Difference between revisions of "Regular boundary point"
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− | + | A point | |
+ | 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, | ||
− | + | $$ | |
+ | \lim\limits _ {\begin{array}{c} | ||
+ | x \rightarrow y _ {0} \\ | ||
+ | x \in D | ||
+ | \end{array} | ||
+ | } \ | ||
+ | u ( x) = f ( y _ {0} ) . | ||
+ | $$ | ||
+ | |||
+ | 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 | ||
− | + | $$ | |
+ | E _ {k} = \{ {x \in D ^ {c} } : {2 ^ {-} k \leq | x - y _ {0} | | ||
+ | \leq 2 ^ {-} k+ 1 } \} | ||
+ | $$ | ||
− | and let | + | 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 | ||
− | + | $$ | |
+ | \sum _ { k= } 1 ^ \infty | ||
+ | 2 ^ {k ( n - 2 ) } | ||
+ | c _ {k} ,\ n \geq 3 , | ||
+ | $$ | ||
− | diverges, or for | + | diverges, or for $ n = 2 $, |
+ | that the series | ||
− | + | $$ | |
+ | \sum _ { k= } 1 ^ \infty 2 ^ {k} c _ {k} $$ | ||
diverges, where | 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). | (Wiener's criterion). | ||
− | For | + | 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) |
Regular boundary point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_boundary_point&oldid=48479