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A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i0526601.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i0526602.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i0526603.png" /> at which there is a continuous boundary function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i0526604.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i0526605.png" /> such that the Perron–Wiener–Brélot generalized solution (cf. [[Perron method|Perron method]]) of the [[Dirichlet problem|Dirichlet problem]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i0526606.png" />, does not take the boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i0526607.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i0526608.png" />, i.e. either the limit
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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/i/i052/i052660/i0526609.png" /></td> </tr></table>
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does not exist, or it does not coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266010.png" />. For domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266011.png" /> in the plane every isolated point of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266012.png" /> is irregular. In the case of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266013.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266015.png" />, it was H. Lebesgue who first discovered that the vertex of a very acute angle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266016.png" /> is an irregular boundary point. E.g., the coordinate origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266017.png" /> is an irregular boundary point if the boundary of the domain has, in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266018.png" />, the shape of the entering acute angle obtained by rotating the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266020.png" />, around the positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266021.png" />-axis (Lebesgue spine). The generalized solution of the Dirichlet problem does not take the boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266022.png" /> at an irregular boundary point if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266023.png" /> is the least upper or greatest lower bound of the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266025.png" />; the classical solution does not exist in this case. The set of irregular boundary points is thin, in a certain sense: it has type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052660/i05266026.png" />, is a [[Polar set|polar set]] and has zero [[Capacity|capacity]]. See also [[Barrier|Barrier]]; [[Regular boundary point|Regular boundary point]].
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A point  $  y _ {0} $
 +
on the boundary  $  \Gamma $
 +
of a domain  $  D $
 +
at which there is a continuous boundary function  $  f ( y) $
 +
on  $  \Gamma $
 +
such that the Perron–Wiener–Brélot generalized solution (cf. [[Perron method|Perron method]]) of the [[Dirichlet problem|Dirichlet problem]],  $  u ( x) $,
 +
does not take the boundary value  $  f ( y _ {0} ) $
 +
at  $  y _ {0} $,
 +
i.e. either the limit
 +
 
 +
$$
 +
\lim\limits _ {\begin{array}{c}
 +
x \rightarrow y _ {0} \\
 +
x \in L
 +
\end{array}
 +
}  u ( x)
 +
$$
 +
 
 +
does not exist, or it does not coincide with $  f ( y _ {0} ) $.  
 +
For domains $  D $
 +
in the plane every isolated point of the boundary $  \Gamma $
 +
is irregular. In the case of a domain $  D $
 +
in a Euclidean space $  \mathbf R  ^ {n} $,  
 +
$  n \geq  3 $,  
 +
it was H. Lebesgue who first discovered that the vertex of a very acute angle in $  D $
 +
is an irregular boundary point. E.g., the coordinate origin 0 \in \mathbf R  ^ {3} $
 +
is an irregular boundary point if the boundary of the domain has, in a neighbourhood of 0 $,  
 +
the shape of the entering acute angle obtained by rotating the curve $  y = e  ^ {-} 1/x $,
 +
$  x > 0 $,  
 +
around the positive $  x $-
 +
axis (Lebesgue spine). The generalized solution of the Dirichlet problem does not take the boundary value $  f ( y _ {0} ) $
 +
at an irregular boundary point if $  f ( y _ {0} ) $
 +
is the least upper or greatest lower bound of the values of $  f ( y) $
 +
on $  \Gamma $;  
 +
the classical solution does not exist in this case. The set of irregular boundary points is thin, in a certain sense: it has type $  F _  \delta  $,  
 +
is a [[Polar set|polar set]] and has zero [[Capacity|capacity]]. See also [[Barrier|Barrier]]; [[Regular boundary point|Regular boundary point]].
  
 
====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)</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)</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)</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)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 22:13, 5 June 2020


A point $ y _ {0} $ on the boundary $ \Gamma $ of a domain $ D $ at which there is a continuous boundary function $ f ( y) $ on $ \Gamma $ such that the Perron–Wiener–Brélot generalized solution (cf. Perron method) of the Dirichlet problem, $ u ( x) $, does not take the boundary value $ f ( y _ {0} ) $ at $ y _ {0} $, i.e. either the limit

$$ \lim\limits _ {\begin{array}{c} x \rightarrow y _ {0} \\ x \in L \end{array} } u ( x) $$

does not exist, or it does not coincide with $ f ( y _ {0} ) $. For domains $ D $ in the plane every isolated point of the boundary $ \Gamma $ is irregular. In the case of a domain $ D $ in a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 3 $, it was H. Lebesgue who first discovered that the vertex of a very acute angle in $ D $ is an irregular boundary point. E.g., the coordinate origin $ 0 \in \mathbf R ^ {3} $ is an irregular boundary point if the boundary of the domain has, in a neighbourhood of $ 0 $, the shape of the entering acute angle obtained by rotating the curve $ y = e ^ {-} 1/x $, $ x > 0 $, around the positive $ x $- axis (Lebesgue spine). The generalized solution of the Dirichlet problem does not take the boundary value $ f ( y _ {0} ) $ at an irregular boundary point if $ f ( y _ {0} ) $ is the least upper or greatest lower bound of the values of $ f ( y) $ on $ \Gamma $; the classical solution does not exist in this case. The set of irregular boundary points is thin, in a certain sense: it has type $ F _ \delta $, is a polar set and has zero capacity. See also Barrier; Regular boundary point.

References

[1] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[2] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)

Comments

See [a2] for an additional classical reference, and [a1] for irregular points in axiomatic potential theory.

References

[a1] L.L. Helms, "Introduction to potential theory" , Wiley (1969) (Translated from German)
[a2] C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)
How to Cite This Entry:
Irregular boundary point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irregular_boundary_point&oldid=47434
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article