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A numerical characteristic of a set of points in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r0825101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r0825102.png" />, closely connected with the [[Capacity|capacity]] of the set.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r0825103.png" /> be a compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r0825104.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r0825105.png" /> be a positive Borel measure concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r0825106.png" /> and normalized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r0825107.png" />. The integral
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r0825108.png" /></td> </tr></table>
+
A numerical characteristic of a set of points in a Euclidean space  $  \mathbf R  ^ {n} $,
 +
$  n \geq  2 $,
 +
closely connected with the [[Capacity|capacity]] of the set.
 +
 
 +
Let  $  K $
 +
be a compact set in  $  \mathbf R  ^ {n} $,
 +
and let  $  \mu $
 +
be a positive Borel measure concentrated on  $  K $
 +
and normalized by the condition  $  \mu ( K) = 1 $.
 +
The integral
 +
 
 +
$$
 +
V ( \mu )  = {\int\limits \int\limits } _ {K \times K }
 +
E _ {n} ( x , y )  d \mu ( x)  d \mu ( y) ,
 +
$$
  
 
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/r/r082/r082510/r0825109.png" /></td> </tr></table>
+
$$
 +
E _ {2} ( x , y )  =   \mathop{\rm ln} 
 +
\frac{1}{| x - y | }
 +
,\ \
 +
E _ {n} ( x , y )  =
 +
\frac{1}{| x - y |  ^ {n-2} }
 +
  \textrm{ for } \
 +
n \geq  3 ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251010.png" /> is the distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251011.png" />, is the energy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251012.png" /> (cf. [[Energy of measures|Energy of measures]]). The Robin constant of the compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251013.png" /> is the lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251014.png" /> over all measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251015.png" /> of the indicate type. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251016.png" />, then this bound is finite and is attained for some (unique) equilibrium, or capacitary, measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251019.png" />, concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251020.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251022.png" /> for all measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251023.png" /> of the indicated type. The Robin constant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251024.png" /> is related to its capacity by the formula
+
and $  | x - y | $
 +
is the distance between two points $  x , y \in \mathbf R  ^ {n} $,  
 +
is the energy of $  \mu $(
 +
cf. [[Energy of measures|Energy of measures]]). The Robin constant of the compact set $  K $
 +
is the lower bound $  \gamma ( K) = \inf  V ( \mu ) $
 +
over all measures $  \mu $
 +
of the indicate type. If $  \gamma ( K) < + \infty $,  
 +
then this bound is finite and is attained for some (unique) equilibrium, or capacitary, measure $  \lambda > 0 $,  
 +
$  \gamma ( K) = V ( \lambda ) $,  
 +
$  \lambda ( K) = 1 $,  
 +
concentrated on $  K $;  
 +
if $  \gamma ( K) = + \infty $,  
 +
then $  V ( \mu ) = + \infty $
 +
for all measures $  \mu $
 +
of the indicated type. The Robin constant of $  K $
 +
is related to its capacity by the formula
  
<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/r/r082/r082510/r08251025.png" /></td> </tr></table>
+
$$
 +
\gamma ( K)  =
 +
\frac{1}{C ( K) }
 +
\  \textrm{ for }  n \geq  3 ,
 +
$$
  
<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/r/r082/r082510/r08251026.png" /></td> </tr></table>
+
$$
 +
\gamma ( K)  = - \mathop{\rm ln}  C ( K) \  \textrm{ for }  n = 2 .
 +
$$
  
If the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251028.png" /> is sufficiently smooth, for example, if it consists of a finite number of pairwise non-intersecting simple closed surfaces (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251029.png" />) or curves (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251030.png" />) of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251032.png" />, then the equilibrium measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251033.png" /> is concentrated on the part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251034.png" /> which forms the boundary of that connected component of the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251035.png" /> which contains the point at infinity. The equilibrium potential, Robin potential or [[Capacity potential|capacity potential]], i.e. the potential of the equilibrium measure
+
If the boundary $  S $
 +
of $  K $
 +
is sufficiently smooth, for example, if it consists of a finite number of pairwise non-intersecting simple closed surfaces (for $  n \geq  3 $)  
 +
or curves (for $  n = 2 $)  
 +
of class $  C ^ {1 , \alpha } $,
 +
$  0 < \alpha < 1 $,  
 +
then the equilibrium measure $  \lambda $
 +
is concentrated on the part $  \widetilde{S}  \subset  S $
 +
which forms the boundary of that connected component of the complement $  C K = \mathbf R  ^ {n} \setminus  K $
 +
which contains the point at infinity. The equilibrium potential, Robin potential or [[Capacity potential|capacity potential]], i.e. the potential of the equilibrium measure
  
<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/r/r082/r082510/r08251036.png" /></td> </tr></table>
+
$$
 +
u ( x)  = \int\limits E _ {n} ( x , y )  d \lambda ( y) ,
 +
$$
  
in this case assumes a constant value on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251037.png" />, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251038.png" />, which allows one to calculate the Robin constant of a compact set in the simplest cases (see [[Robin problem|Robin problem]]). For instance, the Robin constant of a disc of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251040.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251041.png" />, and the Robin constant of a ball of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251042.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251044.png" />, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251045.png" />. In the case of an arbitrary compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251046.png" /> of positive capacity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251047.png" /> everywhere and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251048.png" /> everywhere on the support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251049.png" /> of the equilibrium measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251050.png" />, except possibly at the points of some polar set; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251051.png" />.
+
in this case assumes a constant value on $  \widetilde{S}  $,  
 +
equal to $  \gamma ( K) $,  
 +
which allows one to calculate the Robin constant of a compact set in the simplest cases (see [[Robin problem|Robin problem]]). For instance, the Robin constant of a disc of radius r > 0 $
 +
in $  \mathbf R  ^ {2} $
 +
is $  -  \mathop{\rm ln}  r $,  
 +
and the Robin constant of a ball of radius r > 0 $
 +
in $  \mathbf R  ^ {n} $,  
 +
$  n \geq  3 $,  
 +
is $  1 / r ^ {n-2} $.  
 +
In the case of an arbitrary compact set $  K $
 +
of positive capacity, $  u ( x) \leq  \gamma ( K) $
 +
everywhere and $  u ( x) = \gamma ( K) $
 +
everywhere on the support $  S ( \lambda ) $
 +
of the equilibrium measure $  \lambda $,  
 +
except possibly at the points of some polar set; moreover, $  S ( \lambda ) \subset  K $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251052.png" /> be a domain in the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251053.png" /> containing inside it the point at infinity and having a [[Green function|Green function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251054.png" /> with pole at infinity. Then the following representation holds:
+
Let $  D $
 +
be a domain in the extended complex plane $  \overline{\mathbf C}\; $
 +
containing inside it the point at infinity and having a [[Green function|Green function]] $  g ( z , \infty ) $
 +
with pole at infinity. Then the following representation holds:
  
<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/r/r082/r082510/r08251055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
g ( z , \infty )  =   \mathop{\rm ln}  | z | + \gamma ( D) + \epsilon
 +
( z , \infty ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251056.png" /> is a complex variable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251057.png" /> is the Robin constant of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251059.png" /> is a [[Harmonic function|harmonic function]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251060.png" />; moreover,
+
where $  z = x + i y $
 +
is a complex variable, $  \gamma ( D) $
 +
is the Robin constant of the domain $  D $
 +
and $  \epsilon ( z , \infty ) $
 +
is a [[Harmonic function|harmonic function]] in $  D $;  
 +
moreover,
  
<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/r/r082/r082510/r08251061.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {| z | \rightarrow \infty }  \epsilon ( z , \infty )  = 0 .
 +
$$
  
The Robin constant of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251062.png" />, defined by (1), coincides with the Robin constant of the compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251063.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251064.png" />. If the Green function for the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251065.png" /> does not exist, then one assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251066.png" />.
+
The Robin constant of the domain $  D $,  
 +
defined by (1), coincides with the Robin constant of the compact set $  \partial  D $:  
 +
$  \gamma ( D) = \gamma ( \partial  D ) $.  
 +
If the Green function for the domain $  D $
 +
does not exist, then one assumes that $  \gamma ( D) = + \infty $.
  
By generalizing the representation (1) to a [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251067.png" /> which has a Green function, one can obtain a local representation of the Green function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251068.png" /> with pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251069.png" />:
+
By generalizing the representation (1) to a [[Riemann surface|Riemann surface]] $  R $
 +
which has a Green function, one can obtain a local representation of the Green function $  g ( p , p _ {0} ) $
 +
with pole $  p _ {0} \in 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/r/r082/r082510/r08251070.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
g ( p , p _ {0} )  = \
 +
\mathop{\rm ln} 
 +
\frac{1}{| z - z _ {0} | }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251071.png" /> is a local uniformizing parameter in a neighbourhood of the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251074.png" /> is the Robin constant of the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251075.png" /> relative to the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251076.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251077.png" /> is a harmonic function in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251078.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251079.png" />. For Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251080.png" /> which do not have a Green function one assumes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251081.png" />. In expression (2) the value of the Robin constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251082.png" /> depends now on the choice of the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251083.png" />. However, the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082510/r08251085.png" /> are independent of the choice of the pole. This allows one to use the notion of a Robin constant in the classification of Riemann surfaces (cf. [[Riemann surfaces, classification of|Riemann surfaces, classification of]]).
+
+ \gamma ( R ;  p _ {0} ) + \epsilon ( p , p _ {0} ) ,
 +
$$
 +
 
 +
where $  z = z ( p) $
 +
is a local uniformizing parameter in a neighbourhood of the pole $  p _ {0} $,  
 +
$  z ( p _ {0} ) = z _ {0} $,  
 +
$  \gamma ( R ;  p _ {0} ) $
 +
is the Robin constant of the Riemann surface $  R $
 +
relative to the pole $  p _ {0} $,  
 +
and $  \epsilon ( p , p _ {0} ) $
 +
is a harmonic function in a neighbourhood of $  p _ {0} $;  
 +
moreover, $  \lim\limits _ {p \rightarrow p _ {0}  }  \epsilon ( p , p _ {0} ) = 0 $.  
 +
For Riemann surfaces $  R $
 +
which do not have a Green function one assumes $  \gamma ( R ;  p _ {0} ) = + \infty $.  
 +
In expression (2) the value of the Robin constant $  \gamma ( R ;  p _ {0} ) $
 +
depends now on the choice of the pole $  p _ {0} \in R $.  
 +
However, the relations $  \gamma ( R ;  p _ {0} ) < + \infty $
 +
and $  \gamma ( R ;  p _ {0} ) = + \infty $
 +
are independent of the choice of the pole. This allows one to use the notion of a Robin constant in the classification of Riemann surfaces (cf. [[Riemann surfaces, classification of|Riemann surfaces, classification of]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Nevanilinna,  "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. [S. Stoilov] Stoilow,  "Leçons sur les principes topologiques de la théorie des fonctions analytiques" , Gauthier-Villars  (1938)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Sario,  M. Nakai,  "Classification theory of Riemann surfaces" , Springer  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Nevanilinna,  "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. [S. Stoilov] Stoilow,  "Leçons sur les principes topologiques de la théorie des fonctions analytiques" , Gauthier-Villars  (1938)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Sario,  M. Nakai,  "Classification theory of Riemann surfaces" , Springer  (1970)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
See also the references quoted in [[Capacity|Capacity]]; [[Energy of measures|Energy of measures]]; [[Robin problem|Robin problem]].
 
See also the references quoted in [[Capacity|Capacity]]; [[Energy of measures|Energy of measures]]; [[Robin problem|Robin problem]].

Latest revision as of 10:37, 5 March 2022


A numerical characteristic of a set of points in a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, closely connected with the capacity of the set.

Let $ K $ be a compact set in $ \mathbf R ^ {n} $, and let $ \mu $ be a positive Borel measure concentrated on $ K $ and normalized by the condition $ \mu ( K) = 1 $. The integral

$$ V ( \mu ) = {\int\limits \int\limits } _ {K \times K } E _ {n} ( x , y ) d \mu ( x) d \mu ( y) , $$

where

$$ E _ {2} ( x , y ) = \mathop{\rm ln} \frac{1}{| x - y | } ,\ \ E _ {n} ( x , y ) = \frac{1}{| x - y | ^ {n-2} } \textrm{ for } \ n \geq 3 , $$

and $ | x - y | $ is the distance between two points $ x , y \in \mathbf R ^ {n} $, is the energy of $ \mu $( cf. Energy of measures). The Robin constant of the compact set $ K $ is the lower bound $ \gamma ( K) = \inf V ( \mu ) $ over all measures $ \mu $ of the indicate type. If $ \gamma ( K) < + \infty $, then this bound is finite and is attained for some (unique) equilibrium, or capacitary, measure $ \lambda > 0 $, $ \gamma ( K) = V ( \lambda ) $, $ \lambda ( K) = 1 $, concentrated on $ K $; if $ \gamma ( K) = + \infty $, then $ V ( \mu ) = + \infty $ for all measures $ \mu $ of the indicated type. The Robin constant of $ K $ is related to its capacity by the formula

$$ \gamma ( K) = \frac{1}{C ( K) } \ \textrm{ for } n \geq 3 , $$

$$ \gamma ( K) = - \mathop{\rm ln} C ( K) \ \textrm{ for } n = 2 . $$

If the boundary $ S $ of $ K $ is sufficiently smooth, for example, if it consists of a finite number of pairwise non-intersecting simple closed surfaces (for $ n \geq 3 $) or curves (for $ n = 2 $) of class $ C ^ {1 , \alpha } $, $ 0 < \alpha < 1 $, then the equilibrium measure $ \lambda $ is concentrated on the part $ \widetilde{S} \subset S $ which forms the boundary of that connected component of the complement $ C K = \mathbf R ^ {n} \setminus K $ which contains the point at infinity. The equilibrium potential, Robin potential or capacity potential, i.e. the potential of the equilibrium measure

$$ u ( x) = \int\limits E _ {n} ( x , y ) d \lambda ( y) , $$

in this case assumes a constant value on $ \widetilde{S} $, equal to $ \gamma ( K) $, which allows one to calculate the Robin constant of a compact set in the simplest cases (see Robin problem). For instance, the Robin constant of a disc of radius $ r > 0 $ in $ \mathbf R ^ {2} $ is $ - \mathop{\rm ln} r $, and the Robin constant of a ball of radius $ r > 0 $ in $ \mathbf R ^ {n} $, $ n \geq 3 $, is $ 1 / r ^ {n-2} $. In the case of an arbitrary compact set $ K $ of positive capacity, $ u ( x) \leq \gamma ( K) $ everywhere and $ u ( x) = \gamma ( K) $ everywhere on the support $ S ( \lambda ) $ of the equilibrium measure $ \lambda $, except possibly at the points of some polar set; moreover, $ S ( \lambda ) \subset K $.

Let $ D $ be a domain in the extended complex plane $ \overline{\mathbf C}\; $ containing inside it the point at infinity and having a Green function $ g ( z , \infty ) $ with pole at infinity. Then the following representation holds:

$$ \tag{1 } g ( z , \infty ) = \mathop{\rm ln} | z | + \gamma ( D) + \epsilon ( z , \infty ) , $$

where $ z = x + i y $ is a complex variable, $ \gamma ( D) $ is the Robin constant of the domain $ D $ and $ \epsilon ( z , \infty ) $ is a harmonic function in $ D $; moreover,

$$ \lim\limits _ {| z | \rightarrow \infty } \epsilon ( z , \infty ) = 0 . $$

The Robin constant of the domain $ D $, defined by (1), coincides with the Robin constant of the compact set $ \partial D $: $ \gamma ( D) = \gamma ( \partial D ) $. If the Green function for the domain $ D $ does not exist, then one assumes that $ \gamma ( D) = + \infty $.

By generalizing the representation (1) to a Riemann surface $ R $ which has a Green function, one can obtain a local representation of the Green function $ g ( p , p _ {0} ) $ with pole $ p _ {0} \in R $:

$$ \tag{2 } g ( p , p _ {0} ) = \ \mathop{\rm ln} \frac{1}{| z - z _ {0} | } + \gamma ( R ; p _ {0} ) + \epsilon ( p , p _ {0} ) , $$

where $ z = z ( p) $ is a local uniformizing parameter in a neighbourhood of the pole $ p _ {0} $, $ z ( p _ {0} ) = z _ {0} $, $ \gamma ( R ; p _ {0} ) $ is the Robin constant of the Riemann surface $ R $ relative to the pole $ p _ {0} $, and $ \epsilon ( p , p _ {0} ) $ is a harmonic function in a neighbourhood of $ p _ {0} $; moreover, $ \lim\limits _ {p \rightarrow p _ {0} } \epsilon ( p , p _ {0} ) = 0 $. For Riemann surfaces $ R $ which do not have a Green function one assumes $ \gamma ( R ; p _ {0} ) = + \infty $. In expression (2) the value of the Robin constant $ \gamma ( R ; p _ {0} ) $ depends now on the choice of the pole $ p _ {0} \in R $. However, the relations $ \gamma ( R ; p _ {0} ) < + \infty $ and $ \gamma ( R ; p _ {0} ) = + \infty $ are independent of the choice of the pole. This allows one to use the notion of a Robin constant in the classification of Riemann surfaces (cf. Riemann surfaces, classification of).

References

[1] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[2] S. [S. Stoilov] Stoilow, "Leçons sur les principes topologiques de la théorie des fonctions analytiques" , Gauthier-Villars (1938)
[3] L. Sario, M. Nakai, "Classification theory of Riemann surfaces" , Springer (1970)

Comments

See also the references quoted in Capacity; Energy of measures; Robin problem.

How to Cite This Entry:
Robin constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Robin_constant&oldid=18330
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article