Namespaces
Variants
Actions

Difference between revisions of "Capacity"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
c0202801.png
 +
$#A+1 = 216 n = 0
 +
$#C+1 = 216 : ~/encyclopedia/old_files/data/C020/C.0200280 Capacity
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''of a set''
 
''of a set''
  
 
A set function arising in [[Potential theory|potential theory]] as the analogue of the physical concept of the electrostatic capacity.
 
A set function arising in [[Potential theory|potential theory]] as the analogue of the physical concept of the electrostatic capacity.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c0202801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c0202802.png" /> be two smooth closed hypersurfaces in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c0202803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c0202804.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c0202805.png" /> enclosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c0202806.png" />. Such a system is called a condenser <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c0202807.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c0202808.png" /> be the harmonic function in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c0202809.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028011.png" /> taking the value 1 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028014.png" />. The condenser capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028015.png" /> is the number
+
Let $  S $
 +
and $  S  ^ {*} $
 +
be two smooth closed hypersurfaces in a Euclidean space $  \mathbf R  ^ {n} $,  
 +
$  n \geq  3 $,  
 +
with $  S  ^ {*} $
 +
enclosing $  S $.  
 +
Such a system is called a condenser $  ( S , S  ^ {*} ) $.  
 +
Let $  u (x) $
 +
be the harmonic function in the domain $  D $
 +
between $  S $
 +
and $  S  ^ {*} $
 +
taking the value 1 on $  S $
 +
and  $  0 $
 +
on  $  S  ^ {*} $.
 +
The condenser capacity  $  C ( S , S  ^ {*} ) $
 +
is the number
 +
 
 +
$$ \tag{1 }
 +
C ( S , S  ^ {*} )  = -
 +
\frac{1}{( n - 2 ) \sigma _ {n} }
 +
\int\limits _ {S  ^  \prime  }
 +
 
 +
\frac{\partial  u (x) }{\partial  n }
 +
  d \sigma =
 +
$$
 +
 
 +
$$
 +
= \
 +
 
 +
\frac{1}{( n - 2 ) \sigma _ {n} }
 +
\int\limits _ { D } |  \mathop{\rm grad}  u (x) |  ^ {2}  d \omega ,
 +
$$
 +
 
 +
where  $  \sigma _ {n} = 2 \pi  ^ {n/2} / \Gamma ( n / 2 ) $
 +
is the area of the unit sphere in  $  \mathbf R  ^ {n} $,
 +
$  \partial  u / \partial  n $
 +
is the derivative in the direction of the outward normal to an arbitrary intermediate hypersurface  $  S  ^  \prime  $
 +
lying between  $  S $
 +
and  $  S  ^ {*} $
 +
and enclosing  $  S $,
 +
$  d \sigma $
 +
is the area element on $  S  ^  \prime  $,
 +
and  $  d \omega $
 +
is the volume element. Alternatively, the condenser capacity $  C ( S , S  ^ {*} ) $
 +
may be defined as the infimum of the integrals
 +
 
 +
$$
 +
 
 +
\frac{1}{( n - 2 ) \sigma _ {n} }
 +
\int\limits _ { D }
 +
|  \mathop{\rm grad}  v (x) |  ^ {2}  d \omega
 +
$$
 +
 
 +
in the class of all continuously-differentiable functions  $  v (x) $
 +
in  $  D $
 +
that take the values 1 and 0 on  $  S $
 +
and  $  S  ^ {*} $,
 +
respectively. If  $  S  ^ {*} = S ( 0 , R ) $
 +
is a sphere with centre at the origin and sufficiently large radius  $  R $,
 +
then, letting  $  R \rightarrow \infty $
 +
in (1), one obtains the capacity of the compact set  $  K $
 +
bounded by  $  S $,
 +
also called the harmonic capacity of  $  K $
 +
or the Newtonian capacity of  $  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/c/c020/c020280/c02028016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$
 +
C (K)  = \lim\limits _ {R \rightarrow \infty } \
 +
C ( S , S ( 0 , R ) ) ,
 +
$$
  
<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/c/c020/c020280/c02028017.png" /></td> </tr></table>
+
which always satisfies  $  0 \leq  C (K) < \infty $.  
 +
$  C (K) $
 +
is the analogue of the electrostatic capacity of the isolated conductor  $  K $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028018.png" /> is the area of the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028020.png" /> is the derivative in the direction of the outward normal to an arbitrary intermediate hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028021.png" /> lying between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028023.png" /> and enclosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028025.png" /> is the area element on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028027.png" /> is the volume element. Alternatively, the condenser capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028028.png" /> may be defined as the infimum of the integrals
+
In the case of the plane  $  \mathbf R  ^ {2} $
 +
a condenser  $  ( L , L  ^ {*} ) $
 +
is a system of two non-intersecting smooth simple closed curves  $  L $
 +
and  $  L  ^ {*} $
 +
with  $  L  ^ {*} $
 +
enclosing  $  L $.  
 +
Let  $  u (x) $
 +
be the harmonic function in the domain  $  D $
 +
between $  L $
 +
and $  L  ^ {*} $
 +
taking the value 1 on $  L $
 +
and 0 $
 +
on  $  L  ^ {*} $.  
 +
The condenser capacity $  C ( L , L  ^ {*} ) $
 +
is the number
  
<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/c/c020/c020280/c02028029.png" /></td> </tr></table>
+
$$
 +
C ( L , L  ^ {*} )  = -  
 +
\frac{1}{2 \pi }
 +
\int\limits _ {L  ^  \prime  }
  
in the class of all continuously-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028031.png" /> that take the values 1 and 0 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028033.png" />, respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028034.png" /> is a sphere with centre at the origin and sufficiently large radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028035.png" />, then, letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028036.png" /> in (1), one obtains the capacity of the compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028038.png" /> bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028039.png" />, also called the harmonic capacity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028040.png" /> or the Newtonian capacity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028041.png" />:
+
\frac{\partial  u (x) }{\partial  n }
 +
  ds  =
 +
\frac{1}{2 \pi }
  
<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/c/c020/c020280/c02028042.png" /></td> </tr></table>
+
\int\limits _ { D } |  \mathop{\rm grad}  u (x) |  ^ {2}  d \omega ,
 +
$$
  
which always satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028043.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028044.png" /> is the analogue of the electrostatic capacity of the isolated conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028045.png" />.
+
where  $  ds $
 +
is the element of arc length of a curve  $  L  ^  \prime  $
 +
lying between  $  L $
 +
and  $  L  ^ {*} $
 +
and enclosing  $  L $.
 +
Let  $  L  ^ {*} = S ( 0 , R ) $
 +
be a circle with centre at the origin and sufficiently large radius  $  R $;
 +
then letting  $  R \rightarrow \infty $
 +
in the formula,
  
In the case of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028046.png" /> a condenser <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028047.png" /> is a system of two non-intersecting smooth simple closed curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028049.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028050.png" /> enclosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028051.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028052.png" /> be the harmonic function in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028053.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028055.png" /> taking the value 1 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028058.png" />. The condenser capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028059.png" /> is the number
+
$$
 +
W (K)  = \lim\limits _ {R \rightarrow \infty } \
 +
\left [
 +
\frac{1}{C ( L , S ( 0 , R ) ) }
 +
- \mathop{\rm ln}  R \right ] ,
 +
$$
  
<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/c/c020/c020280/c02028060.png" /></td> </tr></table>
+
gives the Wiener capacity, or the Robin constant, of the compact set  $  K $
 +
bounded by  $  L $;  
 +
the Wiener capacity can take any value  $  - \infty < W (K) < \infty $.
 +
The logarithmic capacity, also called the harmonic capacity or the conformal capacity, is more often used:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028061.png" /> is the element of arc length of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028062.png" /> lying between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028064.png" /> and enclosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028065.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028066.png" /> be a circle with centre at the origin and sufficiently large radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028067.png" />; then letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028068.png" /> in the formula,
+
$$ \tag{2 }
 +
C (K)  = e  ^ {-W(K)}  = \
 +
\lim\limits _ {R \rightarrow \infty }  R e ^
 +
{- 1 / {C ( L , S ( 0 , R ) ) } } ,
 +
$$
  
<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/c/c020/c020280/c02028069.png" /></td> </tr></table>
+
it varies between  $  0 \leq  C (K) < \infty $.
  
gives the Wiener capacity, or the Robin constant, of the compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028070.png" /> bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028071.png" />; the Wiener capacity can take any value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028072.png" />. The logarithmic capacity, also called the harmonic capacity or the conformal capacity, is more often used:
+
The capacity of a compact set $  K $
 +
bounded by a hypersurface  $  S $
 +
for  $  n \geq  3 $
 +
may also be defined rather differently. Let  $  v _ {K} (x) $
 +
be the capacitary, or equilibrium, potential of this compact set (cf. [[Capacity potential|Capacity potential]]), that is, the function harmonic everywhere outside  $  K $,
 +
regular at infinity and taking the value 1 on  $  S $.
 +
Then
  
<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/c/c020/c020280/c02028073.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{3 }
 +
C (K)  = -  
 +
\frac{1}{( n - 2 ) \sigma _ {n} }
  
it varies between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028074.png" />.
+
\int\limits _ { S }
 +
\frac{\partial  v _ {K} (x) }{\partial  n }
 +
  d \sigma =
 +
$$
  
The capacity of a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028075.png" /> bounded by a hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028076.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028077.png" /> may also be defined rather differently. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028078.png" /> be the capacitary, or equilibrium, potential of this compact set (cf. [[Capacity potential|Capacity potential]]), that is, the function harmonic everywhere outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028079.png" />, regular at infinity and taking the value 1 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028080.png" />. Then
+
$$
 +
= \
  
<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/c/c020/c020280/c02028081.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\frac{1}{( n - 2 ) \sigma _ {n} }
 +
\int\limits _ {D  ^  \prime  } |  \mathop{\rm grad}  v _ {K} (x) |  ^ {2}  d \omega ,
 +
$$
  
<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/c/c020/c020280/c02028082.png" /></td> </tr></table>
+
where  $  D  ^  \prime  $
 +
is the exterior of  $  S $.  
 +
Formula (3) shows that the capacity  $  C (K) $
 +
is a positive measure, distributed on  $  S $
 +
and such that the Newtonian potential of the simple layer generated by this measure coincides precisely with the capacitary potential  $  v _ {K} (x) $,
 +
that is,
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028083.png" /> is the exterior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028084.png" />. Formula (3) shows that the capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028085.png" /> is a positive measure, distributed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028086.png" /> and such that the Newtonian potential of the simple layer generated by this measure coincides precisely with the capacitary potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028087.png" />, that is,
+
$$
 +
\left .  
 +
\begin{array}{l}
 +
v _ {K} (x)  = \int\limits _ { S }
 +
\frac{d \mu (y) }{| x - y |  ^ {n-2} }
 +
,\  x \in D  ^  \prime  ;  \\
 +
C (K)  = \int\limits _ { S } d \mu (y)  = \mu (S). \\
 +
\end{array}
 +
  \right \}
 +
$$
  
<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/c/c020/c020280/c02028088.png" /></td> </tr></table>
+
The measure  $  \mu $
 +
is called the capacitary, or equilibrium, measure.
  
The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028089.png" /> is called the capacitary, or equilibrium, measure.
+
In the class of all positive Borel measures  $  \lambda $
 +
on  $  K $
 +
such that  $  \lambda (K) = \mu (S) = C (K) $,
 +
the capacitary measure $  \mu $
 +
minimizes the energy integral
  
In the class of all positive Borel measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028090.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028091.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028092.png" />, the capacitary measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028093.png" /> minimizes the energy integral
+
$$ \tag{4 }
 +
E ( \lambda )  = {\int\limits \int\limits } _ {K \times 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/c/c020/c020280/c02028094.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\frac{d \lambda (x)  d \lambda (y) }{| x - y |  ^ {n-2} }
 +
.
 +
$$
  
In other words, the capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028095.png" /> can be defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028096.png" />, where the infimum is taken over the class of all positive measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028097.png" /> concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028098.png" /> and normalized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c02028099.png" />.
+
In other words, the capacity $  C (K) $
 +
can be defined by the formula $  C (K) = 1 / \inf  E ( \lambda ) $,  
 +
where the infimum is taken over the class of all positive measures $  \lambda $
 +
concentrated on $  K $
 +
and normalized by the condition $  \lambda (K) = 1 $.
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280100.png" />, because of the singular behaviour of the logarithmic potential at infinity, the construction given above for the capacitary potential is possible only for a condenser, for example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280101.png" />, by means of the [[Green function|Green function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280102.png" /> for the interior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280103.png" /> of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280104.png" />, in the form
+
For $  n = 2 $,  
 +
because of the singular behaviour of the logarithmic potential at infinity, the construction given above for the capacitary potential is possible only for a condenser, for example, for $  ( L , S ( 0 , R ) ) $,
 +
by means of the [[Green function|Green function]] $  G ( x , y ) $
 +
for the interior $  \Delta $
 +
of the circle $  S ( 0 , R) $,
 +
in 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/c/c020/c020280/c020280105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
\left .
 +
\begin{array}{l}
 +
u _ {K} ( x ; S ( 0 , R ) )  = \int\limits _ { L }
 +
G ( x , y )  d \mu (y) ,\  x \in D ; \\
 +
C ( L , S ( 0 , R ) )  = \int\limits _ { L } d \mu (y)  = \
 +
\mu (L) ,  \\
 +
\end{array}
 +
  \right \}
 +
$$
  
where the capacitary potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280106.png" /> coincides in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280107.png" /> with the harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280108.png" /> introduced earlier for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280109.png" />. The capacity defined by formula (5) is sometimes called the Green capacity; this construction is possible for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280110.png" />. The formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280112.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280113.png" /> gives the Wiener capacity of the compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280114.png" />, and the energy integral
+
where the capacitary potential $  u _ {K} ( x ;  S ( 0 , R ) ) $
 +
coincides in $  D $
 +
with the harmonic function $  u (x) $
 +
introduced earlier for $  ( L , S ( 0 , R ) ) $.  
 +
The capacity defined by formula (5) is sometimes called the Green capacity; this construction is possible for any $  n \geq  2 $.  
 +
The formula $  W (K) = 1 / \inf  E ( \lambda ) $,  
 +
$  \lambda (K) = 1 $,  
 +
for $  n = 2 $
 +
gives the Wiener capacity of the compact set $  K $,  
 +
and the energy integral
  
<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/c/c020/c020280/c020280115.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
E ( \lambda )  = {\int\limits \int\limits } _ {K \times K }
 +
\mathop{\rm ln}  {
 +
\frac{1}{| x-y | }
 +
}  d \lambda (x)  d \lambda (y)
 +
$$
  
 
is now not always positive.
 
is now not always positive.
  
The capacity of an arbitrary compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280118.png" />, may be defined by means of the above property of minimum energy:
+
The capacity of an arbitrary compact set $  K \subset  \mathbf R  ^ {n} $,  
 +
$  n \geq  3 $,  
 +
may be defined by means of the above property of minimum energy:
  
<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/c/c020/c020280/c020280119.png" /></td> </tr></table>
+
$$
 +
C (K)  =
 +
\frac{1}{\inf  E ( \lambda ) }
 +
,\ \
 +
\lambda (K) = 1 ,\  \lambda \geq  0 ,
 +
$$
  
where the integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280120.png" /> are computed as in formula (4). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280121.png" /> this leads to the definition of the Wiener capacity of an arbitrary compact set:
+
where the integrals $  E ( \lambda ) $
 +
are computed as in formula (4). For $  n = 2 $
 +
this leads to the definition of the Wiener capacity of an arbitrary compact set:
  
<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/c/c020/c020280/c020280123.png" /></td> </tr></table>
+
$$
 +
W (K)  =
 +
\frac{1}{\inf  E ( \lambda ) }
 +
,\ \
 +
\lambda (K) = 1 ,\  \lambda \geq  0 ,
 +
$$
  
where the energy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280124.png" /> is computed as in formula (6). The transition to the logarithmic capacity is effected by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280125.png" />.
+
where the energy $  E ( \lambda ) $
 +
is computed as in formula (6). The transition to the logarithmic capacity is effected by the formula $  C (K) = e  ^ {-W(K)} $.
  
An equivalent method is the construction of a capacitary potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280126.png" /> for an arbitrary compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280127.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280128.png" /> it may be defined as the largest of the potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280129.png" /> of the positive measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280130.png" /> concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280131.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280132.png" />. The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280133.png" /> generating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280134.png" /> is the capacity measure, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280135.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280136.png" />, the construction of the capacitary potential is done as above for a condenser <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280137.png" /> by means of the Green function for the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280138.png" />. The capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280139.png" /> of a compact set is then obtained by limit transition, as in formula (2).
+
An equivalent method is the construction of a capacitary potential $  v _ {K} (x) $
 +
for an arbitrary compact set $  K $.  
 +
For $  n \geq  3 $
 +
it may be defined as the largest of the potentials $  U _  \lambda  (x) $
 +
of the positive measures $  \lambda $
 +
concentrated on $  K $
 +
for which $  U _  \lambda  (x) \leq  1 $.  
 +
The measure $  \mu $
 +
generating $  v _ {K} (x) $
 +
is the capacity measure, $  \mu (K) = C (K) $.  
 +
For $  n = 2 $,  
 +
the construction of the capacitary potential is done as above for a condenser $  ( K , S ( 0 , R ) ) $
 +
by means of the Green function for the disc $  \Delta $.  
 +
The capacity $  C (K) $
 +
of a compact set is then obtained by limit transition, as in formula (2).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280140.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280141.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280142.png" />, the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280143.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280144.png" /> are equivalent. Compact sets of capacity zero play the same role in potential theory as sets of measure zero in integration theory. For example, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280145.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280146.png" /> holds everywhere with the possible exception of a set of points belonging to some compact set of capacity zero. The potential of any positive measure concentrated on a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280147.png" /> of capacity zero is unbounded. In addition, for any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280148.png" /> of capacity zero, there exists a positive measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280149.png" />, concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280150.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280151.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280153.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280154.png" />, that is, any compact set of capacity zero is a [[Polar set|polar set]].
+
If $  v _ {K} (x) = 0 $,  
 +
then $  C (K) = 0 $.  
 +
For $  n = 2 $,  
 +
the equations $  C (K) = 0 $
 +
and $  W (K) = + \infty $
 +
are equivalent. Compact sets of capacity zero play the same role in potential theory as sets of measure zero in integration theory. For example, the equation $  v _ {K} (x) = 1 $
 +
on $  K $
 +
holds everywhere with the possible exception of a set of points belonging to some compact set of capacity zero. The potential of any positive measure concentrated on a compact set $  K $
 +
of capacity zero is unbounded. In addition, for any compact set $  K $
 +
of capacity zero, there exists a positive measure $  \nu $,  
 +
concentrated on $  K $,  
 +
such that $  U _  \nu  (x) = + \infty $
 +
for $  x \in K $
 +
and $  U _  \nu  (x) < + \infty $
 +
for $  x \notin K $,  
 +
that is, any compact set of capacity zero is a [[Polar set|polar set]].
  
Properties of capacitary potentials and capacities of compact sets: 1) The mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280156.png" /> are increasing, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280157.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280158.png" /> everywhere and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280159.png" />; 2) these mappings are continuous on the right, that is, for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280160.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280161.png" /> there exists an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280162.png" /> such that if a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280163.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280164.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280165.png" /> everywhere, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280166.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280167.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280168.png" /> are strongly subadditive as functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280169.png" />, that is,
+
Properties of capacitary potentials and capacities of compact sets: 1) The mappings $  K \rightarrow v _ {K} (x) $
 +
and $  K \rightarrow C (K) $
 +
are increasing, that is, $  K _ {1} \subset  K _ {2} $
 +
implies that $  v _ {K _ {1}  } (x) \leq  v _ {K _ {2}  } (x) $
 +
everywhere and $  C ( K _ {1} ) \leq  C ( K _ {2} ) $;  
 +
2) these mappings are continuous on the right, that is, for any fixed $  x \in \mathbf R  ^ {n} $
 +
and any $  \epsilon > 0 $
 +
there exists an open set $  \omega $
 +
such that if a compact set $  K  ^  \prime  $
 +
satisfies $  K \subset  K  ^  \prime  \subset  \omega $,  
 +
then $  v _ {K  ^  \prime  } (x) - v _ {K} (x) < \epsilon $
 +
everywhere, and $  C ( K  ^  \prime  ) - C (K) < \epsilon $;  
 +
and 3) $  v _ {K} (x) $
 +
and $  C (K) $
 +
are strongly subadditive as functions of $  K $,  
 +
that is,
  
<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/c/c020/c020280/c020280170.png" /></td> </tr></table>
+
$$
 +
v _ {K _ {1}  \cup K _ {2} } (x) + v _ {K _ {1}  \cap K _ {2} }
 +
(x)  \leq  v _ {K _ {1}  } (x) + v _ {K _ {2}  } (x) ,
 +
$$
  
<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/c/c020/c020280/c020280171.png" /></td> </tr></table>
+
$$
 +
C ( K _ {1} \cup K _ {2} ) + C ( K _ {1} \cap K _ {2} )  \leq  C ( K _ {1} ) + C ( K _ {2} ) .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280172.png" /> is an open set lying in a ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280173.png" />, then, by definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280174.png" />. For an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280175.png" />, the inner capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280176.png" /> is defined as the least upper bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280177.png" /> over all compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280178.png" />. The outer capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280179.png" /> is defined as the greatest lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280180.png" /> over all open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280181.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280182.png" /> is called capacitable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280183.png" />. All Borel sets, and even all analytic sets, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280184.png" /> are capacitable. Thus, the capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280185.png" /> is a set function invariant under motions, but, however, not additive. The fact that the capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280186.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280187.png" /> is zero is a very important property of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280188.png" />. In many problems of potential theory sets of capacity zero in the above sense may be neglected. For example, the following strong maximum principle is valid. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280189.png" /> be a subharmonic function bounded from above on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280190.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280191.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280192.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280193.png" /> hold for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280194.png" />, with the possible exception of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280195.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280196.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280197.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280198.png" /> everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280199.png" />, and equality, even at a single point, is possible only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280200.png" />.
+
If $  G $
 +
is an open set lying in a ball $  B = B ( 0 , R ) $,
 +
then, by definition, $  C (G) = C (B) - C ( \overline{B}\; \setminus  G ) $.  
 +
For an arbitrary set $  E $,  
 +
the inner capacity $  \underline{C} (E) $
 +
is defined as the least upper bound $  \underline{C} (E) = \sup  C (K) $
 +
over all compact sets $  K \subset  E $.  
 +
The outer capacity $  \overline{C}\; (E) $
 +
is defined as the greatest lower bound $  \overline{C}\; (E) = \inf  C (G) $
 +
over all open sets $  G \subset  E $.  
 +
A set $  E $
 +
is called capacitable if $  \overline{C}\; (E) = \underline{C} (E) = C (E) $.  
 +
All Borel sets, and even all analytic sets, in $  \mathbf R  ^ {n} $
 +
are capacitable. Thus, the capacity $  C (E) $
 +
is a set function invariant under motions, but, however, not additive. The fact that the capacity $  C (E) $
 +
of a set $  E $
 +
is zero is a very important property of $  E $.  
 +
In many problems of potential theory sets of capacity zero in the above sense may be neglected. For example, the following strong maximum principle is valid. Let $  w (x) $
 +
be a subharmonic function bounded from above on a domain $  G \subset  \mathbf R  ^ {n} $,  
 +
$  n \geq  3 $,  
 +
$  \infty \notin G $;  
 +
let $  \lim\limits _ {x \rightarrow y }  \sup  w (x) \leq  M $
 +
hold for all $  y \in \partial  G $,  
 +
with the possible exception of a set $  E $
 +
with $  C (E) = 0 $,  
 +
$  \infty \notin E $.  
 +
Then $  w (x) \leq  M $
 +
everywhere in $  G $,  
 +
and equality, even at a single point, is possible only if $  w (x) \equiv M $.
  
 
The concept of a capacity can be generalized in various directions. Starting from the concept of a capacitary potential and a capacitary measure or an energy, theories of capacity have been constructed for non-Newtonian potentials, such as for Bessel potentials, non-linear potentials, Riesz potentials, and others (cf. [[Bessel potential|Bessel potential]]; [[Non-linear potential|Non-linear potential]]; [[Riesz potential|Riesz potential]]). In particular, these constructions enable one to vary the concept of a set of capacity zero in accordance with various problems of mathematical physics and analysis (see [[#References|[6]]]).
 
The concept of a capacity can be generalized in various directions. Starting from the concept of a capacitary potential and a capacitary measure or an energy, theories of capacity have been constructed for non-Newtonian potentials, such as for Bessel potentials, non-linear potentials, Riesz potentials, and others (cf. [[Bessel potential|Bessel potential]]; [[Non-linear potential|Non-linear potential]]; [[Riesz potential|Riesz potential]]). In particular, these constructions enable one to vary the concept of a set of capacity zero in accordance with various problems of mathematical physics and analysis (see [[#References|[6]]]).
  
According to G. Choquet, a capacity in an abstract separable topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280201.png" /> is defined axiomatically as a numerical set function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280202.png" /> satisfying the following axioms: it is increasing, continuous on the right and strongly subadditive. A rather different approach to the theory of capacity in abstract spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280203.png" /> can be found within the framework of the general axiomatics of abstract potential theory (cf. [[Potential theory, abstract|Potential theory, abstract]]) or the theory of harmonic spaces (cf. [[Harmonic space|Harmonic space]]). In an abstract theory of capacities, a fundamental result is Choquet's theorem, stating that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280205.png" />-analytic sets, i.e. continuous images of sets of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280206.png" /> in some compact space, are capacitable.
+
According to G. Choquet, a capacity in an abstract separable topological space $  X $
 +
is defined axiomatically as a numerical set function $  K \rightarrow C (K) $
 +
satisfying the following axioms: it is increasing, continuous on the right and strongly subadditive. A rather different approach to the theory of capacity in abstract spaces $  X $
 +
can be found within the framework of the general axiomatics of abstract potential theory (cf. [[Potential theory, abstract|Potential theory, abstract]]) or the theory of harmonic spaces (cf. [[Harmonic space|Harmonic space]]). In an abstract theory of capacities, a fundamental result is Choquet's theorem, stating that $  K $-
 +
analytic sets, i.e. continuous images of sets of type $  K _ {\sigma \delta }  $
 +
in some compact space, are capacitable.
  
In general, in various problems of function theory, mainly concerning approximation in specific classes of functions, it turns out to be useful to introduce an appropriate notion of capacity. For example, the concept of [[Analytic capacity|analytic capacity]] is of great importance in questions of approximation by analytic functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280207.png" /> be a compact set in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280208.png" />-plane, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280209.png" /> be an analytic function outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280210.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280211.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280212.png" />; the analytic capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280213.png" /> is the number
+
In general, in various problems of function theory, mainly concerning approximation in specific classes of functions, it turns out to be useful to introduce an appropriate notion of capacity. For example, the concept of [[Analytic capacity|analytic capacity]] is of great importance in questions of approximation by analytic functions. Let $  K $
 +
be a compact set in the complex $  z $-
 +
plane, let $  f (z) $
 +
be an analytic function outside $  K $,  
 +
$  | f (z) | \leq  1 $,  
 +
$  f ( \infty ) = 0 $;  
 +
the analytic capacity $  \gamma (K) $
 +
is the number
  
<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/c/c020/c020280/c020280214.png" /></td> </tr></table>
+
$$
 +
\gamma (K)  = \sup \
 +
\left |
 +
\frac{1}{2 \pi }
 +
\int\limits _ {L  ^  \prime  } f (z)  d z \right | ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280215.png" /> is a contour enclosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280216.png" /> and the supremum is taken over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280217.png" /> satisfying the stated conditions.
+
where $  L  ^  \prime  $
 +
is a contour enclosing $  K $
 +
and the supremum is taken over all $  f (z) $
 +
satisfying the stated conditions.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)  {{MR|0350027}} {{ZBL|0253.31001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)  {{MR|0106366}} {{ZBL|0084.30903}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Pólya,  G. Szegö,  "Isoperimetric inequalities in mathematical physics" , Princeton Univ. Press  (1951)  {{MR|0043486}} {{ZBL|0044.38301}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Brélot,  "Lectures on potential theory" , Tata Inst.  (1960)  {{MR|0118980}} {{ZBL|0098.06903}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C. Dellacherie,  "Capacités et processus stochastiques" , Springer  (1972)  {{MR|0448504}} {{ZBL|0246.60032}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L. Carleson,  "Selected problems on exceptional sets" , v. Nostrand  (1967)  {{MR|0225986}} {{ZBL|0189.10903}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.G. Vitushkin,  "Uniform approximation by holomorphic functions"  ''J. Soviet Math.'' , '''5''' :  5  (1976)  pp. 607–611  ''Itogi Nauk. i Tekhn. Sovrem. Pobl. Mat.'' , '''4'''  (1975)  pp. 5–12  {{MR|0382722}} {{ZBL|0657.30028}} {{ZBL|0633.30034}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)  {{MR|0350027}} {{ZBL|0253.31001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)  {{MR|0106366}} {{ZBL|0084.30903}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Pólya,  G. Szegö,  "Isoperimetric inequalities in mathematical physics" , Princeton Univ. Press  (1951)  {{MR|0043486}} {{ZBL|0044.38301}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Brélot,  "Lectures on potential theory" , Tata Inst.  (1960)  {{MR|0118980}} {{ZBL|0098.06903}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C. Dellacherie,  "Capacités et processus stochastiques" , Springer  (1972)  {{MR|0448504}} {{ZBL|0246.60032}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L. Carleson,  "Selected problems on exceptional sets" , v. Nostrand  (1967)  {{MR|0225986}} {{ZBL|0189.10903}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.G. Vitushkin,  "Uniform approximation by holomorphic functions"  ''J. Soviet Math.'' , '''5''' :  5  (1976)  pp. 607–611  ''Itogi Nauk. i Tekhn. Sovrem. Pobl. Mat.'' , '''4'''  (1975)  pp. 5–12  {{MR|0382722}} {{ZBL|0657.30028}} {{ZBL|0633.30034}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For capacities within the framework of harmonic spaces see [[#References|[a1]]]; [[#References|[a2]]], [[#References|[a3]]] contain among other things discussions of the [[Robin constant|Robin constant]], where [[#References|[a3]]] focuses on the relation with analytic functions in domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280218.png" />. Recently one has started to study capacities in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280220.png" /> in relation with obtaining bounds on the growth of analytic functions on domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020280/c020280221.png" />, cf. [[#References|[a4]]] and the references given there.
+
For capacities within the framework of harmonic spaces see [[#References|[a1]]]; [[#References|[a2]]], [[#References|[a3]]] contain among other things discussions of the [[Robin constant|Robin constant]], where [[#References|[a3]]] focuses on the relation with analytic functions in domains in $  \mathbf C $.  
 +
Recently one has started to study capacities in $  \mathbf C  ^ {n} $
 +
in relation with obtaining bounds on the growth of analytic functions on domains in $  \mathbf C  ^ {n} $,  
 +
cf. [[#References|[a4]]] and the references given there.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)  {{MR|0419799}} {{ZBL|0248.31011}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Doob,  "Classical potential theory and its probabilistic counterpart" , Springer  (1984)  pp. 390  {{MR|0731258}} {{ZBL|0549.31001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Tsuji,  "Potential theory in modern function theory" , Chelsea, reprint  (1975)  {{MR|0414898}} {{ZBL|0322.30001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Korevaar,  "Green functions, capacities, polynomial approximation numbers and applications in real and complex analysis"  ''Nieuw Archief voor Wiskunde IV'' , '''4''' :  2  (1986)  pp. 133–153  {{MR|0864476}} {{ZBL|0618.32015}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)  {{MR|0419799}} {{ZBL|0248.31011}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Doob,  "Classical potential theory and its probabilistic counterpart" , Springer  (1984)  pp. 390  {{MR|0731258}} {{ZBL|0549.31001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Tsuji,  "Potential theory in modern function theory" , Chelsea, reprint  (1975)  {{MR|0414898}} {{ZBL|0322.30001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Korevaar,  "Green functions, capacities, polynomial approximation numbers and applications in real and complex analysis"  ''Nieuw Archief voor Wiskunde IV'' , '''4''' :  2  (1986)  pp. 133–153  {{MR|0864476}} {{ZBL|0618.32015}} </TD></TR></table>

Latest revision as of 06:29, 30 May 2020


of a set

A set function arising in potential theory as the analogue of the physical concept of the electrostatic capacity.

Let $ S $ and $ S ^ {*} $ be two smooth closed hypersurfaces in a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 3 $, with $ S ^ {*} $ enclosing $ S $. Such a system is called a condenser $ ( S , S ^ {*} ) $. Let $ u (x) $ be the harmonic function in the domain $ D $ between $ S $ and $ S ^ {*} $ taking the value 1 on $ S $ and $ 0 $ on $ S ^ {*} $. The condenser capacity $ C ( S , S ^ {*} ) $ is the number

$$ \tag{1 } C ( S , S ^ {*} ) = - \frac{1}{( n - 2 ) \sigma _ {n} } \int\limits _ {S ^ \prime } \frac{\partial u (x) }{\partial n } d \sigma = $$

$$ = \ \frac{1}{( n - 2 ) \sigma _ {n} } \int\limits _ { D } | \mathop{\rm grad} u (x) | ^ {2} d \omega , $$

where $ \sigma _ {n} = 2 \pi ^ {n/2} / \Gamma ( n / 2 ) $ is the area of the unit sphere in $ \mathbf R ^ {n} $, $ \partial u / \partial n $ is the derivative in the direction of the outward normal to an arbitrary intermediate hypersurface $ S ^ \prime $ lying between $ S $ and $ S ^ {*} $ and enclosing $ S $, $ d \sigma $ is the area element on $ S ^ \prime $, and $ d \omega $ is the volume element. Alternatively, the condenser capacity $ C ( S , S ^ {*} ) $ may be defined as the infimum of the integrals

$$ \frac{1}{( n - 2 ) \sigma _ {n} } \int\limits _ { D } | \mathop{\rm grad} v (x) | ^ {2} d \omega $$

in the class of all continuously-differentiable functions $ v (x) $ in $ D $ that take the values 1 and 0 on $ S $ and $ S ^ {*} $, respectively. If $ S ^ {*} = S ( 0 , R ) $ is a sphere with centre at the origin and sufficiently large radius $ R $, then, letting $ R \rightarrow \infty $ in (1), one obtains the capacity of the compact set $ K $ bounded by $ S $, also called the harmonic capacity of $ K $ or the Newtonian capacity of $ K $:

$$ C (K) = \lim\limits _ {R \rightarrow \infty } \ C ( S , S ( 0 , R ) ) , $$

which always satisfies $ 0 \leq C (K) < \infty $. $ C (K) $ is the analogue of the electrostatic capacity of the isolated conductor $ K $.

In the case of the plane $ \mathbf R ^ {2} $ a condenser $ ( L , L ^ {*} ) $ is a system of two non-intersecting smooth simple closed curves $ L $ and $ L ^ {*} $ with $ L ^ {*} $ enclosing $ L $. Let $ u (x) $ be the harmonic function in the domain $ D $ between $ L $ and $ L ^ {*} $ taking the value 1 on $ L $ and $ 0 $ on $ L ^ {*} $. The condenser capacity $ C ( L , L ^ {*} ) $ is the number

$$ C ( L , L ^ {*} ) = - \frac{1}{2 \pi } \int\limits _ {L ^ \prime } \frac{\partial u (x) }{\partial n } ds = \frac{1}{2 \pi } \int\limits _ { D } | \mathop{\rm grad} u (x) | ^ {2} d \omega , $$

where $ ds $ is the element of arc length of a curve $ L ^ \prime $ lying between $ L $ and $ L ^ {*} $ and enclosing $ L $. Let $ L ^ {*} = S ( 0 , R ) $ be a circle with centre at the origin and sufficiently large radius $ R $; then letting $ R \rightarrow \infty $ in the formula,

$$ W (K) = \lim\limits _ {R \rightarrow \infty } \ \left [ \frac{1}{C ( L , S ( 0 , R ) ) } - \mathop{\rm ln} R \right ] , $$

gives the Wiener capacity, or the Robin constant, of the compact set $ K $ bounded by $ L $; the Wiener capacity can take any value $ - \infty < W (K) < \infty $. The logarithmic capacity, also called the harmonic capacity or the conformal capacity, is more often used:

$$ \tag{2 } C (K) = e ^ {-W(K)} = \ \lim\limits _ {R \rightarrow \infty } R e ^ {- 1 / {C ( L , S ( 0 , R ) ) } } , $$

it varies between $ 0 \leq C (K) < \infty $.

The capacity of a compact set $ K $ bounded by a hypersurface $ S $ for $ n \geq 3 $ may also be defined rather differently. Let $ v _ {K} (x) $ be the capacitary, or equilibrium, potential of this compact set (cf. Capacity potential), that is, the function harmonic everywhere outside $ K $, regular at infinity and taking the value 1 on $ S $. Then

$$ \tag{3 } C (K) = - \frac{1}{( n - 2 ) \sigma _ {n} } \int\limits _ { S } \frac{\partial v _ {K} (x) }{\partial n } d \sigma = $$

$$ = \ \frac{1}{( n - 2 ) \sigma _ {n} } \int\limits _ {D ^ \prime } | \mathop{\rm grad} v _ {K} (x) | ^ {2} d \omega , $$

where $ D ^ \prime $ is the exterior of $ S $. Formula (3) shows that the capacity $ C (K) $ is a positive measure, distributed on $ S $ and such that the Newtonian potential of the simple layer generated by this measure coincides precisely with the capacitary potential $ v _ {K} (x) $, that is,

$$ \left . \begin{array}{l} v _ {K} (x) = \int\limits _ { S } \frac{d \mu (y) }{| x - y | ^ {n-2} } ,\ x \in D ^ \prime ; \\ C (K) = \int\limits _ { S } d \mu (y) = \mu (S). \\ \end{array} \right \} $$

The measure $ \mu $ is called the capacitary, or equilibrium, measure.

In the class of all positive Borel measures $ \lambda $ on $ K $ such that $ \lambda (K) = \mu (S) = C (K) $, the capacitary measure $ \mu $ minimizes the energy integral

$$ \tag{4 } E ( \lambda ) = {\int\limits \int\limits } _ {K \times K } \frac{d \lambda (x) d \lambda (y) }{| x - y | ^ {n-2} } . $$

In other words, the capacity $ C (K) $ can be defined by the formula $ C (K) = 1 / \inf E ( \lambda ) $, where the infimum is taken over the class of all positive measures $ \lambda $ concentrated on $ K $ and normalized by the condition $ \lambda (K) = 1 $.

For $ n = 2 $, because of the singular behaviour of the logarithmic potential at infinity, the construction given above for the capacitary potential is possible only for a condenser, for example, for $ ( L , S ( 0 , R ) ) $, by means of the Green function $ G ( x , y ) $ for the interior $ \Delta $ of the circle $ S ( 0 , R) $, in the form

$$ \tag{5 } \left . \begin{array}{l} u _ {K} ( x ; S ( 0 , R ) ) = \int\limits _ { L } G ( x , y ) d \mu (y) ,\ x \in D ; \\ C ( L , S ( 0 , R ) ) = \int\limits _ { L } d \mu (y) = \ \mu (L) , \\ \end{array} \right \} $$

where the capacitary potential $ u _ {K} ( x ; S ( 0 , R ) ) $ coincides in $ D $ with the harmonic function $ u (x) $ introduced earlier for $ ( L , S ( 0 , R ) ) $. The capacity defined by formula (5) is sometimes called the Green capacity; this construction is possible for any $ n \geq 2 $. The formula $ W (K) = 1 / \inf E ( \lambda ) $, $ \lambda (K) = 1 $, for $ n = 2 $ gives the Wiener capacity of the compact set $ K $, and the energy integral

$$ \tag{6 } E ( \lambda ) = {\int\limits \int\limits } _ {K \times K } \mathop{\rm ln} { \frac{1}{| x-y | } } d \lambda (x) d \lambda (y) $$

is now not always positive.

The capacity of an arbitrary compact set $ K \subset \mathbf R ^ {n} $, $ n \geq 3 $, may be defined by means of the above property of minimum energy:

$$ C (K) = \frac{1}{\inf E ( \lambda ) } ,\ \ \lambda (K) = 1 ,\ \lambda \geq 0 , $$

where the integrals $ E ( \lambda ) $ are computed as in formula (4). For $ n = 2 $ this leads to the definition of the Wiener capacity of an arbitrary compact set:

$$ W (K) = \frac{1}{\inf E ( \lambda ) } ,\ \ \lambda (K) = 1 ,\ \lambda \geq 0 , $$

where the energy $ E ( \lambda ) $ is computed as in formula (6). The transition to the logarithmic capacity is effected by the formula $ C (K) = e ^ {-W(K)} $.

An equivalent method is the construction of a capacitary potential $ v _ {K} (x) $ for an arbitrary compact set $ K $. For $ n \geq 3 $ it may be defined as the largest of the potentials $ U _ \lambda (x) $ of the positive measures $ \lambda $ concentrated on $ K $ for which $ U _ \lambda (x) \leq 1 $. The measure $ \mu $ generating $ v _ {K} (x) $ is the capacity measure, $ \mu (K) = C (K) $. For $ n = 2 $, the construction of the capacitary potential is done as above for a condenser $ ( K , S ( 0 , R ) ) $ by means of the Green function for the disc $ \Delta $. The capacity $ C (K) $ of a compact set is then obtained by limit transition, as in formula (2).

If $ v _ {K} (x) = 0 $, then $ C (K) = 0 $. For $ n = 2 $, the equations $ C (K) = 0 $ and $ W (K) = + \infty $ are equivalent. Compact sets of capacity zero play the same role in potential theory as sets of measure zero in integration theory. For example, the equation $ v _ {K} (x) = 1 $ on $ K $ holds everywhere with the possible exception of a set of points belonging to some compact set of capacity zero. The potential of any positive measure concentrated on a compact set $ K $ of capacity zero is unbounded. In addition, for any compact set $ K $ of capacity zero, there exists a positive measure $ \nu $, concentrated on $ K $, such that $ U _ \nu (x) = + \infty $ for $ x \in K $ and $ U _ \nu (x) < + \infty $ for $ x \notin K $, that is, any compact set of capacity zero is a polar set.

Properties of capacitary potentials and capacities of compact sets: 1) The mappings $ K \rightarrow v _ {K} (x) $ and $ K \rightarrow C (K) $ are increasing, that is, $ K _ {1} \subset K _ {2} $ implies that $ v _ {K _ {1} } (x) \leq v _ {K _ {2} } (x) $ everywhere and $ C ( K _ {1} ) \leq C ( K _ {2} ) $; 2) these mappings are continuous on the right, that is, for any fixed $ x \in \mathbf R ^ {n} $ and any $ \epsilon > 0 $ there exists an open set $ \omega $ such that if a compact set $ K ^ \prime $ satisfies $ K \subset K ^ \prime \subset \omega $, then $ v _ {K ^ \prime } (x) - v _ {K} (x) < \epsilon $ everywhere, and $ C ( K ^ \prime ) - C (K) < \epsilon $; and 3) $ v _ {K} (x) $ and $ C (K) $ are strongly subadditive as functions of $ K $, that is,

$$ v _ {K _ {1} \cup K _ {2} } (x) + v _ {K _ {1} \cap K _ {2} } (x) \leq v _ {K _ {1} } (x) + v _ {K _ {2} } (x) , $$

$$ C ( K _ {1} \cup K _ {2} ) + C ( K _ {1} \cap K _ {2} ) \leq C ( K _ {1} ) + C ( K _ {2} ) . $$

If $ G $ is an open set lying in a ball $ B = B ( 0 , R ) $, then, by definition, $ C (G) = C (B) - C ( \overline{B}\; \setminus G ) $. For an arbitrary set $ E $, the inner capacity $ \underline{C} (E) $ is defined as the least upper bound $ \underline{C} (E) = \sup C (K) $ over all compact sets $ K \subset E $. The outer capacity $ \overline{C}\; (E) $ is defined as the greatest lower bound $ \overline{C}\; (E) = \inf C (G) $ over all open sets $ G \subset E $. A set $ E $ is called capacitable if $ \overline{C}\; (E) = \underline{C} (E) = C (E) $. All Borel sets, and even all analytic sets, in $ \mathbf R ^ {n} $ are capacitable. Thus, the capacity $ C (E) $ is a set function invariant under motions, but, however, not additive. The fact that the capacity $ C (E) $ of a set $ E $ is zero is a very important property of $ E $. In many problems of potential theory sets of capacity zero in the above sense may be neglected. For example, the following strong maximum principle is valid. Let $ w (x) $ be a subharmonic function bounded from above on a domain $ G \subset \mathbf R ^ {n} $, $ n \geq 3 $, $ \infty \notin G $; let $ \lim\limits _ {x \rightarrow y } \sup w (x) \leq M $ hold for all $ y \in \partial G $, with the possible exception of a set $ E $ with $ C (E) = 0 $, $ \infty \notin E $. Then $ w (x) \leq M $ everywhere in $ G $, and equality, even at a single point, is possible only if $ w (x) \equiv M $.

The concept of a capacity can be generalized in various directions. Starting from the concept of a capacitary potential and a capacitary measure or an energy, theories of capacity have been constructed for non-Newtonian potentials, such as for Bessel potentials, non-linear potentials, Riesz potentials, and others (cf. Bessel potential; Non-linear potential; Riesz potential). In particular, these constructions enable one to vary the concept of a set of capacity zero in accordance with various problems of mathematical physics and analysis (see [6]).

According to G. Choquet, a capacity in an abstract separable topological space $ X $ is defined axiomatically as a numerical set function $ K \rightarrow C (K) $ satisfying the following axioms: it is increasing, continuous on the right and strongly subadditive. A rather different approach to the theory of capacity in abstract spaces $ X $ can be found within the framework of the general axiomatics of abstract potential theory (cf. Potential theory, abstract) or the theory of harmonic spaces (cf. Harmonic space). In an abstract theory of capacities, a fundamental result is Choquet's theorem, stating that $ K $- analytic sets, i.e. continuous images of sets of type $ K _ {\sigma \delta } $ in some compact space, are capacitable.

In general, in various problems of function theory, mainly concerning approximation in specific classes of functions, it turns out to be useful to introduce an appropriate notion of capacity. For example, the concept of analytic capacity is of great importance in questions of approximation by analytic functions. Let $ K $ be a compact set in the complex $ z $- plane, let $ f (z) $ be an analytic function outside $ K $, $ | f (z) | \leq 1 $, $ f ( \infty ) = 0 $; the analytic capacity $ \gamma (K) $ is the number

$$ \gamma (K) = \sup \ \left | \frac{1}{2 \pi } \int\limits _ {L ^ \prime } f (z) d z \right | , $$

where $ L ^ \prime $ is a contour enclosing $ K $ and the supremum is taken over all $ f (z) $ satisfying the stated conditions.

References

[1] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian) MR0350027 Zbl 0253.31001
[2] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) MR0106366 Zbl 0084.30903
[3] G. Pólya, G. Szegö, "Isoperimetric inequalities in mathematical physics" , Princeton Univ. Press (1951) MR0043486 Zbl 0044.38301
[4] M. Brélot, "Lectures on potential theory" , Tata Inst. (1960) MR0118980 Zbl 0098.06903
[5] C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) MR0448504 Zbl 0246.60032
[6] L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967) MR0225986 Zbl 0189.10903
[7] A.G. Vitushkin, "Uniform approximation by holomorphic functions" J. Soviet Math. , 5 : 5 (1976) pp. 607–611 Itogi Nauk. i Tekhn. Sovrem. Pobl. Mat. , 4 (1975) pp. 5–12 MR0382722 Zbl 0657.30028 Zbl 0633.30034

Comments

For capacities within the framework of harmonic spaces see [a1]; [a2], [a3] contain among other things discussions of the Robin constant, where [a3] focuses on the relation with analytic functions in domains in $ \mathbf C $. Recently one has started to study capacities in $ \mathbf C ^ {n} $ in relation with obtaining bounds on the growth of analytic functions on domains in $ \mathbf C ^ {n} $, cf. [a4] and the references given there.

References

[a1] C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972) MR0419799 Zbl 0248.31011
[a2] J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 MR0731258 Zbl 0549.31001
[a3] M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975) MR0414898 Zbl 0322.30001
[a4] J. Korevaar, "Green functions, capacities, polynomial approximation numbers and applications in real and complex analysis" Nieuw Archief voor Wiskunde IV , 4 : 2 (1986) pp. 133–153 MR0864476 Zbl 0618.32015
How to Cite This Entry:
Capacity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Capacity&oldid=46200
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article