ) = \int\limits U _ {\lambda _ {0} } (x) d \lambda _ {0} (x) = \ \frac{1}{C (K) } . $$

If the potential $ U _ \mu $ of a measure $ \mu \in {\mathcal E} $ has a square-summable gradient, then

$$ \tag{5 } c (n) \| \mu \| _ {e} = \| U _ \mu \| , $$

where

$$ \| U _ \mu \| = \left ( \int\limits _ {\mathbf R ^ {n} } {\rm grad} ^ {2} U _ \mu (x) d x \right ) ^ {1/2} $$

is the Dirichlet norm and $ c (n) = ( n - 2 ) 2 \pi ^ {n/2} / \Gamma ( n / 2 ) $, $ n \geq 3 $. In fact, (5) remains valid for any measure $ \mu \in {\mathcal E} $, and the Dirichlet norm $ \| U _ \mu \| $ can be defined by an appropriate limit transition.

In the case of the plane $ \mathbf R ^ {2} $, a direct application of (3) with the logarithmic potential (2) for the definition of the energy of measures is not possible because of the singular behaviour of the logarithmic kernel (1) at infinity. Let $ \Omega $ be a bounded domain in $ \mathbf R ^ {n} $, $ n \geq 2 $, admitting a Green function $ g ( x , y ) $, and let $ \mu $ be a Borel measure on $ \Omega $. When one applies Green potentials $ G _ \mu $ and $ G _ \nu $ of the form

$$ G _ \mu (x) = \int\limits g ( x , y ) d \mu (y) $$

instead of Newton potentials $ U _ \mu $ and $ U _ \nu $ in (3), one obtains for $ n \geq 3 $ a definition of the energy of measures on $ \Omega $ that is equivalent to the one given above, but which turns out to be suitable also for $ n = 2 $, with preservation of all properties described above (and $ c (2) = 2 \pi $).

References

[1] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
[2] J. Wermer, "Potential theory" , Lect. notes in math. , 408 , Springer (1974)
[3] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
How to Cite This Entry:
Energy of measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Energy_of_measures&oldid=16347
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article