Regular boundary point

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)