Difference between revisions of "Irregular boundary point"
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| − | does not exist, or it does not coincide with | + | 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> | ||
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====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
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