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A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806801.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806802.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806803.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806805.png" />, at which, for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806806.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806807.png" />, the generalized solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806808.png" /> of the [[Dirichlet problem|Dirichlet problem]] in the sense of Wiener–Perron (see [[Perron method|Perron method]]) takes the boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806809.png" />, that is,
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<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/r080/r080680/r08068010.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
The regular boundary points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068011.png" /> form a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068012.png" />, at the points of which the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068013.png" /> is not a [[Thin set|thin set]]; the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068014.png" /> of irregular boundary points (cf. [[Irregular boundary point|Irregular boundary point]]) is a [[Polar set|polar set]] of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068015.png" />. If all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068016.png" /> are regular boundary points, then the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068017.png" /> is called regular with respect to the Dirichlet problem.
+
A point  $  y _ {0} $
 +
on the boundary $  \Gamma $
 +
of a domain  $  D $
 +
in a Euclidean space  $  \mathbf R  ^ {n} $,
 +
$  n \geq  2 $,  
 +
at which, for any continuous function  $  f ( z) $
 +
on  $  \Gamma $,
 +
the generalized solution  $  u ( x) $
 +
of the [[Dirichlet problem|Dirichlet problem]] in the sense of Wiener–Perron (see [[Perron method|Perron method]]) takes the boundary value  $  f ( y _ {0} ) $,  
 +
that is,
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068018.png" /> to be a regular boundary point it is necessary and sufficient that in the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068020.png" /> with any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068022.png" /> there is a superharmonic [[Barrier|barrier]] (a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068025.png" />, Lebesgue's criterion for a barrier). It was first shown by H. Lebesgue in 1912 that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068026.png" /> the vertex of a sufficiently acute angle lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068027.png" /> need not be a regular boundary point.
+
$$
 +
\lim\limits _ {\begin{array}{c}
 +
x \rightarrow y _ {0} \\
 +
x \in D
 +
\end{array}
 +
} \
 +
u ( x)  = f ( y _ {0} ) .
 +
$$
 +
 
 +
The regular boundary points of  $  D $
 +
form a set  $  R $,
 +
at the points of which the complement  $  D  ^ {c} = \mathbf R  ^ {n} \setminus  D $
 +
is not a [[Thin set|thin set]]; the set  $  \Gamma \setminus  R $
 +
of irregular boundary points (cf. [[Irregular boundary point|Irregular boundary point]]) is a [[Polar set|polar set]] of type  $  F _  \sigma  $.  
 +
If all points of  $  \Gamma $
 +
are regular boundary points, then the domain  $  D $
 +
is called regular with respect to the Dirichlet problem.
 +
 
 +
For  $  y _ {0} \in \Gamma $
 +
to be a regular boundary point it is necessary and sufficient that in the intersection $  U _ {0} = U \cap D $
 +
of $  D $
 +
with any neighbourhood $  U $
 +
of $  y _ {0} $
 +
there is a superharmonic [[Barrier|barrier]] (a function $  \omega ( x) > 0 $
 +
in $  U _ {0} $
 +
such that $  \lim\limits _ {x \rightarrow y _ {0}  }  \omega ( x) = 0 $,  
 +
Lebesgue's criterion for a barrier). It was first shown by H. Lebesgue in 1912 that for $  n \geq  3 $
 +
the vertex of a sufficiently acute angle lying in $  D $
 +
need not be a regular boundary point.
  
 
Let
 
Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068028.png" /></td> </tr></table>
+
$$
 +
E _ {k}  = \{ {x \in D  ^ {c} } : {2  ^ {-} k \leq  | x - y _ {0} |
 +
\leq  2  ^ {-} k+ 1 } \}
 +
$$
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068029.png" /> be the [[Capacity|capacity]] of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068030.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068031.png" /> to be a regular boundary point it is necessary and sufficient that the series
+
and let $  c _ {k} = C ( E _ {k} ) $
 +
be the [[Capacity|capacity]] of the set $  E _ {k} $.  
 +
For $  y _ {0} \in \Gamma $
 +
to be a regular boundary point it is necessary and sufficient that the series
  
<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/r080/r080680/r08068032.png" /></td> </tr></table>
+
$$
 +
\sum _ { k= } 1 ^  \infty 
 +
2 ^ {k ( n - 2 ) }
 +
c _ {k} ,\  n \geq  3 ,
 +
$$
  
diverges, or for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068033.png" />, that the series
+
diverges, or for $  n = 2 $,  
 +
that the series
  
<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/r080/r080680/r08068034.png" /></td> </tr></table>
+
$$
 +
\sum _ { k= } 1 ^  \infty  2  ^ {k} c _ {k}  $$
  
 
diverges, where
 
diverges, 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/r080/r080680/r08068035.png" /></td> </tr></table>
+
$$
 +
E _ {k}  = \left \{ {
 +
x \in D } : {2  ^ {k} \leq  \mathop{\rm ln} 
 +
\frac{1}{| x - y _ {0} | }
 +
 
 +
\leq  2  ^ {k+} 1 } \right \}
 +
$$
  
 
(Wiener's criterion).
 
(Wiener's criterion).
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068036.png" />, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068037.png" /> is a regular boundary point if there is a continuous path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068039.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068041.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068042.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068043.png" />, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068044.png" /> is a regular boundary point if it can be reached by the vertex of a right circular cone belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068045.png" /> in a sufficiently small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068046.png" />. In the case of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068047.png" /> in the compactified space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068049.png" />, the point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068050.png" /> is always a regular boundary point; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068051.png" />, the point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068052.png" /> is a regular boundary point if there is a continuous path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068054.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068055.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068056.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068057.png" />.
+
For $  n = 2 $,  
 +
a point $  y _ {0} \in \Gamma $
 +
is a regular boundary point if there is a continuous path $  x ( t) $,  
 +
0 \leq  t \leq  1 $,  
 +
such that $  x( 1) = y _ {0} $,  
 +
and $  x ( t) \in D  ^ {c} $
 +
for  $  0 \leq  t < 1 $.  
 +
When $  n \geq  3 $,  
 +
a point $  y _ {0} \in \Gamma $
 +
is a regular boundary point if it can be reached by the vertex of a right circular cone belonging to $  D  ^ {c} $
 +
in a sufficiently small neighbourhood of $  y _ {0} $.  
 +
In the case of a domain $  D $
 +
in the compactified space $  \overline{\mathbf R}\; {}  ^ {n} $,  
 +
$  n \geq  3 $,  
 +
the point at infinity $  \infty \in \Gamma $
 +
is always a regular boundary point; when $  n = 2 $,  
 +
the point at infinity $  \infty \in \Gamma $
 +
is a regular boundary point if there is a continuous path $  x ( t) $,
 +
0 \leq  t < 1 $,  
 +
such that $  x ( t) \in D  ^ {c} $
 +
for  $  0 \leq  t < 1 $,  
 +
and $  \lim\limits _ {t \uparrow 1 }  x ( t) = \infty $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.V. Keldysh,  "On the solvability and stability of the Dirichlet problem"  ''Uspekhi Mat. Nauk'' , '''8'''  (1941)  pp. 171–232  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.K. Hayman,  P.B. Kennedy,  "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.V. Keldysh,  "On the solvability and stability of the Dirichlet problem"  ''Uspekhi Mat. Nauk'' , '''8'''  (1941)  pp. 171–232  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.K. Hayman,  P.B. Kennedy,  "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:10, 6 June 2020


A point $ y _ {0} $ on the boundary $ \Gamma $ of a domain $ D $ in a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, at which, for any continuous function $ f ( z) $ on $ \Gamma $, the generalized solution $ u ( x) $ of the Dirichlet problem in the sense of Wiener–Perron (see Perron method) takes the boundary value $ f ( y _ {0} ) $, that is,

$$ \lim\limits _ {\begin{array}{c} x \rightarrow y _ {0} \\ x \in D \end{array} } \ u ( x) = f ( y _ {0} ) . $$

The regular boundary points of $ D $ form a set $ R $, at the points of which the complement $ D ^ {c} = \mathbf R ^ {n} \setminus D $ is not a thin set; the set $ \Gamma \setminus R $ of irregular boundary points (cf. Irregular boundary point) is a polar set of type $ F _ \sigma $. If all points of $ \Gamma $ are regular boundary points, then the domain $ D $ is called regular with respect to the Dirichlet problem.

For $ y _ {0} \in \Gamma $ to be a regular boundary point it is necessary and sufficient that in the intersection $ U _ {0} = U \cap D $ of $ D $ with any neighbourhood $ U $ of $ y _ {0} $ there is a superharmonic barrier (a function $ \omega ( x) > 0 $ in $ U _ {0} $ such that $ \lim\limits _ {x \rightarrow y _ {0} } \omega ( x) = 0 $, Lebesgue's criterion for a barrier). It was first shown by H. Lebesgue in 1912 that for $ n \geq 3 $ the vertex of a sufficiently acute angle lying in $ D $ need not be a regular boundary point.

Let

$$ E _ {k} = \{ {x \in D ^ {c} } : {2 ^ {-} k \leq | x - y _ {0} | \leq 2 ^ {-} k+ 1 } \} $$

and let $ c _ {k} = C ( E _ {k} ) $ be the capacity of the set $ E _ {k} $. For $ y _ {0} \in \Gamma $ to be a regular boundary point it is necessary and sufficient that the series

$$ \sum _ { k= } 1 ^ \infty 2 ^ {k ( n - 2 ) } c _ {k} ,\ n \geq 3 , $$

diverges, or for $ n = 2 $, that the series

$$ \sum _ { k= } 1 ^ \infty 2 ^ {k} c _ {k} $$

diverges, where

$$ E _ {k} = \left \{ { x \in D } : {2 ^ {k} \leq \mathop{\rm ln} \frac{1}{| x - y _ {0} | } \leq 2 ^ {k+} 1 } \right \} $$

(Wiener's criterion).

For $ n = 2 $, a point $ y _ {0} \in \Gamma $ is a regular boundary point if there is a continuous path $ x ( t) $, $ 0 \leq t \leq 1 $, such that $ x( 1) = y _ {0} $, and $ x ( t) \in D ^ {c} $ for $ 0 \leq t < 1 $. When $ n \geq 3 $, a point $ y _ {0} \in \Gamma $ is a regular boundary point if it can be reached by the vertex of a right circular cone belonging to $ D ^ {c} $ in a sufficiently small neighbourhood of $ y _ {0} $. In the case of a domain $ D $ in the compactified space $ \overline{\mathbf R}\; {} ^ {n} $, $ n \geq 3 $, the point at infinity $ \infty \in \Gamma $ is always a regular boundary point; when $ n = 2 $, the point at infinity $ \infty \in \Gamma $ is a regular boundary point if there is a continuous path $ x ( t) $, $ 0 \leq t < 1 $, such that $ x ( t) \in D ^ {c} $ for $ 0 \leq t < 1 $, and $ \lim\limits _ {t \uparrow 1 } x ( t) = \infty $.

References

[1] M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" Uspekhi Mat. Nauk , 8 (1941) pp. 171–232 (In Russian)
[2] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[3] W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)

Comments

The polarity of the set of irregular boundary points is contained in the Kellogg–Evans theorem. See, e.g., [a1] for irregular boundary points in abstract potential theory.

References

[a1] J. Bliedtner, W. Hansen, "Potential theory. An analytic and probabilistic approach to balayage" , Springer (1986)
[a2] H. Lebesgue, "Sur des cas d'impossibilité du problème de Dirichlet ordinaire" C.R. Séances Soc. Math. France , 41 (1913) pp. 17
[a3] H. Lebesgue, "Conditions de régularité, conditions d'irrégularité, conditions d'impossibilité dans le problème de Dirichlet" C.R. Acad. Sci. Paris , 178 (1924) pp. 349–354
[a4] N. Wiener, "The Dirichlet problem" J. Math. Phys. , 3 (1924) pp. 127–146
[a5] M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975)
[a6] J. Wermer, "Potential theory" , Lect. notes in math. , 408 , Springer (1981)
How to Cite This Entry:
Regular boundary point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_boundary_point&oldid=15319
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article