# 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)